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The role of internal stresses on the plastic deformation of the Al–Mg–Si–Cu alloy AA6111 Poole, Warren J.; Proudhon, H.; Wang, X.; Brechet, Y. 2008-12-31

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December 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111Philosophical Magazine,Vol. 00, No. 00, DD Month 200x, 1?26The role of internal stresses on the plastic deformation of the Al?Mg?Si?Cu alloyAA6111H. Proudhon?, W.J. Pooleasteriskmath?, X. Wang?, Y. Br? het??Dept. of Materials Engineering, The University of British Columbia,309-6350 Stores Road, Vancouver BC V6T1Z4, Canada?Dept. of Material Science and Engineering, Master University,1280 Main St. West, Hamilton ON L8S4L7, Canada?LTPCM, INP Grenoble, BP75, 38402 St. Martin d?Heres, France(Received 00 Month 200x; in final form 00 Month 200x)In this work, we have investigated the internal stress contribution to the flow stress for a commercial 6xxx aluminium alloy (AA6111).In contrast to stresses from forest and precipitation hardening, the internal stress cannot be assessed properly with a uniaxial tensiletest. Instead, tension-compression tests have been used to measure the Bauschinger stress and produce a comprehensive study whichexamines its evolution with i) the precipitation structure and ii) a wide range of applied strain. A large set of ageing conditions wasinvestigated to explore the effect of the precipitation state on the development of internal stress within the material.It is shown that the Bauschinger stress generally increases with the applied strain and critically depends on the precipitate averageradius and is thus linked to the shearable/non shearable transition. Further work in the case of non-shearable particles shows that higherstrain eventually lead to particle fracture and the Bauschinger stress then rapidly decreases. Following the seminal work of Brown et al, aphysically based approach including plastic relaxation and particle fracture is developed to predict the evolution of the internal stress asa function of the applied strain. Knowing the precipitation structure main characteristics ?such as the average precipitate radius, lengthand volume fraction? allows one to estimate accurately the internal stress contribution to the flow stress with this model.asteriskmathCorresponding author. E-mail: warren.poole@ubc.caPhilosophical MagazineISSN 1478-6435 print/ISSN 1478-6443 online xa9200x Taylor & Francishttp://www.tandf.co.uk/journalsDOI: 10.1080/1478643YYxxxxxxxxDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa61112 H. Proudhon et al.1 IntroductionA complete understanding of the plastic behaviour of two phase materials requires a multi-faceted approachas the presence of a second phase introduces a variety of new possible deformation mechanisms comparedto single phase materials. At a general level, the second phase modifies the work hardening of materialdue to partitioning of stress and strain between the two phases. The modeling of these stresses for a widerange of volume fractions and particle shapes can be examined using by continuum models such as loadtransfer models, Eshelby and finite element method calculations [1, 2]. Alternatively, dislocation basedmodels can be considered. Using the concept of geometrically necessary dislocations, the seminal work ofAshby examined the work hardening of Cu?SiO2 single crystals [3]. In this work, an intrinsic length scale,?the geometric slip distance?, can be introduced to account for the effects of particle size, shape (plates vs.spheres) and volume fraction. In recent years, it has become possible to examine plasticity problems usingdiscrete dislocation models; this seems to be promising in complex situations such as two phase systemse.g. see Cleveringa and Van Der Giessen [4], but much more work is needed here before commerciallyrelevant systems can be explored.Each approach has its limitations. For example, dislocation models have difficulty in predicting internalstresses such as observed during strain path changes (e.g. Bauschinger tests) while continuum models areinherently scale independent and cannot account for particle size effects. As a result, modelling approacheswhich incorporate aspects of both continuum and dislocation based approaches have been developed [5].For example, Brown and Stobbs developed a comprehensive framework to describe the work hardeningbehaviour for the Cu?SiO2 single crystals system [6,7]. Most recently, it has been shown that it is possibleto extend continuum plasticity models by using higher order plasticity theories such as strain gradientplasticity which explicitly include a length scale [8].It is also important to consider the possible interactions between plasticity and the second phase. First,for small particles (e.g. precipitates) which are sheared by dislocations, the increase in interfacial areacan lead to the dissolution of the second phase [9, 10]. Alternatively, deformation may lead to dynamicprecipitation [11, 12]. Second, the build of internal stresses can lead to the intervention of alternativeDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 3processes such as particle fracture, interfacial decohesion or in the case of high aspect particles undercompressive loading, elastic buckling and the formation of shear bands [13]. In general, these processesdepend critically on the stress-state, e.g. the magnitude and direction of the hydrostatic and deviatoricstresses.Precipitation hardening aluminum alloys represent a particularly interesting and challenging two-phasesystem to examine. During precipitation, the volume fraction, size, shape and the composition of thesecond phase changes (i.e. a series of metastable precipitates is often observed). The early work of Byrneand co-workers [14] on Al?Cu single crystals found that there was a substantial change in macroscopicwork hardening behaviour as a function of the precipitate state. A fundamental question that arose fromthese studies and the later work of Moan and Embury [15] was whether the enhanced work hardeningbehaviour observed in the Al?Cu samples (which have plate shaped precipitates) was best understood inthe framework of additional storage of geometrically necessary dislocations (Ashby) or in terms of thedevelopment of long range back stress [6,16,17]. It is not possible to differentiate these theories based onmonotonic tests but the use of strain reversal, i.e. Bauschinger tests, is particularly useful for evaluatinginternal stresses [15,18,19].The current work examines the development of internal stresses as a function of precipitate state for thecommercial aluminum alloy AA6111 using Bauschinger experiments. This system is a good candidate fordetailed examination as it has recently been well characterized in terms of the nature of the precipitates,their size, shape and volume fraction [20?22] as well as the macroscopic work hardening behaviour [23] fora wide range of heat treatments. It is the objective of the current work to develop a comprehensive modelframework for understanding the development of internal stresses in this alloy system for deformation atambient temperature over a wide range of strains (i.e. strains from 0.01 to 0.6).2 Experimental MethodsA commercial aluminium alloy AA6111, provided by Novelis Global Technology Centre (Kingston, ON)was used for this study. The material was obtained in form of a rolled sheet of 10 mm thickness andDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa61114 H. Proudhon et al.Table 1. Chemical composition of the AA6111 alloy (wt.%)Alloy Mg Si Cu Fe Mn Cr TiAA6111 0.8 0.6 0.7 0.25 0.2 0.05 0.06Table 2. Details of the five heat treatments applied to the specimensCondition label solutionizing quench ageingUnder aged 5M180 10 min at 560 x89 WQ 5 minutes at 180 x89 in oil bathPeak aged 7H180 10 min at 560 x89 WQ 7 hours at 180 x89 in oil bathOver aged 1 60D180 10 min at 560 x89 WQ 1 hour at 250 x89 in oil bath, then 60 days at 180 x89 in air furnaceaOver aged 2 6H250 10 min at 560 x89 WQ 6 hours at 250 x89 in oil bathOver aged 3 7D250 10 min at 560 x89 WQ 1 hour at 250 x89 in oil bath, then 7 days at 250 x89 in air furnaceaaFor long duration heat treatments, an air furnace was used rather than an oil bath; but to ensure a consistent heating rate inall experiments, the first hour was carried out in an oil bath like for the other treatments.its chemical composition was determined as shown in Table 1. For the tension-compression tests, axi-symmetrical specimens were machined in the sheet thickness with a 20 mm gage length and a diameter ratioof 1.63 (the diameter is 9 mm in the grip section v.s. 5.5 mm in the gage length section). Specimens wereall solution treated for 10 minutes in a salt bath at 560 x89. In order to study the effect of the precipitationstructure, various ageing stages have been investigated. Based on previous work [24], 5 conditions have beenselected from under-aged to massively over-aged, as shown in Table 2. For the sake of clarity, each conditionwill be hereafter identified by the combination of annealing time and temperature after quenching; e.g.6H250 for 6 hours at 250 x89, see Table 2 for full details).Mechanical tests were conducted on a computer controlled MTS servo-hydraulic machine. Tests havebeen conducted with a nominal strain rate of 10-3 s-1 at room temperature. One specimen for each con-dition was tested in tension to failure to record the tensile properties. Then for each ageing condition,tension-compression tests were conducted in the following way: during the first straining phase, the spec-imen was deformed under tension to a certain amount of plastic strain. A clip-on extensometer was usedto monitor the total strain, the target value having been determined from the tensile test. The strainwas then reversed to apply at least 6% plastic strain in compression and at this point the specimen isunloaded. Tests were conducted on different samples with four different amounts of plastic strain in thetension loading phase (1%, 2%, 4% and 6%) to capture the evolution of the Bauschinger effect as a functionof strain.It will be shown that for the particular case of 7 days at 250 x89, it is of interest to gather information onDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 5Table 3. Yield stress measured with 0.02% and 0.2% plastic strain off-sets for all investigated materials.Ageing 5M180 7H180 60D180 6H250 7D250sigmaY 0.02% (MPa) 110 310 285 210 140sigmaY 0.2% (MPa) 140 335 300 245 155the Bauschinger effect at larger strains (between 10% and 60%). With the method described above, thestrain is limited to moderate values due to either necking or buckling of the sample. Therefore, additionalspecimens were processed in the following way: four 50 mm?100 mm?10 mm plates of the as receivedsheet were cut and prepared with the 7D250 condition. Three of these samples were rolled to 10, 20 and40% reduction (the fourth sample was not rolled and was used to ensure there was no variation in the heattreatment due to the bulky geometry of the plate compared to the dogbone specimens). Specimens werethen machined from the four plates, and both tension and compression tests were conducted as previouslydescribed.The samples for transmission electron microscopy (TEM) were prepared as follows: mechanical polishingdown to approximately 100 xb5m thickness and then jet polishing in a solution of 10% perchloric acid and 90%methanol at -35 x89. TEM observations were conducted using a PHILIPS CM12 transmission microscopeoperating at 120 kV.3 ResultsAs shown by Poole and co-workers in the same alloy, the yield and work hardening behaviour dependscritically on the heat treatment history of the material [24]. From the tensile behaviour, the yield stressvalues corresponding to 0.2% and 0.02% of permanent deformation sigma0.2% and sigma0.02% respectively, weredetermined and reported in Table 3. Tensile curves for a large set of ageing conditions in this material canbe found elsewhere [25].Fig. 1 illustrates how the data recorded during the tensile testing was analysed. The process has threesteps: (i) the true stress v.s. true strain curve is plotted; (ii) the compressive data is reversed and plottedas a function of cumulative strain; (iii) the elastic strain is removed and the true stress v.s. cumulativetrue strain can be compared to the monotonic behaviour.December 19, 2007 17:43 Philosophical Magazine bauschinger?aa61116 H. Proudhon et al.Figure 1. Illustration of the stress-strain data processing in a three steps way in the case of 7H180 material with 2% forward strain:(a) true stress v.s. true strain curve; (b) reversed stress data plotted v.s. cumulative true strain (c) the elastic strain is removed and thecurve is compared with the monotonic behavior.Fig. 2 shows the forward-reverse behaviour obtained for different heat treatments after 1% plastic strainapplied in the forward direction. Similar data have been obtained for forward strain values of 2%, 4%and 6%. One can readily observe that the reverse parts of the curves can be split in two; first a transientregion where there is a very steep apparent work hardening rate, followed by a steady state one. Thematerials with different heat treatments show very different behaviours. For the under aged and peak agedmaterial, the transient is rather short and the flow stress quickly comes back to the value reached duringthe forward straining. On the other hand for the over aged materials, the transient progressively becomeslonger and more complicated. In particular for the materials aged at 250 x89, the transient part seems muchmore complicated with an inflexion in the reverse flow stress curve. While an in-depth investigation of themicroscopic processes responsible of this inflexion is beyond the scope of this paper, it should be notedthat this behaviour has been reported before [26,27] and was attributed to the recovery of the dislocationstructures formed during the forward straining.3.1 Bauschinger stress analysisAs previously suggested by Wilson [18] and followed by Atkinson et al. [16], the evolution of the Bauschingerstress sigmab as a function of the reverse plastic strain can be calculated by halving the difference betweenthe monotonous and the reversed stress-strain data. The corresponding curves are shown in Fig. 3. Thisrepresentation is useful to compare between different materials and can give important information on theDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 7Figure 2. Forward-reverse stress/strain curves, obtained by a Bauschinger test with a 1% forward plastic strain, for all materials.evolution of the internal structure as the reverse strain develops. Fig. 3 presents a summary of the resultsof all Bauschinger tests. One can see that the results obtained for the 1% forward strain (see Fig. 2) aresimilar for the other values of 2%, 4% and 6%; however it is clear that for a given material, the generalvalue of the Bauschinger stress increases as a function of the forward strain.The difference in terms of the evolution of the Bauschinger stress as the reverse strain increases issignificant. On the one hand, the under aged and peak aged materials (and to a lesser extend for the60D180 material which appears to be an intermediate case), undergo a short transient stage (approximately1% plastic strain) where the Bauschinger stress decreases very rapidly. After this transient decrease, theBauschinger stress, if any, decreases to a low value. On the other hand, the over aged materials 6H250 and7D250 show a more complicated behaviour with higher values of Bauschinger stress. The transient is stillobserved (approximately 1% plastic strain) followed by a regime where the Bauschinger stress decreasesin a linear manner with increasing plastic strain.December 19, 2007 17:43 Philosophical Magazine bauschinger?aa61118 H. Proudhon et al.Figure 3. Bauschinger stress as a function of reverse plastic strain. For each ageing condition, 4 different forward strains were tested(1%, 2%, 4% and 6%); the curves for rolled samples are shown in the bottom right graph and have been annoted ?CR?.To further analyse these data, it is useful to choose a plastic strain offset to characterise the Bauschingerstress sigmab. This is a rather important issue as the choice of the offset significantly affects the results.Examination of the literature shows that there is no general agreement on this choice. In the work ofBrown and coworkers on copper-silica, the internal stress is defined as being ?permanent softening? andthus measured when the reverse curve becomes parallel to the tensile behaviour. This method does notseem suitable in our case since permanent softening is not readily observed. Alternatively, Lloyd chose touse a 3% plastic strain offset [28], while Moan and Embury focused on the transient behavior and haveused very small offset values ?between 0.001% and 0.1% plastic strain? to characterize the Bauschingereffect [15].Table 4 presents the results showing the effect of choosing different reverse strains, i.e. 0.5, 1 and 2%,to estimate the Bauschinger stress from the experimental results. For the cases of 5M180 and 7H180, theDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 9Table 4. Measure of the Bauschinger stress (MPa) in the tension-compression tests presented on Fig. 3 with three different plastic strainoffsets.Forward strain 5M180 7H180 60D180 6H250 7D250plastic strain offset = 0.5%1% 6.5 10.7 31.3 45.8 37.92% 8.6 19.8 49.8 50.3 42.44% 10.1 26.0 53.7 47.3 46.46% 11.0 26.5 47.7 44.6 50.6plastic strain offset = 1%1% 4.0 3.4 7.9 10.8 20.72% 5.8 11.5 28.9 37.0 36.34% 5.5 14.7 39.3 37.8 41.46% 6.4 14.7 33.8 36.3 45.4plastic strain offset = 2%1% 2.1 1.0 4.4 7.3 14.72% 5.1 5.7 9.9 16.9 28.24% 4.2 6.1 23.7 26.6 34.96% 4.2 7.3 22.6 29.6 36.9choice of reverse strains has almost no practical effect, i.e. in all case the estimated stress is small, rangingfrom 1?8% of the yield stress. The value of 8% will be taken as a worst case estimate for the resolution ofthis technique. On the other hand, in the case of 6H250 or 7D250 the effect of the choice for reverse strainsis more important, especially for low forward strains. By far the biggest difference in the estimate occurswhen the reverse strain is increased from 0.5 to 1%. This is consistent with the data plotted in Fig. 3dand 3e which shows that the estimate of Bauschinger stress is changing rapidly in this range of the reversestrain. After 1% reverse strain the rate of change of Bauschinger stress is much lower in all cases.It is well known that during the initial stage of the strain reversal, the dislocation structure is evolvingin a complicated manner. For example, even single phase materials such as copper exhibit significanttransient effects [29]. Considering the strain resulting from the local motion of a dislocation sampling anarray of point obstacles when the strain is reversed, Brown estimated that the transient strain would beapproximately 1% plastic strain [19]. This would be consistent with the present results which show thatthe region where the reverse stress is evolving the most rapidly also corresponds to approximately 1%plastic strain. As a result, the subsequent discussion will use the results obtained at 1% plastic reversestrain sigmar 1%, as an estimate of the internal stress, see Eq. (1). While this is not completely satisfying froma theoretical point of view, we feel that in the absence of direct measurements for the stress in the secondphase (by for example in situ X-ray diffraction), this represents a reasonable approach.December 19, 2007 17:43 Philosophical Magazine bauschinger?aa611110 H. Proudhon et al.The Bauschinger stress, measured with a 1% plastic strain offset sigmab1%, is thus used to estimate the valueof the internal stress angbracketleftsigmaangbracketright stored within the material at the point of reversal (where sigmaf0 denotes the flowstress reached at this point). From now on we shall use the term internal stress regardless whether weare refering to the experimental Bauschinger results or to the stress actually stored within the material.Hence:angbracketleftsigmaangbracketright = sigmab1% = sigmaf0 -|sigmar 1%|2 (1)The measured internal stress increases in all cases with plastic strain as shown in Fig. 4a; but clearly,the five studied materials break into two groups. The under aged and peak aged materials show very littleinternal stress and the over aged materials show a significant level. The internal stress angbracketleftsigmaangbracketright first increaseswith a very steep slope and then remains more or less stable, as shown by the upper dashed line on thefigure.Having characterised the internal stress, it is interesting to be able to estimate the fraction of the workhardening which can be attributed to the internal stress. This can be done by plotting the ratio of theinternal stress to the level of work hardening: angbracketleftsigmaangbracketright/(sigmaf0 - sigmaY 0.02%) as shown in Fig. 4b. Again the fivematerials separate into two groups, underaged and peak aged materials have less than 20% of the workhardening which can be attributed to internal stress build-up while for the overaged materials, more than50% of the work hardening (up to 86% for 7D250) can be attributed to the evolution of internal stress.3.2 Large strain experiments with material aged 7D250As shown by the results in Fig. 4, the material aged 7D250 exhibits a very large internal stress when it isplastically deformed. It is clear that the internal stress increases with the applied forward plastic strain butit is of interest to see if there is a limit to this increase. This cannot be answered from these experimentsbecause we were limited to approximately 7?8% plastic strain due to either necking or buckling. MuchDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 11Figure 4. Internal stress measurements in AA6111, a) angbracketleftsigmaangbracketright measured after 1% reverse strain, b) angbracketleftsigmaangbracketright expressed as a percentage of thework hardening of the materials; symbols denote the same materials in both pictures.Table 5. Bauschinger stress measurements in materialover aged 7 days at 250 x89, the tension-compression re-sults have been put together with rolling-compressiontests to achieve very large forward straining.Plastic strain (%) Bauschinger stress sigmab (MPa)1 20.72 36.34 41.46 45.412 31.725.6 23.858.7 16.6higher values of strain can be achieved by cold rolling the material1. Considering the von Mises equivalentstress, rolling to 10%, 20% and 40% reduction lead to forward strain values of 0.12, 0.26 and 0.58. Theexperimental results for the Bauschinger stress as a function of reverse strain are presented in Fig. 3fand the corresponding results in terms of internal stress measured after 1% reverse plastic strain aresummarized in Table 5 together with the other values measured with the tension-compression tests for7D250.It is clear that the increase of the internal stress observed before 10% strain does not continue. In fact,it is observed that after 10% plastic strain, the internal stress start decreasing. Fig. 5 presents some TEMmicrographs of 7D250 material which provides evidence as to what is happening. Fig. 5a is a dark fieldimage of a sample deformed 8% in tension. The dislocation structure is complex but one should observethat the precipitates are still intact and very straight. Fig. 5b and Fig. 5c are bright field images of cold1Note: to a first approximation, one does not expect a large difference in the internal structure for specimens submitted to rolling vs.tensile deformationDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa611112 H. Proudhon et al.Figure 5. Observation of the microstructure of material 7D250 by TEM: a) dark field image of a sample deformed 8% in tension, b)bright field image after 20% cold rolling, c) bright field image after 40% cold rolling; arrows indicate examples of precipitates fracture.rolled specimens for 20% and 40% reduction. One should note in the latter two cases that the precipitateshave fractured; this has actually been observed also for the 10% reduction cold rolled specimens.4 DiscussionThe results presented in the previous section have shown that depending on the thermal history of thematerial, very different behaviours are observed in terms of internal stress build-up. This section deals withanalysing the results regarding the interaction of the dislocations with the precipitation structure. First wereview the importance of the shearable/non-shearable transition and then the intervention of alternativeprocesses is considered for the case of non-shearable particles.4.1 The shearable/non-shearable transitionThere is a general agreement about the precipitation sequence of strengthening alloys such as AA6111.That sequence can be written as follow:SSS arrowright clusters/GP zones arrowright betaprimeprime precursor of Q arrowright equilibrium Q + Mg2SiHere, SSS denotes the supersaturated solid solution; betaprimeprime are Mg5Si6 needle shaped precipitates lying inthe angbracketleft100angbracketright direction of the aluminium matrix; Q are lath shaped precipitates whose chemical compositionDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 13Table 6. Summary of the main information describing the precipitates structure in the 5studied materials.material nature of precipitates average radius precipitates are...5M180 betaprimeprime + Qprime <1.2 nm shearable7H180 betaprimeprime + Qprime 1.8 nm shearable60D180 Q 3.3 nm mixed6H250 Q similar 5 nm mixed, mostly non-shearable7D250 Q 6.6 nm non-shearableis likely to be Al4Cu2Mg8Si7; and Mg2Si is a plate shaped equilibrium phase (for full details with TEMobservations and references on the precipitation sequence, see [20]).Table 6 provides a summary of the precipitates structure and the nature of the dislocations/precipitatesinteraction (based on monotonic work hardening and TEM studies) for the 5 different materials examinedin this work. According to Table 6, one can correlate the observed behaviour for the internal stress buildup during deformation with the nature of the dislocations/precipitates interactions. When the dislocationscan pass through the precipitates (5M180 and 7H180), no Orowan loop is left around the precipitatesand little or no internal stress is found within the material. As the size of precipitates increases, at somepoint it will become impossible for the dislocation to pass through. At this point, the dislocation will bowbetween the precipitates and then by-pass the precipitates, leaving one dislocation loop around the particlefor each mobile dislocation. This mechanism is responsible for the internal stress measured in the cases ofmaterials 6H250 and 7D250. In the case of 60D180, it is believed that the distribution is mixed ?see [24]for high resolution TEM evidence that some precipitates are still sheared by the dislocations? so bothmechanisms coexist which is consistent with the intermediate value of internal stress that was observed.4.2 Deformation mechanisms at larger strainsThe mechanical tests carried out on the rolled 7D250 samples have shown that the internal stress firstincreases with forward strain and then after approximately 10%, starts to decrease as a function of theamount of reduction. In other words, increasing the deformation in the specimen actually reduces theinternal stress stored inside the material.At first sight, this behavior seems inconsistent with the mechanisms we have just described, where theBauschinger stress increases with epsilonp. But this can be understood by the fact that the storage of more andDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa611114 H. Proudhon et al.more loops around the particle correspondingly increases the axial stress on it. Thus, it is reasonable thatat some point, the number of loops around the particle is large enough that the particle will fracture. Thishypothesis is supported by TEM observation of 7D250 deformed by rolling shown previously on Fig. 5where the fracture of the precipitates can readily be observed.4.3 Experimental determination of the unrelaxed plastic strainIt is of interest to extend furter the analysis of the Bauschinger stress data. For this, only the over agedmaterials will be considered since they are the only ones to contain a population of non-shearable particles.It has been shown for those materials that the internal stress first increases linearly (see Fig. 4a) with aslope Theta = dangbracketleftsigmaangbracketright/depsilon. Although more data points would be needed to measure these slopes precisely, onecan estimate (?200 MPa) the following values: Theta60D180 = 1800 MPa, Theta6H250 = 2200 MPa, Theta7D250 = 2400MPa (this would be consistent with a slight increase in the volume fraction of precipitates as a functionof ageing time).Those experimental values can now be used to calculate the unrelaxed plastic strain epsilonasteriskmathp at the end of theforward loading for each experimental test. Because in the linear region (i.e. epsilonp <=< 2%) the work hardeningprocess is predominantly associated with storing more and more dislocation loops around the precipitates,the unrelaxed plastic strain is equal to the plastic strain:epsilonasteriskmathp = epsilonp for epsilonp <=< 2% (2)and one can now calculate the unrelaxed plastic strain epsilonasteriskmathp for the data in Fig. 4a as:epsilonasteriskmathp = angbracketleftsigmaangbracketrightTheta (3)December 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 15The experimental values will be shown later on (see Fig. 6), together with the predictions achieved by theequation developed in section 5.1.5 Modeling5.1 A constitutive equation for the unrelaxed plastic strain, epsiloncurrency1pBrown and coworkers derived an expression for the internal stress starting with the unrelaxed shearplastic strain gammaasteriskmathp [7]. For spherical particles of radius r, gammaasteriskmathp = nb/2r where n is the number of dislocationsresponsible for the deformation, and b represents the magnitude of their Burgers? vector. We shall considera similar approach, while taking into consideration the particular geometry and size of the precipitates.In this case, the precipitates are lath shaped, aligned along the angbracketleft100angbracketright directions and the dislocation glidealong slip planes ({111} in FCC). Thus the precipitate length intercepting the slip planes l costheta replacesthe sphere diameter 2r in the previous expression, with theta = 54.74xb0:gammaasteriskmathp = nbl costheta (4)The actual deformation in the direction of interest must be related to the effective Burgers? vector in thatdirection: b sintheta. Introducing the Taylor factor M for multi-slip systems (gamma = M epsilon) leads to the followingexpression:epsilonasteriskmathp = nb sinthetaM l costheta (5)Now the rate of creation of these dislocation loops dn/dgamma, as the shear strain gamma operates, must beassessed. Because the loops will be stored along the axial direction of the precipitates, it should be propor-tional to l costheta/(b sintheta), the number of slip planes intercepting the precipitate. However, this dislocationDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa611116 H. Proudhon et al.accumulation around a precipitate cannot increase without limit. Either the internal stress will be suficientto fracture the particles and provide new free surfaces where the loops can annihilate, or the stresses atthe interface will trigger recovery mechanisms preventing further accumulation. The net effect is that thenumber of loops that can be stored must have an upper limit nasteriskmath. That can be described easily if one pic-tures this effect as the maximum number of ?available sites? nasteriskmath. If n dislocations are stored, the proportionof sites unoccupied on which further storage is possible is (1-n/nasteriskmath). Accounting of this effect amounts tomultiply the ?geometric storage rate? by this factor as expressed in Eq. (6). A similar approach as beenrecently used successfully to describe the storage of dislocations at grain boundaries [30]. Finally, one canwrite the rate of Orowan loops formation as:dndgamma =l costhetab sintheta?1 - nnasteriskmath?(6)Using the Taylor factor and reordering give:d(n/nasteriskmath)1 -n/nasteriskmath =M l costhetanasteriskmath b sinthetadepsilon (7)Integration of Eq. (7) leads to:ln?1 - nnasteriskmath?= -M l costheta epsilonnasteriskmath b sinthetaThen n can be written:December 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 17Figure 6. Prediction of the unrelaxed plastic strain by Eq. (9) (with nasteriskmath = 60) compared to the values issued from the experiments andcalculated with Eq. (3); as before, the error bars are based on a 200 MPa uncertainty in the measure of Theta.n = nasteriskmath?1 -exp?-M l costheta epsilonnasteriskmath b sintheta??(8)and finally, combining with Eq. (5), the unrelaxed plastic strain comes to:epsilonasteriskmathp = nasteriskmath b sinthetal costheta?1 -exp?-M l costheta epsilonnasteriskmath b sintheta??(9)From a physical point of view, nasteriskmath is the maximum number of dislocation loops which can be stored aroundthe precipitate. One should note that, if epsilon arrowright0, we have epsilonasteriskmathp arrowrightepsilonp; which is consistent with the experimentsas presented in xa74.3 (this behaviour is observed for epsilonp <=< 2%). The predictions for the unrelaxed plasticstrain are presented together with the experimental values in Fig. 6.5.2 Modeling the internal stressIn order to model the internal stress, we will use a combined continuum/dislocation based approach.As shown in Fig. 5, the precipitates can be considered as fibres oriented along the angbracketleft100angbracketright directions inan aluminum matrix (note the precipitates are lath shaped but we shall consider a simpler view whereDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa611118 H. Proudhon et al.their cross section is represented by a circle with an equivalent radius r0 = 6.6 nm). The average initialprecipitate length l0 is approximately 300 nm, although there is some uncertainty here as it is difficult tomeasure in the TEM.The internal stress angbracketleftsigmaangbracketright is associated with the dislocations stored around the precipitates. According tothis view, one can write:angbracketleftsigmaangbracketright = alphaf sigmappt (10)f being the volume fraction of the precipitates, sigmappt the average stress in the precipitates, and alpha a geo-metrical factor relative to the orientation of the precipitates with respect to the tensile axis. Because theexperimental results have shown that it is important to account for the particle fracture, we shall adopttwo different views depending wether or not the precipitate has been fractured or not:? For unfractured precipitates, we use Eshelby?s solution for ellipsoidal particles of high aspect ratio, i.e.the stress is uniform within the inclusion. Then we have sigmappt = sigmappt = E epsilonasteriskmathp, where E is the modulus ofelasticity of the precipitates in the deformation direction. While we can now estimate epsilonasteriskmathp from Eq. (9),determining the stiffness of the precipitates requires special attention and will be addressed in paragraphsection 5.2.1.? For fractured precipitates, we consider that the fracture process creates new free surfaces at the precip-itates ends where the stress must now be zero. We then take the simplest possible view that the stressin the particles increases linearly from the end to the centre using a simple shear lag model which ispresented in section the elastic constants of the Q phase. The Q phase precipitates are lath shaped, perfect HCPcrystals of Al4Cu2Mg8Si7, aligned along the angbracketleft100angbracketright direction of the Al matrix. A critical piece of informationneeded is the elastic modulus of the crystal along its c axis. Unfortunately, no experimental values areknown. As a result, we will use the first-principles calculation by Wang and Wolverton [31]. The calculationDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 19Table 7. Elastic constants (unit is GPa) obtained by first-principles calculationsfor Al3Cu2Mg9Si7.C11 = C22 C12 C13 C33 C44 = C55 C66 = (C11 -C12)/2147.3 36.8 31.0 128.3 36.4 55.5is made using the Vienna ab initio simulation package (VASP) with Vanderbilt ultrasoft pseudopotentialsand the generalized gradient approximation (GGA). The stress-strain approach is employed with the 0Klowest energy DFT (density functional theory) stoichiometry Al3Cu2Mg9Si7 to give the elastic constantslisted in Table 7.Now, using the relations for hexagonal crystals (see for instance [32]), one can derive the stiffness coef-ficient Sij from the elastic constants. In particular, this leads to S33 = 0.00848 GPa-1. Then the Youngmodulus can be derived:E = 1S33= 118 GPa (11)5.2.2 A simple plastic shear lag model for fractured precipitates. The Q phase precipitates are es-sentially perfect crystals, and as such will fail when the theoretical tensile strength sigmau, is reached. Weshall consider that the particles are brittle, so once the load reaches sigmau, the particle fractures into twopieces of equal length. The experimental results suggest that the fracture stress of the precipitates shouldbe within a 2500?3000 MPa range which is approximately E/40. This is within a factor of 4 theoreticaltensile strength [33].Once fractured, the ends of the particles are unbonded with the matrix and the stress transfer fromthe inclusion to the matrix is expected to be very different from the non-broken particles. We shall adopta simple view where the stress follows a triangular evolution from zero at both ends and reaching itsmaximum in the middle of the precipitates as depicted on Fig. 7.With this view the stress increases linearly with the following rate [34]:dsigmadx =2taur =sigmafr (12)December 19, 2007 17:43 Philosophical Magazine bauschinger?aa611120 H. Proudhon et al.Figure 7. Stress distribution in the simple shear lag model and illustration of the break up when the load reaches the tensile strengthof the precipitate; x denotes the position along the precipitate length.tau being the shear stress at the interface (2tau = sigmaf). Finally the average stress in the fibre is simply half themaximum stress:sigmappt = sigmappt2 = l4 sigmafr (13)According to this model, the breakup of precipitates will lower the maximum stress (and thus the averagestress) and the internal stress calculated by Eq. (10) will decrease. The internal stress drop is highlightedby the hatched area on Fig. 7.Of course, the fracture process is much more complex than the simple view we are developping here. Firstof all, the dislocation loops around the precipitates may not be perfectly equispaced, such that fracturemay not occur at the centre of the precipitate. In addition, a precipitate will not be perfectly aligned withthe angbracketleft001angbracketright direction of the Al matrix and its geometrical parameter be different from the average ones.To account for all these discrepancies, we have chosen to use a non unique fracture stress. A Gaussiandistribution is used, centered on the mean value sigmau and with a standard deviation fu.A consequence of using a Gaussian distribution for the fracture stress is the possibility to resolve readilythe fraction of broken precipitates, which will be given exactly by the Gaussian cumulative distributionfunction. This is due to the fact that for a given stress sigma = E epsilonasteriskmathp all the precipitates with sigmau <=< sigma will bebroken. The total fraction of broken precipitates p1 (the fraction of unbroken precipitates being 1 - p1),December 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 21can be obtained by:p1 = 12?1 + erf?E epsilonasteriskmathp -sigmaufuradical2??(14)where erf is the Gauss error function: erf(x) = 2/radicalpi Rx0 exp-t2dt. After the first fracture, the fractionof precipitates broken i times (i > 1) among the ones broken i -1 times1, is:pi = 120B@1 + erf0B@sigmafrl04i -sigmaufuradical21CA1CA (15)5.2.3 Integration of the models. In order to fully describe the evolution of the internal stress carriedout by the precipitates over a wide range of strain, we need to combine the models. The full model isconstituted by a small computer program implementing the unrelaxed plastic strain expression stated byEq. (9), the plastic shear lag model with the particle fracture. For completeness, the parameters used inthe model have all been gathered in Table 8. Although all parameters have a physical meaning, some ofthem, namely nasteriskmath and fu are not precisely known and the values have been chosen to fit the experimentaldata. nasteriskmath controls the rate of the elastic build up of dislocation loops and the maximum stress reachedin the precipitates. A value of nasteriskmath = 60 was chosen to fit the data presented on Fig. 6. Although noprecise measure is available in the literature to check this value, it is worth noting that preliminary TEMobservations counting the number of dislocation loops around the precipitates (in dark field contrast)gave approximately 40 loops per precipitate after 8% tensile strain. Given the difficulty in making thesemeasurements, we beleive this is in reasonable agreement with the maximum value of nasteriskmath = 60. Now, withsigmau similarequal E/40, the precipitate fracture occurs approximately for epsilonasteriskmathp = 0.025 which according to Fig. 6 willhappend slightly before a strain of 0.1. This is consistent with the experimental observations. In the end,a fracture stress variance of 170 MPa is chosen to fit the internal stress drop measured experimentally.The matrix flow stress evolution is taken from the experimental tensile behavior. Finally, combining1In Eq. (15) the fraction pi is relative. For instance, since to break twice, a precipitate must first break once, the actual fraction ofprecipitate broken twice is p1 p2.December 19, 2007 17:43 Philosophical Magazine bauschinger?aa611122 H. Proudhon et al.Table 8. Physical parameters used in the internal stressmodel.Parameter Symbol ValueVolume fraction f 0.019Young Modulus E 118 GPaOrientation factor alpha 1Burger?s vector b 0.286 nmMaximum number of loops nasteriskmath 60Initial length l0 300 nmAverage radius r 6.6 nmMean fracture stress sigmau 2900 MPaFracture stress standard dev. f 170 MPaEq. (14) and (15) the general form of the internal stress is given by:angbracketleftsigmaangbracketright = (1 -p1)f E epsilonasteriskmathp| {z }unbroken precipitates+f p1(1 -p2) l04 sigmafr| {z }broken once+f p1 p2 l08 sigmafr| {z }broken twice+? ? ? (16)The output of the simulation has been plotted in Fig. 8. One can see a very good agreement betweenthe prediction by the model and the experimental values of internal stress measured by the Bauschingertests. Fig. 8 shows that the initial internal stress carried out by the precipitates almost linearly increaseswith strain (initially epsilonasteriskmathp = epsilon). This is followed by a regime where plastic relaxation occurs and then theprecipitates start to fracture at a strain just below 0.1 resulting in a rapid drop of the internal stress. Dueto increasing flow stress of the alloy sigmaf as the straining continues, most of the precipitates will fracture oncemore, further decreasing the stress borne by the precipitates. It is worth noting that with the current valuesof the parameters, a small fraction of the precipitates never fracture. Physically, this can be interpretedthat some of the precipitates are oriented such that the load transfer is ineffective.We are here standing at the limit of the continuum/discrete description of the deformation. The re-sults of Fig. 8 show that adopting a physically based model taking into account the discrete nature ofthe dislocation/precipitates interaction is capable of describing micro-mechanical phenomena such as theinternal stress build up and decrease, without having to explicitly account for the discrete view of thedislocation motion. However there could be a value for a discrete dislocation simulation to push furtherthe physical description of those mechanisms by adopting a multiscale modeling framework. For example,discrete dislocation simulation could help to shape fundamental equations such as the rate of dislocationloop formation and provide physical input for the fitting parameters such as the fracture stress variance.December 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 23Figure 8. Results of the internal stress model: a) internal stress predicted by the model, as a comparison, the calculation for f E epsilon andf E epsilonasteriskmathp have also been plotted b) evolution of the fraction of broken precipitates.Yet, some points remain unclear at the moment. One is that with alpha = 1, the internal stress contributedfrom the precipitates has the form of f E epsilonasteriskmathp; this implicitly supposes that all precipitates fully participatein the process. While this may be true in the case of spherical inclusion (such as the copper silica system),it is not true in our case since the precipitates are distributed with different orientations in the materialwith respect to the loading direction. There are several explanations possible to this issue. First one canargue that during deformation, the fibre-like precipitates tend to align with the loading direction. Thiswill tend to lower down the orientation effect. Second, there are some uncertainties in the microstructuralparameters values used in the model. Specifically, the Young modulus of the precipitates was evaluated fromfirst principles and as such there may be some uncertainty in its value. Third, as previously discussed thereis also a degree of uncertainty from the offset chosen to evaluate the internal stress from the Bauschingertests. A direct measurement of the stress in the precipitates using for example, X-ray diffraction would bevaluable.6 ConclusionsThe plastic deformation of a commercial precipitation strengthening aluminium alloy (AA6111) has beeninvestigated for a large range of ageing conditions. The main results can be summarized as follows.The development of internal stresses critically depends on the precipitates size and volume fraction. ItDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa611124 H. Proudhon et al.has been shown that for the case of shearable particles, the development of long range internal stress israther small and the work hardening is dominated by dislocation-dislocation interactions. On the otherhand, for non-shearable precipitates, a significant fraction of the work hardening can be attributed to thedevelopment of internal stress. This is interpreted by long range elastic stresses produced by dislocationloops stored around the particles, and can be characterized by tension-compression test. It was found thatplastic relaxation occurs after approximately 2% plastic strain. At higher strains, the dislocation loopstorage is such that axial loading of the precipitates reaches a new regime where the precipitates fracture,reducing the internal stress in the material.Following Brown and coworkers, the framework for modelling deformation in two phase materials hasbeen revisited with the insight from the current experiments. First, a physically based expression for theunrelaxed plastic strain has been derived. This expression has been shown to accurately predict the plasticrelaxation measured from Bauschinger experiments. To fully describe the internal stress evolution with theongoing strain, a composite view of the material has been adopted. Within this view, a simple shear lagmodel has been used to compute the internal stress of the broken precipitates, allowing a good predictionof the internal stress development, up to strains of 0.6. This is, to the best of the authors knowledge, thefirst time that this regime is clearly identified and successfully modelled.AcknowledgementThe authors greatfully acknowledge the help of C. Wolverton and Y. Wang for providing the elasticconstants of the Q phase via first principle calculations. The authors are thankful to Pr. C. W. Sinclairfor helpful discussion and comments on the present work. H. P. would like to thank the French ministryof foreign affairs for the Lavoisier fellowship during his work at the University of British Columbia.References[1] J. D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Societyof London, 241(1226):376?396, August 1957.December 19, 2007 17:43 Philosophical Magazine bauschinger?aa6111The internal stress in AA6111 25[2] T.W. Clyne and P.J Withers. An Introduction To Metal Matrix Composites. Cambridge Solid State cience Series. CambridgeUniversity Press, Cambridge, 1993.[3] M. F. Ashby. Strengthening Methods in Crystals. John Wiley and Sons, New York, 1971.[4] H. H. M. Cleveringa, E. Van Der Giessen, and A. Needleman. Comparison of discrete dislocation and continuum plasticity predictionsfor a composite material. Acta Materialia, 45(8):3163?3179, August 1997.[5] J. D. Embury. Plastic flow in dispersion hardened materials. Metall. Trans. A, 16(12):2191, 1985.[6] L. M. Brown and W. M. Stobbs. The work-hardening of copper-silica I. A model based on internal stresses, with no plastic relaxation.Philosophical Magazine, 23(185):1185?1199, 1971.[7] L. M. Brown and W. M. Stobbs. The work-hardening of copper-silica II. The role of plastic relaxation. Philosophical Magazine,23(185):1201?1233, 1971.[8] N. A. Fleck, G. M. Muller, M. F. Ashby, and J. W. Hutchinson. Strain gradient plasticity: Theory and experiment. Acta Metallurgicaet Materialia, 42(2):475?487, February 1994.[9] C. M. Sargent and G. R. Purdy. Dissolution of small precipitates. Scripta Metallurgica, 8:569?572, 1974.[10] I. Gutierrez-Urrutia, M.A. Munoz-Morris, and D.G. Morris. The effect of coarse second-phase particles and fine precipitates onmicrostructure refinement and mechanical properties of severely deformed al alloy. Materials Science and Engineering A, 394(1-2):399?410, March 2005.[11] A. Deschamps, F. Bley, F. Livet, D. Fabregue, and L. David. In-situ small-angle X-ray scattering study of dynamic precipitation inan Al?Zn?Mg?Cu alloy. Philosophical Magazine, 83(6):677?692, February 2003.[12] A. Kelly and R. B. Nicholson. Guinier preston zones in an aluminium-silver alloy. Acta Metallurgica, 12(2):277, February 1964.[13] S. Tao and J. Embury. The mechanical behavior of directionally solidified Al?Ni in compression. Metallurgical and Materials Trans-actions A, 24(3):713?719, March 1993.[14] J. G. Byrne, M. E. Fine, and A. Kelly. Precipitate hardening in an aluminum-copper alloy. Philosophical Magazine, 6:1119?1145,1961.[15] G. D. Moan and J. D. Embury. A study of the Bauschinger effect in aluminum-copper alloys. Acta Metallurgica, 27(5):903?914, May1979.[16] J.D. Atkinson, L. M. Brown, and W. M. Stobbs. The work-hardening of copper-silica IV. The Bauschinger effect and plasticrelaxation. Philosophical Magazine, 30:1247?1280, 1974.[17] L. M. Brown and D. R. Clarke. The work hardening of fibrous composites with particular reference to the copper-tungsten system.Acta Metallurgica, 25(3):563?570, May 1977.[18] D. V. Wilson. Reversible work hardening in alloys of cubic metals. Acta Metallurgica, 13(7):807?814, July 1965.[19] L. M. Brown. Orowan?s explanation of the Bauschinger effect. Scripta Metallurgica, 11:127?131, 1977.[20] X. Wang, W. J. Poole, S. Esmaeili, D. J. Lloyd, and J. D. Embury. Precipitation strengthening of the aluminum alloy AA6111.Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, 34(12):2913?2924, December 2003.[21] S. Esmaeili, X. Wang, D.J. Lloyd, and W.J. Poole. On the precipitation-hardening behavior of the Al?Mg?Si?Cu alloy AA6111.Metallurgical and Materials Transactions A, 34:751?763, 2003.[22] X. Wang, S. Esmaeili, and D. J. Lloyd. The sequence of precipitation in the Al?Mg?Si?Cu alloy AA6111. Metallurgical and MaterialsTransactions A, 37(9):2691?2699, 2006.[23] L.M. Cheng, W.J. Poole, J.D. Embury, and D.J. Lloyd. The influence of precipitation on the work hardening behavior of theDecember 19, 2007 17:43 Philosophical Magazine bauschinger?aa611126 H. Proudhon et al.aluminum alloys AA6111 and AA7030. Metallurgical and Materials Transactions A, 34:2913?2924, 2003.[24] W. J. Poole, X. Wang, D. J. Lloyd, and J. D. Embury. The shearable/non-shearable transition in Al?Mg?Si?Cu precipitationhardening alloys: Implications on the distribution of slip, work hardening and fracture. Phil Mag., 85(26-27):3113?3135, September2005.[25] W. J. Poole and D. J. Lloyd. Modelling the stress-strain behaviour for aluminum alloy AA6111. In J. F. Nie, A. J. Morton, and B. C.Muddle, editors, Proceedings of the 9th International Conference on Aluminium Alloys, pages 939?945, Brisbane, 2004. Institute ofMaterials Engineering Australasia Ltd.[26] R. E. Stoltz and R. M. Pelloux. Cyclic deformation and Bauschinger effect in Al?Cu?Mg alloys. Scripta Metallurgica, 8:269?276,1974.[27] R. J. Asaro. Elastic-plastic memory and kinematic-type hardening. Acta Metallurgica, 23:1255?1265, October 1975.[28] D. J. Lloyd. The Bauschinger effect in polycrystalline aluminium containing coarse particles. Acta Metallurgica, 25(4):459?466, 1977.[29] O. B. Pendersen, L. M. Brown, and W. M. Stobbs. The Bauschinger effect in copper. Acta Metallurgica, 29:1843?1850, 1981.[30] C. W. Sinclair, W. J. Poole, and Y. Br? het. A model for the grain size dependent work hardening of copper. Scripta Materialia,55(8):739?742, August 2006.[31] C. Wolverton. Crystal structure and stability of complex precipitate phases in Al?Cu?Mg?(Si) and Al?Zn?Mg alloys. Acta Materialia,49:3129?3142, 2001.[32] W. F. Hosford. The Mechanics Of Crystals And Textured Polycrystals. Oxford University Press, 1993.[33] A. Kelly and N.H. Macmillan. Strong Solids. Clarendon press, Oxford, 3rd edition edition, 1986.[34] H. Lilholt. Hardening in two phase materials - II. plastic strain and mean stress hardening rate. Acta Metallurgica, 25:587?593, 1977.


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