An Age-Hardening Model for Al-Mg-Si Alloys Considering
Needle-Shaped Precipitates
A. BAHRAMI, A. MIROUX, and J. SIETSMA
In the present study, an age-hardening model for Al-Mg-Si alloys was developed considering cylindrical morphology with constant aspect ratio for precipitates. It is assumed that the pre-cipitate distribution during underaging is controlled by simultaneous nucleation and growth, and after peak age, the process becomes coarsening controlled. The transition from the nucleation/growth regime to the coarsening regime takes place when the equilibrium fraction of the precipitating phase is reached. The microstructural model is combined with a precipitation-strengthening model to predict the evolution of yield strength of Al-Mg-Si alloys during aging. The predictions of the model on the evolution of yield strength and length, radius, and volume fraction of precipitates are presented and compared with experimental data.
DOI: 10.1007/s11661-012-1211-8
The Author(s) 2012. This article is published with open access at Springerlink.com
I. INTRODUCTION
SIMULATIONS
of precipitation and strengthening during aging of heat-treatable aluminum alloys have gained considerable interest during the past decades.[1–13] Numerous attempts were made to develop age-hardening models for different applications: isothermal aging of naturally aged,[2] preaged,[1] and predeformed alloys;[14]nonisothermal aging;[3]precipitation reactions during aging;[15]and multistage aging.[16,17]The various previously developed age-hardening models use the simple assumption of spherical particles in a metal matrix. Nevertheless, the presence of the elongated needlelike b¢¢-precipitates and rod-shaped b¢-precipi-tates, both oriented in 001h iAldirections, is thought to be the main source of hardening.[18–20]Kelly[21]showed that the hardening due to nonshearable rod-shaped precipitates is greater than that produced by an equiv-alent density of spherical precipitates with the same volume. Mathematical solutions to the problem of the diffusion-controlled growth of precipitates with a parab-oloid shape were proposed.[8–10] However, very few applications of these models exist for precipitation in aluminum alloys. Most studies deal with the precipita-tion of platelike precipitates,[22] and only one attempt was made to model precipitation of elongated precipi-tates in Al-Mg-Si alloys to predict the strength evolution during aging.[23]The latter model, however, relies on a series of assumptions, including constant supersatura-tion and nonshearable precipitates, and was fitted on the peak-age strength. A process model for aging,considering elongated precipitates with a more accurate strength model, was also proposed[24]but uses the semi-phenomenological Johnson–Mehl–Avrami equation to simulate the precipitation kinetics. The present article proposes a coupled precipitation and strength model applicable to isothermal aging of Al-Mg-Si alloys. The model is able to predict the evolution of the radius of precipitates as well as their number density and volume fraction. The precipitates are assumed to be cylindrical with constant aspect ratio. The microstructural reac-tions are divided into two parts: (1) simultaneous nucleation and growth during underaging and (2) coarsening during overaging. In Sections II through IV, the model principles and relations are described. The model is applied to simulate precipitation in the Al-Mg-Si alloy AA6061. The results are compared with experimental data in SectionV, and finally conclusions regarding the relevance of considering the precipitate shape for aging simulation are discussed in SectionVI.
II. MICROSTRUCTURE MODEL
Precipitation is a process in which the initial super-saturated alloy is decomposed into matrix and a new phase, generally containing a higher concentration of solute atoms. The precipitation is traditionally catego-rized into three stages: nucleation, growth, and coars-ening. In this model, simultaneous nucleation and growth is assumed to take place as long as the equilibrium volume fraction has not been reached. After reaching the equilibrium value, coarsening starts. In the microstructure model, a single type of precipitate is considered. These assumptions already were successfully applied to modeling the precipitation hardening in Al-Mg-Si alloys.[3–5]For a complete model description, there are additional assumptions, as follows.
(1) Precipitates have cylindrical morphology, with h being the half-length and r being the radius. A. BAHRAMI, Postdoctoral Researcher, and A. MIROUX, Senior
Researcher, are with the Materials innovation institute (M2i) and Department of Materials Science and Engineering, Delft University of Technology, 2628 CD Delft, the Netherlands. Contact e-mail: a.n.bahrami@gmail.com J. SIETSMA, Professor, Department of Materials Science and Engineering, is with the Delft University of Technology.
Manuscript submitted January 26, 2012. Article published online August 9, 2012
(2) The aspect ratio of precipitates, defined as the ratio of half-length over radius (A = h/r), is constant during aging.
(3) The stoichiometry of precipitates is Mg2Si.
(4) The interfacial energies at the tip and at the rim of precipitates are assumed to be identical. The inter-facial energy is constant during aging, and it is not dependent on the aging time, aging temperature, and alloy composition.
(5) Mg diffusion in the matrix controls the kinetics of aging.
Assuming that precipitates nucleate with a cylindrical morphology, the change in the Gibbs free energy of the system due to nucleation can be written as
DGnucl¼ 2pr2hD Gvþ 4prhc þ 2pr2c ½1 where DGmis the driving force per unit volume to form a precipitate from supersaturated solid solution and c is the interfacial energy between the precipitate and the matrix. Using the definition of the aspect ratio (A = h/r), the critical radius of precipitates, rcr, corresponding to the maximum of DGnucl, can be expressed as
rcr¼ 2 3 2Aþ 1 A c DGv ½2
Inserting the expression of DGmgiven in Reference23 into Eq. [2] yields
rcr¼ 2 3 Vmc RT 2Aþ 1 A ln Cm Cabe 1 ½3
where Cm is the mean concentration of Mg in the matrix, Cab
e is the equilibrium concentration of Mg in the matrix at the aging temperature, and Vm is the molar volume of precipitate. Provided that the incuba-tion time can be neglected, i.e., assuming steady-state nucleation, the nucleation rate J is expressed as[7]
J¼ J0 exp DG RT Qd RT ½4
where J0is a pre-exponential term, and Qd is the acti-vation energy for bulk diffusion of Mg. DG; the acti-vation energy for nucleation, can be obtained from Eqs. [1] and [3] and is given as
DGhet¼8p 27 2Aþ 1 ð Þ3 A2 V2mc3 RT ð Þ2 ln Cm=Cabe h i2 ½5
Myhr et al.[16]proposed an approximate expression to calculate the energy barrierDG;given by
DGhet¼ ðA0Þ 3 RT ð Þ2 ln Cm=Cabe h i2 ½6
where A0 is treated as a fitting parameter. Peripheral and longitudinal growth rates of a needlelike precipi-tate of radius r and half-length h are determined by
the composition gradient outside the precipitate; the bulk diffusivity of Mg, D; and the aspect ratio of pre-cipitates. Extending the approach of Ferrante and Doherty[22] and Liu et al.,[23] the diffusion-controlled thickening and lengthening of precipitates approxi-mately obey r¼2 3 Cm Cabr Cba Cab r 1=2 D pAt 1=2 ½7
where Cbais the concentration of Mg inside the precip-itate, and Cab
r is the equilibrium interface concentra-tion around the precipitates, taking into account the Gibbs–Thomson effect,[22]given as
Cabr ¼ Cab e 1þ 1 þ 1 A cVm RTr 1 Cab e Cba ½8
Since both Cmand Cabr are a function of the degree of precipitation, the aging time during underaging is divided into a series of small time intervals (Dt). At each time-step, a new population of JDt precipitates nucleates, having a radius equal to rcr (Eq. [3]). In the same time interval, other previously formed precipitates grow. Thus, a precipitate size distribution evolves. If Dt is chosen to be very small, one can assume that the change in supersaturation during the time period Dt is negligible. Therefore, the increment in the radius of precipitates group j, having the radius rj, is calculated as
Dr¼1 3 Cm Cabrj Cba Cab rj !1=2 D pA 1=2 t1=2Dt ½9
being the time derivative of Eq. [7]. Consequently, the radius of precipitates in group j in the ith time-step is written as rj i¼ rj i1þ1 3 Cm Cabrj Cba Cab rj !1=2 D pA 1=2 t1=2i Dt ½10
By keeping track of the growth of each group of precipitates formed during aging, one can calculate the mean radius of precipitates, rm. At the end of each time-step, the mean concentration of alloying element in the matrix is updated using the mass balance. The aging time increases step-by-step until the calculated mean concentration of Mg in the matrix becomes equal to the equilibrium interface concentration of precipitates of smallest size. The highest equilibrium interfacial con-centration in the matrix occurs at the smallest precip-itates (due to the Gibbs–Thomson effect). This group of precipitates, therefore, is the first group reaching the criterion Cm<Cabrj ; when the growth stops (Eq. [10]). This is a numerically imposed criterion. From the moment that Cm<Cabrj ; the nucleation/growth stage is terminated and coarsening starts. According to the Gibbs–Thomson equation, during coarsening, the sta-bility of precipitates increases by increasing their size, meaning that fine precipitates dissolve and bigger precipitates grow to reduce the free energy of the
system. Therefore, the driving force for coarsening is provided by the difference between the size-dependent interfacial concentrations of alloying element and the average concentration of alloying element in the matrix. Assuming that during coarsening precipitates of the average size are in equilibrium with the matrix (Crm¼ Cm), the supersaturation during coarsening for a precipitate of size r can be written as[22]
X¼Crm C ab r Cba Cab e Crm C ab r Cba ½11
where Crm is the equilibrium interfacial concentration around the precipitate of mean radius rm. The value of Crm can be obtained from Eq. [8]. Inserting the values of Cabr and Crm into Eq. [11] yields
X¼cVm RT 1þ 1 A Cab e 1 Cabe Cba ð Þ2 1 rm 1 r ½12
This equation clearly shows that for precipitates larger than rm, the supersaturation is positive, meaning that during coarsening, when r is bigger than rm, precipitates grow. From Eq. [7], the coarsening rate can be written as dr dt¼ 2 9 D pA X r ½13
Inserting the value of supersaturation from Eq. [12] into the coarsening rate equation yields
dr dt ¼ 2 9 D p cVm RT 1þ A A2 Cab e 1 Cabe Cba ð Þ2 r rm r2r m ½14
Using the approximation that the mean radius changes at the same rate as the maximum thickening rate, when r = 2rm,[22] the coarsening rate can be written as drm dt ¼ 1 18 D p cVm RT 1þ A A2 Cab e 1 Cabe ðCbaÞ2 1 r2 m ½15
Integrating this equation gives an analytical equation for precipitates radius during coarsening as follows:
Zrm rPA r2mdrm¼ 1 18 D p cVm RT 1 A 1þ 1 A Cab e 1 Cabe Cba ð Þ2 Zt tPA dt ½16 r3m¼ r3 PAþ 1 6 D p cVm RT 1 A 1þ 1 A Cab e 1 Cabe Cba ð Þ2 ðt tPAÞ ½17
with rPA being the mean radius at peak age and tPA being the time to peak age, which corresponds to the time when coarsening starts.
III. STRENGTH MODEL
The strength model is a framework in which the overall strength of the artificially aged alloy can be obtained by the addition of the intrinsic strength of aluminum, the solid solution strength contribution, and the precipitate strength contribution.[4] Assuming that different strengthening contributions to the overall strength can be added linearly, the yield strength of Al-Mg-Si alloys can be expressed as
ry¼ riþ rssþ rppt ½18 where riis the yield strength of pure aluminum chosen as 10 MPa,[3]and rssis the solid solution strengthening term, given by[3]
rss¼
X
j¼Mg;Si;Cu
kjC2=3j ½19
where kj is a constant with a specific value for element j. The effect of precipitates on the strength is given by rppt. The precipitation-strengthening model correlates the size and volume fraction of precipitates with strength. Two main mechanisms exist for precipitation hardening. When a dislocation encounters a precipi-tate, it will either cut through it, a mechanism known as shearing, or bypass the precipitate by looping around it. Shearing is more common in coherent pre-cipitates, which have an orientation relationship with the matrix, whereas in the case of larger precipitates, coherency usually breaks down and looping occurs instead. Initial precipitation hardening involves strengthening of the alloy due to the formation of a high density of small coherent precipitates. These pre-cipitates are sheared during deformation by moving dislocations. By increasing the aging time, the precipi-tates become larger and stronger. Precipiprecipi-tates larger than a transition radius cannot be sheared anymore and a dislocation can only bypass by looping around them. This leaves an Orowan loop around the precipi-tates, which enhances the strength of the alloy. Let F and l denote mean precipitate strength and effective mean interprecipitate distance along the dislocation line, respectively. Then, precipitate strengthening is given by[3]
rppt¼
MF
bl ½20
where M is the Taylor factor and b is the length of the Burgers vector. There are two influencing parameters in precipitation strengthening: (1) mean precipitates strength, F, and (2) effective precipitates spacing, l. The mean precipitate strength is defined as the interac-tion force between the precipitate and the dislocainterac-tion. Experimental observation of dislocation-precipitate interaction in Al-Mg-Si alloys reveals that precipitates are still shearable at the peak-age condition,[2] and a part of them remains shearable even after a long-time overaging.[24] Based on this observation, Esmaeili et al.[2] divided the aging process into three parts: (1) underaging up to peak age when precipitates are shea-rable; (2) the stage between peak age to the transition
point where precipitates are still shearable; and (3) after the transition point, where precipitates behave as nonshearable particles. The transition radius, rtrans, is defined as the radius where the strengthening mecha-nism is changing from shearing to bypassing. The mean precipitate strengths at these three stages are given as[2] F¼ 2bGb2 rm1PA rm transrm rm<rPA ða) F¼ 2bGb2 rm rtrans m rPA<rm<rtrans ðb) F¼ 2bGb2 r trans<rm ðc) 8 > > < > > : ½21
where b is a constant equal to 0.5, and m is 0.6. The effective precipitate spacing is dependent on the precip-itate strength.[24] Precipitate spacing can be estimated based on the planar center-to-center distance between precipitates. Since the most important precipitate spe-cies, b¢¢, is elongated in h100iAl direction, it is very important to consider the effect of its needle-like morphology and its orientation on the effective obstacle spacing.
Considering the orientation relationship between the needle-shape precipitates along theh100iAldirection and the {111}Al slip planes in aluminum, the effective obstacle spacing between precipitates is calculated as[25,26]
l¼ 2pffiffi3p C Ff
1=2
rm for shearable precipitates: ða)
l¼ 2p f
1=2
rm for non-shearable precipitates: ðb)
8 > < > : ½22
where C is the dislocation line tension and f is the vol-ume fraction of precipitates. Knowing the values of effective precipitate spacing for shearable and nonshea-rable precipitates as well as the values of mean obsta-cle strength (Eq. [21]), the contribution of precipitates to the overall yield strength can be calculated as
rppt¼ 2bGbM pfffiffi3p 1=2r3ðm1Þ 2 PA r 3m 2 trans r1=2m rm<rPA ða) rppt¼ 2bGbM pfPAffiffi3p 1=2 r 3m 21 ð Þ m r 3m 2 trans rPA<rm<rtrans ðb) rppt¼ 2bGbM f2pPA 1=2 1 rm rtrans<rm ðc) 8 > > > > > > > < > > > > > > > : ½23
where fPAis the volume fraction of precipitates at peak age.
IV. MODEL IMPLEMENTATION
In this model, it is assumed that all precipitates are cylindrical with a constant aspect ratio, which is a parameter for which different choices can be considered. The alloy used to check the validity of the model is AA6061 (1.12 wt pct Mg, 0.57 wt pct Si, 0.25 wt pct Cu). Assuming that all Si atoms take part in precipitation, the
chemical composition of the alloy is balanced. The equilibrium interfacial concentration of Mg for a flat interface, Ce, is given by[3]
Cabe ¼ 970 wt pctð Þ exp Qs RT
½24
where Qs is the apparent solvus boundary enthalpy. In the strength model, there are two important radii: rPA and rtrans. The value rPA is dependent on the aging temperature and chosen aspect ratio. The microstruc-ture model is developed in such a way that peak age correlates to the time when the precipitation reaction is changing from the nucleation-growth regime to the coarsening regime. Consequently, at peak age, the pre-cipitate density, NPA, is maximum. rPAcan be approxi-mately determined, assuming that the amount of solute elements left in the matrix is negligible com-pared to the amount used to form the precipitates by
rPA¼
C0
2pANPACba
1=3
½25
As is seen, rPA is a function of aspect ratio; the maximum number density of precipitates, which is obtained from the simulation; the initial concentration of alloying elements, C0; and the concentration in the precipitate. Another important parameter in the strength model is the transition radius, rtrans, at which the strengthening mechanism changes from shearing to bypassing. Cheng et al.[27] proposed that for the alloy AA6111, the shearable to nonshearable transition occurs when the yield strength is equal to 0.8 rPA(rPA is the yield strength at peak age). Assuming that this is also the case in alloy AA6061 and knowing that the peak strength of alloy AA6061 is in the range of 250 to 270 MPa, rppt at the transition point, rtransppt ; assuming that the contribution from solid solution hardening is negligible, is given by
rtransppt ¼ 0:8rPA ri ½26
Using Eq. [23], rtranscan be calculated as
rtrans¼ 2bGbM rtrans ppt fPA 2p 1=2 ½27
Assuming that the volume per atom is constant, the final volume fraction of precipitates at the peak age can, by the lever rule of phase equilibria, be approximated as
fPA¼
C0 Cabe
Cba Cab e
½28
Depending on the chosen values, rtransvaries between 3.0 and 4.0 nm. In the developed model, the approxi-mate value of 3.5 nm is used for all conditions. TableI shows the input data used in the microstructure and strength model.
V. MODELING RESULTS
Figure1 shows the simulated length (2h) of precipi-tates in alloy AA6061 during aging at 463 K (190C) together with experimental data. The aspect ratio is adjusted to obtain the correct value for the length of precipitates at peak age (10 ks). The aspect ratio, which gives the correct prediction of the length at peak age, is A = 10. In the underage regime, the model slightly overestimates the length of precipitates, showing the highest overestimation in the beginning of aging. By increasing the aging time, the difference between the modeling results and experimental data becomes smal-ler. In the overage regime, the modeling results are in good agreement with the experimental data.
Figure2 shows the prediction of the volume fraction of precipitates in alloy AA6061 aged at 463 K (190C) together with experimental data. As is seen, the model shows a faster aging kinetics during underaging com-pared to the real aging kinetics. This is possibly due to the assumption of the incubation time being zero. Besides, the maximum precipitate volume fraction predicted by the model is slightly higher than the experimental maximum value. This is due to the fact that it is assumed that all Si atoms are partitioned to the precipitates during precipitation. In reality, a part of Si content of the alloy is used by Fe- and Mn-containing particles.
Figure3shows the reproduction of the yield strength of alloy AA6061 at different temperatures, using A =10, compared with experimental data. Except for aging temperature 443 K (170C), in which there is reasonably good agreement between model and exper-iment, the model overestimates the yield strength of alloy AA6061 in both underage and overage regimes. The overestimation, however, is more pronounced in the underage regime.
Figure4shows the comparison between the modeling results for A = 1 with the case when A = 10. Obvi-ously, for A = 1, the model yields a better fit for the yield strength of the alloy in the underage regime. However, the peak strength is predicted more accurately when A = 10. Also, it is interesting to note that the yield strength at the early stage of coarsening is predicted very well when A = 10, while when A = 1,
the model underestimates the yield strength in the beginning of coarsening. This changes when the over-aging time is increased: the model with A = 1 becomes more accurate, while the model with A = 10 gradually loses its accuracy.
Figure5 shows the effects of aspect ratio on the evolution of number density, mean radius, volume fraction, and precipitate effective spacing during aging at 463 K (190C). The model shows that precipitates with A = 1 have maximum values of both number density and mean radius compared to precipitates of A =20 and 50. Figure5(c) shows that even though both the number density and thickness of precipitates in case A = 1 are higher compared to those of precipitates with A = 20 and 50, the volume fraction of precipitates in the underage regime (when A = 1) shows the lowest magnitude. This is due to the fact that longer precip-itates have larger volumes (and, therefore, lower number density). This also has an effect on the precipitate effective spacing, as seen in Figure5(d), where initially
0 30 60 90 120 150 0.1 1 10 100 1000 Aging Time (ks) Mean Length (nm) Modeling Experimental
Fig. 1—Length of precipitates in the alloy AA6061 aged at 463 K (190C). (Experimental data are from Ref. [23]; the error bar is approximately the size of symbols.).
0 0.01 0.02
0.1 1 10 100 1000
Aging Time/ks
Precipitate Volume Fraction
Modeling Experimental
Fig. 2—Prediction of the precipitate volume fraction in alloy AA6061 aged at 463 K (190C). (Experimental data are from Ref. [23]; the error bar is in the range of the size of symbols.).
Table I. Summary of Input Data Used for the Precipitate and Strengthening Model[3]
Input Parameter Value Input Parameter Value Cbaðwt pctÞ 63.0 c (J/m2 ) 0.26 D0(m 2 /s) 2.2 9 104 rtrans(nm) 3.5 Qd(kJ/mol) 130 M 2.6 to 3.1 Qs(kJ/mol) 45.35 b 0. 5 j0(#/m3s) 3.07 9 1036 b(m) 2.84 9 1010 Vm(m3/mol) 7.62 9 105 G(N/m2) 2.7 9 1010 kSi(MPa/ wt pct2/3) 66.3 kMg(MPa/ wt pct2/3) 29.0 kCu(MPa/ wt pct2/3) 46.4 A0(kJ/mol) 18
the mean precipitate distance is largest for precipitates of A = 1. The difference between the mean precipitate distance for precipitates of different aspect ratios decreases up to the peak age. At peak age, the mean precipitate distance values for different aspect ratios are similar.
Figure6 shows the effects of aspect ratio on the predicted yield strength of alloy AA6061. Clearly, the higher the aspect ratio, the faster the hardening kinetics. Also, the yield strength increases over the entire time range with increasing aspect ratio. This is more pro-nounced in the underage regime.
VI. DISCUSSION
The precipitation sequence in Al-Mg-Si alloys is very complex with a series of precipitates having different morphologies. b¢¢ phase, the most important precipitate in terms of strengthening, is a needlelike precipitate. Apart from that, other precipitates including pre-b¢¢, b¢, and Q¢ have elongated cylindrical morphologies. There-fore, a model that is able to predict the evolution of precipitate radius and length is more realistic than an
aging model based on the assumption of spherical morphology for precipitates. An age-hardening model was developed considering a cylindrical morphology for precipitates. The model is fitted for alloy AA6061 in such a way that it gives a correct prediction of the precipitate mean length at peak age at 463 K (190C). With the fitted value of aspect ratio equal to A = 10, the model yields a satisfactory prediction of precipitate mean length in the overage regime (Figure1). However, it overestimates the mean length in the underage regime. The reason for this overestimation is that not all precipitates formed in the beginning of aging have a needlelike morphology. Especially, a large fraction of GP zones have spherical morphology. In addition, the aspect ratio of b¢¢ phase changes during the aging process in such a way that it initially increases to a maximum value and thereafter decreases,[23] meaning that in the beginning of aging and in the very end of overage regime, the value A = 10 is too high for the aspect ratio of precipitates. For this reason, the yield strength in the beginning of aging and in the very end of overaging is predicted more accurately when A = 1 (Figure4). The aspect ratio of precipitates is influenced not only by aging time but also by aging temperature. At 0 50 100 150 200 250 300 0.01 1 100 10000 Aging time (ks) YS (MPa) Experimental Modeling 0 50 100 150 200 250 300 0.01 1 100 10000 Aging time (ks) YS (MPa) Experimental Modeling
(a)
443 K (170 °C)(b)
463 K (190 °C) 50 100 150 200 250 300 0.01 1 100 10000 Aging time (ks) YS (MPa) Experimental Modeling 0 50 100 150 200 250 300 0.01 1 100 10000 Aging time (ks) YS (MPa) Experimental Modeling(c)
478 K (205 °C)(d)
503 K (230 °C)Fig. 3—Prediction of yield strength of alloy AA6061 aged at different temperatures assuming an aspect ratio A = 10. (Experimental data are from Ref. [23]; the error bar is in the range of the size of symbols.).
higher aging temperatures, the aspect ratio becomes lower due to a smaller lengthening driving force and the readily occurring loss of coherency or semicoherency of peripheral plane.[23] This would result in a stronger deviation of predicted yield strength from the experi-mental values in the case of A = 10, when the aging temperature increases (Figure 3).
The shape of precipitates influences the strength evolution both through its effects on precipitation kinetics (size and density) and by altering the precipi-tate-dislocation interaction. In order to separate these two effects, we will assume that the number density and volume fraction of precipitates are not dependent on the chosen value of the aspect ratio; i.e., at any time, the volume fraction and number density of precipitates for the case A = 1 are supposed to be the same as for cases A =20 and 50. The only difference is the difference between the radius and length of precipitates, i.e., shape difference. Figure 7shows the evolution of rpptin alloy AA6061 for the different aspect ratio and similar number density and volume fraction. It is seen that in the absence of shape effect on precipitation kinetics, the
evolution of strength up to peak age is hardly dependent on aspect ratio. This implies that in the underage regime, the effects of precipitate thickness and number of precipitate/slip-plane intersections cancel. When precip-itates become more elongated, they also become thinner, and therefore weaker (due to lower F). However, as they also become longer, the number of intersections with slip planes increases and consequently l decreases. So, in the underage regime, when precipitates are shearable, the morphology by itself does not have a significant effect on the yield strength. The difference in the yield strength of the alloy at the underage regime, as seen in Figure6, is a kinetics effect (Figure5). Longer precip-itates, due to the faster growth kinetics in the underage regime (Figure5(c)), have more intersections with slip planes (smaller effective precipitate distances, as shown in Figure5(d)), making their contributions to yield strength higher. By entering the overaging regime, the strength becomes independent of F. It is also obvious that the aging kinetics does not have any influence on the yield strength, since both number density and volume fraction are the same, independent of aspect ratio (Figures5(a) and (c)). This implies that the only influencing parameter in the overage regime is the shape of the precipitates. The longer the precipitate, the higher the density of precipitate/slip-plane intersections, lead-ing to higher strength.
VII. CONCLUSIONS
A precipitation-strengthening model is developed for Al-Mg-Si alloys, assuming cylindrical morphology with a constant aspect ratio for precipitates. The model is applied to alloy AA6061, and the obtained results are compared with experimental data. Based on the obtained results, the following conclusions can be drawn.
1. Choosing a constant value for the aspect ratio based on peak age leads to an overestimation of the mean length of precipitates in the underage regime. This is due to the fact that the aspect ratio of pre-cipitates in the underage regime varies from values close to unity to its maximum value.
2. Needlelike precipitates with higher aspect ratio have greater strengthening effects due to yield strength compared to precipitates of lower aspect ratio. 3. For an aspect ratio A = 1, the model yields a
bet-ter prediction for the yield strength of the alloy in the underage regime. However, a good fit of peak strength requires A = 10.
4. The yield strength at the early stage of coarsening is predicted very well when A = 10, while when A =1, the model underestimates the yield strength in the beginning of coarsening.
5. By increasing the overaging time, the model with A =1 becomes more accurate, while the model with A = 10 gradually loses its accuracy.
6. Improving aging prediction would require the devel-opment of models capable of simulating noncon-stant aspect ratio and, therefore, a better knowledge of interfacial energy and interface reaction kinetics
0 50 100 150 200 250 300 0.01 1 100 10000 Aging time (ks) 0.01 1 100 10000 Aging time (ks) YS (MPa) Experimental Modeling-A=10 Modeling-A=1
(a)
0 50 100 150 200 250 300 YS (MPa) Experimental Modeling-A=10 Modeling-A=1(b)
Fig. 4—Prediction of yield strength of alloy AA6061 aged at (a) 463 K (190C) and (b) 478 K (205 C) using A = 1 and 10. (Experimental data is taken from Ref. [23].).
as a function of the precipitate-matrix interface character. Such information, however, is currently not available in sufficient detail to be incorporated in numerical models.
ACKNOWLEDGMENTS
This research was carried out as part of the innova-tion program of the Materials Innovainnova-tion Institute 0 2 4 6 8 10 12 0.01 0.1 1 10 100 1000 Aging time (ks)
No. Density (#/m3)*1e22
A=1 A=20 A=50 0 2 4 6 8 10 0.01 0.1 1 10 100 1000 Aging time (ks) Mean Radius (nm) A=1 A=20 A=50
(b)
(a)
0 0.005 0.01 0.015 0.02 0.025 0.01 0.1 1 10 100 1000 Aging time (ks)Precipitate Volume Fraction
A=1 A=20 A=50 0 50 100 150 200 250 300 0.01 0.1 1 10 100 1000 Aging time (ks) Precipitate Distance (nm) A=1 A=20 A=50
(d)
(c)
Fig. 5—Prediction of (a) precipitate number density, (b) mean radius, (c) volume fraction, and (d) precipitate mean distance for the AA6061 aged at 463 K (190C), assuming different values for aspect ratio.
0 50 100 150 200 250 300 0.0 1 0.1 110 100 1000 Aging time/ks YS (MPa) A=1 A=20 A=50
Fig. 6—Effects of aspect ratio on the predicted yield strength of alloy AA6061 aged at 463 K (190C).
0 50 100 150 200 250 0.0 1 0.1 110 100 1000 Aging time (ks)
Precipitate Strengthening (MPa)
A=1 A=20 A=50
Fig. 7—Precipitate strengthening in alloy AA6061 at 463 K (190C) for different values of aspect ratio, assuming similar number density and volume fraction.
M2i (formerly, the Netherlands Institute for Metals Research) on ‘‘Microstructural control during aging of extruded AA6xxx for forming and crash applications,’’ Project No. MC4.05213 (www.m2i.nl). The authors thank M2i for funding this project.
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NOMENCLATURE
A0 parameter related to the energy barrier for nucleation (J/mol)
A aspect ratio of precipitates (=h/r)
C0 initial concentration of Mg in the alloy (wt pct) Cm mean concentration of Mg in the matrix
(wt pct)
Cba concentration of Mg inside the precipitate
(wt pct) Cab
e equilibrium concentration of Mg in the matrix
(wt pct)
Cabr equilibrium interface concentration around the precipitates (wt pct)
Crm equilibrium interfacial concentration around the
precipitate of mean radius (wt pct) D bulk diffusion coefficient of Mg (m2/s) F interaction force between dislocations and
precipitates (N)
f volume fraction of precipitates
fPA volume fraction of precipitates at peak age h half-length of precipitate (nm)
J0 pre-exponential term in the nucleation equation J nucleation rate (#/m3s)
l effective mean interprecipitate distance (nm) M Taylor factor
NPA precipitate number density at peak age (#/m3) Qd activation energy for bulk diffusion of Mg
(J/mol)
QS apparent solvus boundary enthalpy (J/mol) r radius of precipitate (nm)
rcr critical nucleation radius (nm) rm mean radius of precipitate (nm) rPA mean radius at peak age (nm) tPA time to peak age (s)
rtrans shearable to nonshearable transition radius (nm)
Vm molar volume of precipitate (m 3
/mol) c interfacial energy (J/m2)
X supersaturation during coarsening ri yield strength of pure aluminum (MPa) rppt precipitation strengthening term (MPa) rss solid solution strengthening term (MPa)
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