# Modelling of Creep in Alloys Strengthened by Rod-Shaped Particles: Al-Cu-Mg Age-Hardenable Alloys

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{2}CuMg), while Boag et al [16] found a multitude of phases of different chemistry, including S and θ (Al

_{2}Cu) phases and other particles containing Cu, Fe, Mn, Si and Mg. These studies specifically considered the corrosion properties of the alloy and were thus mainly focused on coarse particles. By contrast, creep response is dramatically influenced by the presence of nano-sized precipitates, while it is only marginally affected by coarse particles, which are mostly located on grain boundaries. Ageing at low temperature results in the formation of GPB zones, S’-metastable phase and, only in a later stage, of the stable S-phase [17]. In this context, the selection of a high temperature for creep testing in Reference [14] was focused on easing the transition of the intermediate phases to their equilibrium S-phase, possibly even during the heating stage of the experiments, before loading. With the same goal in sight, additional variable load experiments were carried out. In these tests, the initial stress was maintained until in proximity or well within the minimum creep rate range (as usual under constant load condition, the steady state was replaced by a more or less extended range during which the strain rate reached its minimum value) and then abruptly increased. Figure 1a plots the minimum creep rate dependence on the applied stress of the AA2024-T3 alloy at 315 °C in Reference [14].

_{0L}and Q

_{L}are, respectively, the pre-exponential factor and the activation energy in the Arrhenius equation describing the temperature dependence of the vacancy diffusion coefficient. The strengthening term σ

_{p}in Equation (1) (frequently identified under the name of threshold stress) represents the effect of the interaction between particles and dislocations. It cannot be directly measured, but can be estimated by a best fitting procedure of experimental creep data, or, from the yield stress values. If the particle strengthening term is properly calculated, when the minimum creep rate is plotted as a function of the effective stress (σ

_{e}= σ − σ

_{p}), the experimental data align on a straight line of slope close to 4–5, as in pure Al (Figure 1b). Details about the calculation of the particle strengthening term are given in Reference [14], but it must be mentioned here that in the quoted work, it was assumed that fine precipitates consisted of nearly equiaxed particles as observed in References [7,8,9], which could not be actually the case in a material tested in T3 state. An analysis of Figure 1b, on the other hand, clearly shows that the strain rate in the alloy is orders of magnitude lower than the one measured in the pure metal [18] or in an Al-0.5%Mg alloy under an applied stress equivalent to the effective stress acting on AA2024. The most widely invoked explanation for this behavior [19] is that alloying severely reduces the stacking fault energy of the alloy [21,22]. Since the parameter A is traditionally considered to be proportional to the third-power of the stacking fault energy [20], a large reduction of this latter parameter [21,22] could indeed result in a decrease of one or even two orders of magnitude in the strain rate for a given effective stress. Yet, recent investigations present a quite different picture of the situation, since alloying actually only slightly reduces stacking fault energy, and this reduction is by far too modest to account for the dramatic reduction in the parameter A [23]. Thus, by Equation (1), one can still obtain an excellent description of the experimental data, but this description largely remains of a phenomenological nature, and any possible correlation with the microstructural evolution of the material is still based on weak bases.

## 2. Materials and Methods

## 3. The Model

#### 3.1. Equations for Dislocation Creep in Al: Strengthening Terms

_{ρ}= αmGbρ

^{1/2}is the dislocation hardening term. The term σ

_{i}represents the stress required to move a dislocation in the absence of other dislocations, and incorporates the effect of grain size, while α is a constant (in the following, α = 0.3 will be considered). The internal stress in pure Al σ

_{i}was calculated by the equation

_{Al}is the creep stress needed to obtain a given strain rate in coarse grained annealed pure Al under a given strain rate, d

_{g}is the grain size, k

_{hp}is the Hall-Petch constant and A

_{y}= 0.0042 [18]. Equation (3) was here rewritten as

_{p}and σ

_{ss}are the particle and solid solution strengthening terms respectively (see below). The stress term in the first part of the equation was depurated from the effect of the grain size strengthening to avoid considering the same phenomenon twice. Once the particle and solid solution strengthening terms are known, Equation (4) can be simply handled, as soon as the Hall-Petch constant is determined (see Appendix A for details on this calculation).

_{ss}thus represents the stress required for dislocations to move through the viscous drag due to solute atoms (Cu and Mg in AA2024). The equation for drag stress was given in Reference [26], in the form

_{d}is the dislocation velocity, c is Mg atomic concentration, D

_{a}= D

_{0Mg}·exp(−Q

_{a}/RT) is the diffusivity of the solid solution elements in Al. The term I(z

_{0}) can be calculated by numerical integration of

_{m}is the mobile dislocation density, a fraction of the free dislocations. Equation (8) has an obvious important implication. In the AA2024 T3 alloy, for a given applied stress (or even effective stress), the strain rate is at least two orders of magnitude lower than the one observed in an Al-Mg alloy with a comparable content of elements in solid solution (roughly estimated in 0.5% at). Thus, the drag stress is correspondingly lower, that is, quite negligible in comparison with the other terms in Equation (2). On this basis, the assumption σ

_{ss}≅ 0 will be used in the following.

_{p}represents the strengthening contribution due to the interaction between fine particles and dislocations.

#### 3.2. Nature of Particle Strengthening in 2024 Alloy

_{ss}→ Cu:Mg co-clusters/GPB zones → GPB2/S’’ phase → S’/S phase

_{ss}is the super-saturated solid solution, GPB are the Guiner-Preston-Bagaryatsky zones and the S’’ (GPB2) is an intermediate phase preceding the formation of the semicoherent S’ phase [29]. S’ and S phase have a very similar composition and crystal structure, but S’’ phase is thought to have limited influence on the strength of AA2024. Thus, following Khan et al. [29], it is possible to apply a simplified approach in which the precipitation sequence becomes

_{ss}→ Cu:Mg co-clusters → S phase

#### 3.3. Models for Particle-Dislocation Interaction

_{p}, oriented in {100} directions [29], the Orowan stress can be calculated as

_{Or}, dislocations can overcome particles by climb (see Reference [20] and References [30,31] for detailed reviews of the different theories on this subject). Figure 2 shows a schematic depiction of a dislocation negotiating a particle array. In Figure 2a, the dislocation climbs over equiaxed particles of size d

_{s}. The climb resistance of the particles, a parameter which describes the rate of increase of the line length as the dislocation climbs over the precipitate, for general climb of spherical particles, can be expressed as [30,31]

_{p}in the following) then assumes the form [30]

_{p}= 0.023 for f = 1%). This is in marked contrast with the older theories of local climb, which, although unlikely to occur due to the sharp bending of the dislocation at the particle interface, predicted a α

_{p}one order of magnitude higher [30]. This higher α

_{p}value, on the other hand, is much closer to the values calculated from experiments.

_{d}) can then expressed by an equation in the form

_{d}is the relaxation factor for the attractive dislocation/particle interaction [34,35]. This detachment stress could reasonably be assumed to coincide with the threshold stress estimated in several alloys, which in general is much higher than that the term associated to general climb, and much closer to the local climb one. In fact, the attractive interaction between particle and dislocation stabilizes the sharp curvatures of the dislocation line that are more typical of local climb model. In any case, we again observe a threshold stress proportional to the Orowan stress.

_{r}= 1/3 d

_{s}, with the same volume and interparticle spacing on the slip plane of the equiaxed precipitates in Figure 2a. The rod axis forms an angle ϕ with the slip plane. With the above-mentioned size of the transversal section, the rods have a length equivalent to 12 d

_{s}. Creep exposure of the 2024 T3 alloy actually results in the formation of even longer rods (Figure 3). At a glance, climb of a dislocation over these long rods requires a much higher increase of the line length than over an equiaxed precipitate of the same volume. Although this model did not predict a significant effect of the particle shape, at least for moderately elongated precipitates, Rösler observed that climb becomes more “local” as the dislocation climbs the rod. Thus, a dislocation whose slip plane cuts the rods halfway along its length will require a higher stress to overcome the obstacle, than another dislocation whose slip plane is closer to the rod tip. If the traditional view, in which σ

_{p}is proportional to σ

_{Or}, is maintained, one should thus reasonably, albeit qualitatively, conclude that, in the presence of strongly elongated precipitates, α

_{p}should be substantially higher than the value predicted for general climb of spherical particles, and much closer to the value for local climb.

_{p}value that is either proportional to the applied stress or to the Orowan stress. Yet, the proportionality constants, as they result from modelling, are temperature-independent, which is in contrast with experimental evidence that rather points toward a marked dependence on T of the particle-strengthening term. Thus, the models mentioned above require a fitting procedure of the experimental data to determine either α

_{p}or k

_{d}at a given temperature and for a given alloy. This fact implies that none of the models here presented have the nature of a model that does not requires any data fitting.

#### 3.4. Rate Equation and Temperature Dependence

_{l}is the dislocation line tension (τ

_{l}= 0.5Gb

^{2}), M is the dislocation mobility and L is the dislocation mean free path, i.e., the distance travelled by a dislocation before it undergoes a reaction. The first term on the right-hand side of Equation (18) represents the strain hardening effect due to dislocation multiplication, which is more rapid when L and, consequently, C

_{L}assume low values and/or the dislocation density is high. The second and third terms on the right-hand side of Equation (18) describe the effect of recovery. At high temperature, the last term in Equation (18) largely predominates on the second term. Since the main emphasis in this study is to describe what happens at high temperatures, Equation (18) can be simplified to

_{max}is the maximum strength of the alloy, tentatively quantified in pure Al as in Reference [18] as 1.5 times the ultimate tensile strength (R

_{UTS}) of the alloy at room temperature.

#### 3.5. Solid Solution Strengthening Effect on Temperature Dependence of the Strain Rate

## 4. Modelling of the Minimum Creep Rate Dependence on Applied Stress in AA2024 T6-T3 Alloy

#### 4.1. The Role of Particle Size and Distribution

_{0p}, after an exposure of duration t at a given temperature T, has the form

_{g}is a constant, and Q

_{a}is the activation energy for the diffusion of the constituent elements in the matrix. The experimental data in Reference [38] can be easily used to estimate k

_{g}at 190 °C. Then, taking Q

_{a}= 134 kJ/mol, which is the activation energy for the diffusion of Cu in Al [41], Equation (25) can be directly used to estimate the size of the rod diameter after prolonged exposure to high temperature. A long-time exposure would result in S-particle coarsening, and, if the volume fraction of intragranular precipitates does not change, in pronounced softening, i.e., in a reduction of the yield strength.

_{0}= 3.4 nm [38].

#### 4.2. Quantification of Particle Strengthening Effect

_{p}= 1). Taking ${\mathsf{\sigma}}_{\mathrm{y}}^{a}$ = 75 MPa (yield strength of Alclad AA2024-O from [36]), the curves in Figure 4 were obtained with f = 0.9 and 0.6% at 260 and 315 °C respectively. The slight overestimation of the strength after very long exposure is an effect of the corresponding overestimation of the strength of the matrix, since even a moderate grain growth should result in a ${\mathsf{\sigma}}_{\mathrm{y}}^{a}$ somewhat lower than 75 MPa. The agreement of the model curves with the data for the T6 state is remarkable, albeit one could wonder if the computed volume fraction of intragranular precipitates is realistic, or rather grossly underestimated. As a matter of fact, the volume fraction of the S-phase could be as high as 4% at 260 °C [37] and be accompanied by substantial amounts of other intermetallic phases. Experimental evidence indeed demonstrates that holding at high temperature results in a strong increase in the number of coarse particles at grain boundaries. Figure 3a shows chains of coarse precipitates, formed during creep exposure, which clearly delineate the grain boundaries. These coarse particles are partly equilibrium S-phase precipitates [15], which in this form do not effectively contribute to the strength. One can thus conclude that this approach gives a reliable description of the softening phenomena associated with intragranular-particle coarsening.

_{p}= 1 results in strongly overestimated values of the yielding (again the strength of the matrix was assimilated to that of an Alclad AA2024-O alloy tested at the same temperatures after similar durations of high-T exposure [36]). The curves in Figure 5 were then obtained with α

_{p}= 0.7 and 0.55 with f = 0.9 and 0.6% at 260 °C and 315 °C respectively. These values of α

_{p}are much higher than the one predicted by the local climb model of equiaxed particles, in agreement with the qualitative conclusions drawn in Section 3.3.

#### 4.3. Effect of Stress on Particle Evolution

_{σ}is a constant. Figure 6 plots the variation of K

_{g}with stress at 190 °C as well as the curve that describes Equation (29). The correlation is excellent (correlation coefficient 0.998). Thus, Equation (29) could be used to estimate the particle size at the time at which the minimum in strain rate is measured.

#### 4.4. Description of the AA2024 T3 Constant and Variable Load Experiments

_{m}).

_{m}, assuming, for example, d

_{0}≅ 3.4 nm as in Reference [38]. To take into account the effect of heating and soaking times, a permanence of 1.5 h at the testing temperature before loading was considered, giving an estimate of the diameter of the rods at the beginning of the test of 4 and 9 nm at 250 and 315 °C respectively. The volume fraction of the intragranular precipitates was assumed to be f = 0.9 and 0.6% and α

_{p}= 0.7 and 0.55, respectively, as in the previous sections. This quantity is obviously only a mere fraction of the total amount of particles, since, as mentioned above, creep exposure resulted in a massive precipitation of Al-Cu-Mg or Al-Cu phases on grain boundaries.

_{max}, here tentatively quantified, for each single test in correspondence to the minimum in creep rate, as

_{p}) model curves, where P represents a given combination of the size and volume fraction of the intragranular precipitates, that is, a given value of the particle-strengthening contribution. The Figure clearly shows that each single datum for the CLE lies on a different iso-P curve. By contrast, the VLE, for which the time of exposure is roughly coincident with the duration of the first stage of the test, carried out under the lower stress, actually lies on the same iso-P curve (broken lines in Figure 7). This fact easily explains the difference in strain rate between the CLE and VLE. In addition, the model also gives a reason for the lower creep rate measured in the AA2024 T3 alloy, for a given effective stress, when compared with pure Al and Al-Mg alloys (the effect shown in Figure 1b). The magnitude of the minimum strain rate in particle-strengthened alloys is influenced by two different parameters, namely the particle strengthening term σ

_{p}, which was already considered in the phenomenological Equation (1), and the dislocation mean free path L, which does not appear in the traditional phenomenological models. This latter microstructural features is substantially coincident with the interparticle spacing in the AA2014-T3 alloy, which, in turn, is orders of magnitude shorter than the dislocation mean free path in single phase metals and alloys.

- The model does not require any fitting of the experimental creep data (none of the creep data were used to fit the equations, once the model was properly tuned by other microstructural results and tensile testing, required to provide an estimate of α
_{p}). - The model is self-consistent, since it easily explains the lower creep rate observed when comparing the alloy tested under a given effective stress and the pure metal under an equivalent applied stress.
- Although the model considers only rod-shaped particles, the observed presence of equiaxed precipitates does not seem to significantly affect the accuracy of the description.
- The main drawback of the model here presented, is that the quantification of the particle-strengthening term still needed to be evaluated from experimental data (the yield stress). Thus, the determination of the threshold stress originated by particle-dislocation interaction remains phenomenological in nature.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{hp}, which can be considered a measure of the effectiveness of grain boundaries in increasing material strength, strongly decreases with increasing temperature [43] and decreasing strain rate [44]. This temperature dependence of the Hall-Petch “constant” can be described by the phenomenological equation [44]

_{hp}is a constant and

_{0}does not change much in fcc metals such as Cu or Pb (β

_{0}≅ 0.003 K

^{−1}), while β

_{1}ranges between 2 × 10

^{−4}(for Cu) and 2.4 × 10

^{−3}K

^{−1}(for Pb) [44]. The constants B

_{hp}and β

_{1}were therefore calculated for Al by considering that, at 350 °C and 6.67 × 10

^{−4}s

^{−1}, k

_{hp}= 0.65 MPa·mm

^{−1/2}[43], while, at room temperature and under the usual conditions for tensile testing (strain rates ranging between 1 × 10

^{−4}s

^{−1}and 1 × 10

^{−3}s

^{−1}), k

_{hp}ranged between 2.0–2.5 [45] and 2.8 MPa·mm

^{−1/2}[42]. This calculation gave B

_{hp}= 86.5 MPa

^{2}·mm

^{−1}and β

_{1}= 8 × 10

^{−4}K

^{−1}(Figure A1).

_{hp}at room temperature given in [42], and the curves obtained by Eqns. (A1) and (A2). The Figure clearly shows that most experimental data are close to the model curves for 10

^{−4}and 10

^{−3}s

^{−1}, which, as mentioned above, roughly correspond to the usual strain rates for constant strain rate testing. The k

_{hp}values given by Equation (A1) and (A2) could then be used to estimate the grain-boundary strengthening effect in Equation (3). Yet, since the value of internal stress, which is required to estimate the minimum strain rate, contains the grain-size strengthening term that depends on the strain rate, an iterative process was required to properly obtain the model curves presented in Figure 7 and Figure 8.

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**Figure 1.**(

**a**) Experimental values of the minimum creep rate for tests on AA2024 T3 at 315 °C [14]. CLE stands for constant load experiments. VLE stands for variable load experiments (tests during which an initial value of the applied stress, 15 or 25 MPa, was increased at the early beginning or in proximity of the minimum creep rate range). (

**b**) Strain rate as a function of the effective stress σ-σ

_{0}for the data (

**a**) compared with the response of pure Al [18] and Al-0.5Mg [19].

**Figure 2.**Schematic depiction of a dislocation negotiating a particle array. In (

**a**) the dislocation climbs over equiaxed particles of size d

_{s}. In (

**b**), the dislocation climbs over rods, of diameter d

_{r}= 1/3 d

_{s}, with the same volume and interparticle spacing on the slip plane of the equiaxed precipitates in (a). The rod axis forms an angle ϕ with the slip plane.

**Figure 3.**Microstructure of the sample tested under 25 MPa at 315 °C up to rupture. The distribution of precipitates was quite inhomogeneous in the various grains. The grain on the right in (

**a**), in particular, was one of those with a higher volume fraction of particles. Semi-continuous chains of coarse particles precipitated during creep highlight grain boundaries. In (

**b**), elongated particles can be easily observed, alongside small circular precipitates (most easily rods observed in a transversal section) and slightly coarser equiaxed particles.

**Figure 4.**Calculated variation of the yield stress for AA2024 T6; the symbols represent the experimental data from Reference [36] for the T3 and T6 states, tested at room temperature. Initial rod diameter 3.4 nm, volume fraction of the intragranular S-phase 0.9% at 250 °C and 0.6% at 315 °C respectively. The figure also reports the yield strength at room temperature for the unexposed T3 and T6 states [36].

**Figure 5.**Calculated variation of the yield stress for AA2024 T6 at 260 and 315 °C, after different durations of annealing at these temperatures; the symbols represent the experimental data from Reference [36] for the T3 and T6 states Initial rod diameter 3.4 nm, volume fraction of the intragranular S-phase 0.9% at 250°C and 0.6% at 315 °C respectively. The figure also reports the yield strength at room temperature for the T3 and T6 states [36].

**Figure 6.**Coarsening constant at 190 °C as a function of applied stress (experimental data from Reference [38]).

**Figure 7.**Minimum creep rate dependence on applied stress. The solid curves connect the modelled values of the strain rate for each single CSE; the broken curves connect the strain rate values modelled by assuming that the time of exposure does not change appreciably for the tests carried out under different loads. The latter condition is comparable to the conditions experienced during the VLE. The grain size used in estimating the internal stress was 50 μm.

**Figure 8.**Model curves representing the strain rate variation with stress for microstructures that experienced different durations of exposure under a given load; (

**a**) 250 °C; (

**b**) 315 °C. Each curve is thus relative to a microstructure equivalent to that observed at t

_{m}under a given applied stress (iso-P curves). The open data in (

**b**) are from the VLE (f = 0.5 and 0.22% at 250 and 315 °C respectively).

CLE | |||||

T (°C) | σ (MPa) | t_{m} (h) | ${\dot{\mathsf{\epsilon}}}_{\mathbf{m}}$(s^{−1}) | ||

250 | 60 | 200 | 9.4 × 10^{−9} | ||

100 | 10 | 7.5 × 10^{−8} | |||

120 | 7 | 3.3 × 10^{−7} | |||

155 | 1 | 6.0 × 10^{−7} | |||

175 | 0.35 | 1.5 × 10^{−6} | |||

200 | 0.05 | 5.0 × 10^{−6} | |||

315 | 15* | 337 | 4.0 × 10^{−9} | ||

25 | 50 | 5.0 × 10^{−8} | |||

40 | 8 | 5.0 × 10^{−7} | |||

60 | 0.6 | 1.6 × 10^{−6} | |||

VLE | |||||

T (°C) | σ (First Loading) (MPa) | σ (Final Loading) (MPa) | t Under First Load (h) | t_{m} Second Load (h) | |

315 | 25 | 50 | 15 | 1 | |

25 | 60 | 15 | 0.08 | ||

25 | 65 | 15 | 0.03 | ||

15* | 35 | 337 | 1 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Paoletti, C.; Regev, M.; Spigarelli, S.
Modelling of Creep in Alloys Strengthened by Rod-Shaped Particles: Al-Cu-Mg Age-Hardenable Alloys. *Metals* **2018**, *8*, 930.
https://doi.org/10.3390/met8110930

**AMA Style**

Paoletti C, Regev M, Spigarelli S.
Modelling of Creep in Alloys Strengthened by Rod-Shaped Particles: Al-Cu-Mg Age-Hardenable Alloys. *Metals*. 2018; 8(11):930.
https://doi.org/10.3390/met8110930

**Chicago/Turabian Style**

Paoletti, Chiara, Michael Regev, and Stefano Spigarelli.
2018. "Modelling of Creep in Alloys Strengthened by Rod-Shaped Particles: Al-Cu-Mg Age-Hardenable Alloys" *Metals* 8, no. 11: 930.
https://doi.org/10.3390/met8110930