Study on Mechanical Relaxations of 7075 (Al–Zn–Mg) and 2024 (Al–Cu–Mg) Alloys by Application of the Time-Temperature Superposition Principle

Research Article Study on Mechanical Relaxations of 7075 (Al–Zn–Mg) and 2024 (Al–Cu–Mg) Alloys by Application of the Time-Temperature Superposition Principle Jose I. Rojas,1,2 Jorge Nicolás,2and Daniel Crespo2,3 1Department of Physics, Division of Aerospace Engineering, Universitat Polit` ecnica de Catalunya, Barcelona, Spain 2Escola d’Enginyeria de Telecomunicaci´ o i Aeroespacial de Castelldefels, Universitat Polit` ecnica de Catalunya, c/Esteve Terradas 7, 08860 Castelldefels, Spain 3Department of Physics, Universitat Polit` ecnica de Catalunya, Barcelona, Spain Correspondence should be addressed to Jose I. Rojas; josep.ignasi.r[email protected] Received 1 January 2017; Accepted 13 March 2017; Published 5 April 2017 Academic Editor: Ming-Xing Zhang Copyright © 2017 Jose I. Rojas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The viscoelastic response of commercial Al–Zn–Mg and Al–Cu–Mg alloys was measured with a dynamic-mechanical analyzer (DMA) as a function of the temperature (from 30 to 425∘C) and the loading frequency (from 0.01 to 150 Hz). The time-temperature superposition (TTS) principle has proven to be useful in studying mechanical relaxations and obtaining master curves for amorphous materials. In this work, the TTS principle is applied to the measured viscoelastic data (i.e., the storage and loss moduli) to obtain the corresponding master curves and to analyze the mechanical relaxations responsible for the viscoelastic behavior of the studied alloys. For the storage modulus it was possible to identify a master curve for a low-temperature region (from room temperature to 150∘C) and, for the storage and loss moduli, another master curve for a high-temperature region (from 320 to 375∘C). These temperature regions are coincidental with the stable intervals where no phase transformations occur. The different temperature dependencies of the shift factors for the identified master curves, manifested by different values of the activation energy in the Arrhenius expressions for the shift factor, are due to the occurrence of microstructural changes and variations in the relaxation mechanisms between the mentioned temperature regions. 1. Introduction Comprehensive research efforts have been devoted to characterizing the mechanical properties of metals. However, their viscoelastic response, a manifestation of internal friction phenomena under dynamic loading, has received much less attention. The comprehension of the underlying physics of viscoelasticity is of high interest because it enables a deeper understanding of important properties like mechanical damping and yielding [1] and because metals are subjected to dynamic loads in most of their structural applications. Indeed, fatigue is a consequence of microstructural changes induced under dynamic loading, and these phenomena must have an effect on the viscoelastic response [2], as it is also intimately linked to the microstructure [1]. Thus, the study of the viscoelastic behavior offers a different approach for analyzing the microstructure and the fatigue behavior of a material. The former has been shown in metallic glasses, for which structural relaxations, the glass transition, and crystallization processes have been studied with dynamicmechanical analyzers (DMA) [3]. Also, the time-temperature superposition (TTS) principle has proven useful in studying themechanicalrelaxationsandinobtainingmastercurvesfor amorphous materials [4, 5]. In this work, a similar approach is applied to commercial 7075 (Al–Zn–Mg) and 2024 (Al–Cu–Mg) alloys. Particularly, the TTS principle is applied to experimental viscoelastic data, that is, the storage modulus 𝐸󸀠and the loss modulus 𝐸󸀠󸀠. These data were measured with a DMA as a function of the temperature (ranging from 30 to 425∘C) and the loading frequency 𝑓(ranging from 0.01 to 150 Hz) [6–8]. The objectives of this research are first to obtain the corresponding master Hindawi Advances in Materials Science and Engineering Volume 2017, Article ID 2602953, 12 pages https://doi.org/10.1155/2017/2602953 2 Advances in Materials Science and Engineering curvesandsecondtoanalyzethemechanicalrelaxations responsible for the viscoelastic behavior of Al–Zn–Mg and Al–Cu–Mg alloys. These alloys feature excellent mechanical properties after proper age hardening and for this reason they are used in a number of industrial applications, especially in the aerospace sector and transport industry. The ageing path, phase transformations, and precipitate types for these alloys have been extensively studied. A summary of this research as reported in the literature can be found in [6–8]. Pure aluminum and aluminum alloys show varied mechanical relaxation phenomena. Namely, Al–Zn alloys may exhibit high internal friction due to large-scale discontinuous precipitation. In particular, the proposed relaxation mechanisms consist of shear stress relaxations across the network of matrix-precipitate interfaces [1]. In contrast with Al–Zn alloys, Al–Cu alloys usually present these phenomena: (i) A relaxation effect associated with atom groupings within individual clusters [9]. (ii) The Zener peak observed at 173–175∘Cat1Hzin solution treated and quenched samples [1]. This peak declines as Guinier-Preston (GP) II zones precipitate, while it is stable provided that excess vacancies are removed previously (e.g., with a short annealing), since GP-II zones form only if large amounts of quenched-in vacancies are present. Hence, this peak was associated with the alloy with all the Cu in solution. The activation energy 𝐸𝐴of this relaxation is 1.32±0.08eV/atom (but may change with composition), and the frequency factor 𝑘0is 1015.6±1 s–1. (iii) A broad relaxation peak at 135∘Cassociatedwith𝜃󸀠 phase[1].Thenucleationandgrowthof𝜃󸀠particles modify the internal friction behavior in two ways: (1) It causes the appearance and growth of a peak slightly below the Zener peak temperature. The magnitude of the former peak is approximately proportional to 𝜃󸀠phase fraction. Thus, it declines as 𝜃󸀠phase transforms into 𝜃phase and vanishes completely when no 𝜃󸀠phase is present. This is why it is named 𝜃󸀠peak. For this peak, the width is about 3 times larger than that for a process having a single relaxation time, 𝐸𝐴 is 0.95±0.05eV/atom, and the relaxation time 𝜏0is 10–12.5±0.8 s. (2) It causes a further decline of the Zener peak mentioned above (ultimately, this peak vanishes when 𝜃󸀠phase formation is complete). (iv) A relaxation peak associated with 𝜃phase [10]. This relaxation has a maximum at 0.1 Hz and disappears after annealing at 550∘C.Thisiswhyitisassociated with 𝜃phase. This relaxation is not thermally activated [11]. Moreover, Al–Zn–Mg and Al–Cu–Mg alloys and pure aluminum may exhibit relaxations associated with dislocations and grain boundaries [12, 13]. Namely, dislocation motion explains some internal friction peaks associated with semicoherent precipitates [14] and the Bordoni peak, which has been extensively studied in cold-worked pure aluminum [1]. Finally, polycrystalline aluminum presents a peak related to grain boundaries at about 300∘C at 1 Hz [1]. Some of these and other internal friction mechanisms suggested for aluminum and related metallic materials are explained in [15, 16], like a peak in Al–Mg alloys at 100–200∘Calsocontrolledbythe Zener mechanism. The dynamic properties of viscoelastic solids can be described using the complex compliance approach. In this case, for a solid showing a single relaxation process, the real and imaginary parts of the compliance, 𝐽1and 𝐽2,aregivenby the Debye equations [1]: 𝐽1(𝑇,Ω)=𝐽 𝑢+𝛿𝐽 1+(Ω𝜏)2 𝐽2(𝑇,Ω)=𝛿𝐽 Ω𝜏 1+(Ω𝜏)2,(1) where 𝐽𝑢(𝑇) is the unrelaxed compliance, 𝛿𝐽(𝑇) = 𝐽𝑟− 𝐽𝑢is the relaxation of the compliance, 𝐽𝑟(𝑇) is the relaxed compliance, Ω=2𝜋𝑓is the loading frequency, and 𝜏(𝑇)is the average mechanical relaxation time of the process/processes occurring at a given temperature. The complex Young’s modulus 𝐸=𝐸 󸀠+𝑖𝐸󸀠󸀠 is also commonly used to describe the tensile dynamic-mechanical behavior, where 𝐸󸀠(𝑇,Ω)=𝐸󸀠󸀠 𝐽1 𝐽2(2) 𝐸󸀠󸀠 (𝑇,Ω)=𝐽2 𝐽12+𝐽22.(3) In this case, 𝐸𝑢(𝑇) = 1/𝐽𝑢is the unrelaxed modulus (i.e., 𝐸 for Ω→∞), 𝐸𝑟(𝑇)=1/𝐽𝑟is the relaxed modulus (i.e., 𝐸for Ω→0), and 𝛿𝐸(𝑇)=𝐸𝑢−𝐸𝑟is the relaxation of the modulus. Indeed,relaxationprocessesareusuallycharacterizedfrom the analysis of 𝐸󸀠󸀠.Particularly,𝐸󸀠󸀠 peaks (internal friction peaks) are defined by four main characteristics: the intensity (𝛿𝐸) of the peak, its bluntness (or broadness), and the power laws defining the low- and high-frequency tails [17]. The empirical Havriliak–Negami (HN), Cole–Cole (CC), and Cole–Davidson (CD) functions are used to describe therelaxationresponseinthefrequencydomain[18].The HN model is 𝐸(𝑇,Ω) = 𝐸𝑢[1−(1+(𝑖Ω𝜏)𝛼)−𝛾],where the exponents 𝛼and 𝛾describe the broadness of the peak (associated with the distribution of relaxation times) and the asymmetry of the peak, respectively [19]. The CC and CD functions are derived from the HN function; namely, the CC model corresponds to the HN model with 𝛾=1and 0<𝛼<1, while the CD model is the HN model with 𝛼=1 and 0<𝛾<1. The CD function can fit asymmetric peaks, with the shape being dictated by 𝛾, while the CC function describes symmetric peaks, with the broadness being dictated by 𝛼. The latter parameter, which often increases with temperature, can give information about how distributed are the relaxation times. Classical anelastic relaxations in crystalline metals are restricted to a small volume, for example, defects Advances in Materials Science and Engineering 3 Table 1: Mechanical properties of the as-received, commercial aluminum alloys (AA) 7075-T6 and 2024-T3. Aluminum alloy Yield stress UTS % area reduction Brinell hardness AA 7075-T6 502 MPa 583 MPa 12% HB 161 AA 2024-T3 377 MPa 485 MPa 15% HB 123 Table 2: Chemical compositions of the as-received, commercial aluminum alloys (AA) 7075-T6 and 2024-T3. Aluminum alloy Units Si Fe Cu Mn Mg Zn Ti Cr Al AA 7075-T6 wt.% 0.06 0.15 1.50 0.01 2.58 6.00 0.05 0.19 89.46 at.% 0.06 0.08 0.67 0.01 2.99 2.59 0.03 0.10 93.48 AA 2024-T3 wt.% 0.18 0.28 4.46 0.64 1.35 0.04 0.05 0.01 92.98 at.% 0.18 0.14 1.95 0.32 1.54 0.02 0.03 0.01 95.81 like dislocations and grain boundaries [1]. Accordingly, the relaxations have a much smaller magnitude and the peaks are veryclosetoaDebyerelaxation,with𝛼≈1and 𝛾≈1[17]. The distribution of relaxation times can also be related to a spectrum of activation energies. The time-temperature relaxation spectrum 𝐸󸀠󸀠(Ω,𝑇)canbemodeledbycombining a frequency response function (e.g., HN, CD, CC, or other models) with a temperature dependence of the main relaxation time 𝜏(𝑇), in what is called the TTS principle [4, 5]. In this approach, the shape of 𝐸󸀠󸀠(Ω,𝑇)describes the influence of the relaxation time spectrum, that is, the deviation from a Debye process [17]. For the HN, CD, and CC models, the shape is determined by 𝛼and/or 𝛾, which are obtained from fitting of the model to the experimental 𝐸󸀠󸀠(Ω,𝑇)data. Regarding the temperature dependence of the average or main relaxation time of the process 𝜏(𝑇),itisoftenvalidto assume an Arrhenius-type temperature dependence for the relaxation rate 1/𝜏(𝑇) (e.g., when the rate-limiting step of the considered relaxation is that of movement over an energy barrier) [20]: 1 𝜏=𝑘0exp (−𝐸𝐴 𝑘𝐵𝑇),(4) where 𝑘0=1/𝜏 0is the frequency factor (or preexponential coefficient), 𝐸𝐴istheactivationenergyassociatedwiththe relaxation, and 𝑘𝐵is the Boltzmann constant. This is why the dynamic response functions may be treated as if they were functions of temperature (while frequency is constant) instead of functions of frequency (while temperature, and thus 𝜏(𝑇), are constant) [1]. This is the basis of the TTS principle, which establishes the relationship between temperature and time on the relaxation and deformation behavior of viscoelastic materials under constant stress/strain or dynamic loading [4, 5, 21]. For instance, the effect of increasing temperature on the viscoelastic response is similar to reducing frequency. Accordingly, master curves can be obtained by superimposing the modulus spectra (𝐸󸀠and 𝐸󸀠󸀠) at different temperatures. This is achieved by shifting horizontally (i.e., along the frequency axis) the isothermal curves of 𝐸󸀠and 𝐸󸀠󸀠 using a shift factor 𝑎𝑇. This factor defines the ratio of relaxation time at a given temperature 𝑇to its value at an arbitrary chosen reference temperature 𝑇0andreflectsthesegmentaloratomic mobility associated with configurational rearrangements in materials [4, 22]. The temperature dependence of 𝑎𝑇has been successfully described by the Williams–Landel–Ferry equation (e.g., for amorphous polymers) [22], the equivalent Vogel–Fulcher–Tammann–Hesse equation, or an Arrheniustype expression [5]: ln (𝑎𝑇)=𝐸𝐴 𝑘𝐵(1 𝑇−1 𝑇0). (5) Amastercurvecanbeusedtotransformmaterialproperties from the frequency domain to the temperature domain, and vice versa [23, 24]. In principle, it allows extrapolating the viscoelastic behavior to any temperature or frequency. However, a master curve associated with a given 𝑎𝑇may only be valid within a limited temperature range, because 𝑎𝑇depends also on ageing (i.e., on the microstructure) and on the relaxation mechanisms of the viscoelastic spectrum. Namely, 𝑎𝑇may exhibit different temperature dependencies from a temperature interval to another (and thus there will be a different master curve in each of these intervals), if phase transformations occur in between or different relaxation mechanisms intervene. This can be used to detect the occurrence of transformations or changes in the dominant relaxation mechanisms based on DMA data, while 𝐸𝐴in (5) gives useful information on the mechanical relaxation processes [5]. 2. Materials and Methods The specimens were prepared from sheet of commercial aluminum alloys (AA) 7075-T6 and 2024-T3. The T6 temper consists in solution heat-treatment at 480∘C for 1 h, followed by rapid water quenching to room temperature (RT) and artificial ageing at 120∘C for 24 h. The T3 temper consists in solution heat-treatment at 480∘C for 1 h, followed by rapid water quenching to RT, cold-working, and natural ageing. Tables 1 and 2 show, respectively, the mechanical properties and compositions in wt.% and at.% for AA 7075-T6 and 2024-T3, as provided by the supplier (Alu-Stock, SA). A first set of specimens was machine cut from the as-received alloys to rectangular plates 60 mm long, 10 mm wide, and 2 mm thick. A second set of identical plates was solution treated at 477∘Cfor 4 Advances in Materials Science and Engineering 50000 55000 60000 65000 70000 75000 0 0.5 1 1.5 2 log(f) Storage modulus E(MPa) 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C (a) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 log(f) Loss modulus E (MPa) 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C (b) Figure 1: Isothermal storage modulus 𝐸󸀠(a) and loss modulus 𝐸󸀠󸀠 (b) versus logarithm of the frequency 𝑓from DMA tests on AA 7075-T6 at frequencies ranging from 1 to 100 Hz and from 50 to 375∘C. 30 min to remove the T6 temper, rapidly quenched in water to RT and tested immediately after. A TA Instruments Q800 DMA was used to measure the viscoelastic response of the studied materials (i.e., 𝐸󸀠and 𝐸󸀠󸀠; more details can be found in [6–8]) in N2atmosphere. The 3-point bending clamp was used, with length between unmovable supports of 50 mm and preload force of 0.10 N, and the DMA was set to sequentially apply dynamic loading with frequencies ranging from 0.01 to 150 Hz, at temperatures from 30 to 425∘C in step increments of 5∘C. 3. Results and Discussion 3.1. Experimental Results for 𝐸󸀠and 𝐸󸀠󸀠.Figures 1(a) and 1(b) show, respectively, isothermal curves of 𝐸󸀠and 𝐸󸀠󸀠 versus frequency obtained from DMA experiments for AA 7075-T6 at various temperatures, while Figures 2(a) and 2(b) show the same curves for solution treated AA 7075. Figures 3(a) and 3(b) show, respectively, isothermal curves of 𝐸󸀠and 𝐸󸀠󸀠 versus frequency obtained from DMA experiments for AA 2024-T3 at various temperatures, while Figures 4(a) and 4(b) show the same curves for solution treated AA 2024. The results confirm that the viscoelastic response of the studied alloys depends on the temperature and the loading frequency. The larger decrease (increase) of 𝐸󸀠(𝐸󸀠󸀠)with temperature observed for lower frequencies (see Figures 1–4) is due to the Arrhenius-type behavior of the relaxation rate, where the mechanical relaxation time diminishes as temperature increases [1]. The decrease of 𝐸󸀠with temperature (observed as early as at RT) is explained by the dependence of the elastic moduli on temperature, a wellknown phenomenon for metals, for example, for the static elasticmoduliofpurealuminumandAA2024[25–27]and for the dynamic elastic moduli of pure aluminum and AA 6061 reinforced with Al2O3[28]. As shown also in Figures 1–4, 𝐸󸀠and 𝐸󸀠󸀠 depend more significantly on the frequency at higher temperatures. This is also explained by the Arrheniustype behavior of the relaxation rate [6–8]. 3.2. Master Curves for 𝐸󸀠and 𝐸󸀠󸀠.The isothermal curves shown in Figures 1–4, as obtained from the DMA data, were superimposed by shifting along the frequency axis using an Arrhenius-type shift factor 𝑎𝑇((5)) [5]. The arbitrary chosen reference temperature 𝑇0was 215∘C. Using this approach, that is, applying the TTS principle, it was possible to identify a master curve for 𝐸󸀠in a low-temperature region, as shown in Figure 5 for AA 7075-T6 and 7075, and in Figure 7 for AA 2024-T3 and 2024. Also, for all the studied alloys, it was possible to identify master curves for both 𝐸󸀠and 𝐸󸀠󸀠 in a hightemperature region, as shown in Figure 6 for AA 7075-T6 and Advances in Materials Science and Engineering 5 30000 40000 50000 60000 70000 80000 90000 0 0.5 1 1.5 2 Storage modulus E(MPa) log(f) −1−1.5 −0.5−2 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C 390∘C 405∘C 425∘C (a) 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 log(f) Loss modulus E (MPa) 0 0.5 1 1.5 2 −1−1.5 −0.5−2 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C 390∘C 405∘C 425∘C (b) Figure 2: Isothermal storage modulus 𝐸󸀠(a) and loss modulus 𝐸󸀠󸀠 (b) versus logarithm of the frequency 𝑓from DMA tests on solution treated AA 7075 at frequencies ranging from 0.01 to 150 Hz and from 50 to 425∘C. 7075, and in Figure 8 for AA 2024-T3 and 2024. These temperature regions, coincidental with the stable intervals where nophasetransformationsoccur,canalsobeobservedinDSC thermograms(seeTables3and4in[7]).Thelow-temperature stable region ranges approximately from RT to 150∘C, which is reasonable assuming that formation of GP zones and/or Guinier-Preston-Bagariastkij (GPB) zones is already complete in our samples prior to testing (this is justified in [7]) and that dissolution of GP zones and GPB zones has not begun. Only the high-frequency data at 50, 65, 80, and 95∘Cforsolution treated AA 7075 seems to fall a bit far from the master curve, suggesting that maybe a different type of relaxation is playing an important role in these cases or that GP zones formationwasstillnotcompletedforthisalloyatthesetemperatures. The high-temperature stable region ranges approximately from 320 to 375∘C, before the onset of the dissolution of equilibrium phases. It was not possible to obtain any master curve for 𝐸󸀠and 𝐸󸀠󸀠 between 150 and 300∘Cortosuperpose these data in any of the master curves mentioned above. The reason is that this is not a stable temperature interval, that is, it is overlapped by on-going transformations like dissolution of GP zones and/or GPB zones and precipitation of metastable andequilibriumphases.Theactivationenergiesusedin(5) to obtain the master curves shown in Figures 5–8 were established by empirical fit (see Tables 3 and 4). The absolute values of 𝐸󸀠and 𝐸󸀠󸀠 measured by the DMA may differ noticeably from one test to another (e.g., due to instrument error, imperfections of the samples, or errors in measuring their dimensions). That is why no attempt is made to provide microstructural explanations for the differences observed between the studied alloys in, for instance, the absolute values of 𝐸󸀠,asshowninFigures5and7.Namely, these differences fall within instrument error. The different temperature dependencies of 𝑎𝑇for the identified master curves (manifested by different values of 𝐸𝐴in the Arrhenius expression for 𝑎𝑇,with𝐸𝐴being smaller for the low-temperature region compared to the hightemperature region) are due, on the one hand, to the occurrence of microstructural changes between the mentioned temperature regions associated with the phase transformations mentioned above. On the other hand, the differences are also caused by the different relaxation mechanisms acting in one region with respect to the other (the relaxation mechanisms change as the material changes). Hence, from the TTS 6 Advances in Materials Science and Engineering log(f) Storage modulus E(MPa) 21.510.50 75000 70000 65000 60000 55000 50000 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C (a) log(f) Loss modulus E (MPa) 21.510.50 4500 4000 3500 3000 2500 2000 1500 1000 500 0 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C (b) Figure 3: Isothermal storage modulus 𝐸󸀠(a)andlossmodulus𝐸󸀠󸀠 (b) versus logarithm of the frequency 𝑓from DMA tests on AA 2024-T3 at frequencies ranging from 1 to 100 Hz and from 50 to 375∘C. Table 3: Activation energy 𝐸𝐴in the Arrhenius-type equation for the shift factor 𝑎𝑇for Al–Zn–Mg alloys. Temperature region 𝐸𝐴in Arrhenius-type equation for 𝑎𝑇 From 𝐸󸀠data [eV/atom] From 𝐸󸀠󸀠 data [eV/atom] Low temperature (AA 7075-T6) 0.30 No master curve was obtaineda High temperature (AA 7075-T6) 0.60 0.60 Low temperature (AA 7075) 0.30 No master curve was obtaineda High temperature (AA 7075) 0.60 0.60 aOnly values of activation energy below 0.01 eV/atom (with little physical significance) resulted in an acceptable degree of superpositionof 𝐸󸀠󸀠 curves. Therefore, this result was disregarded. Table 4: Activation energy 𝐸𝐴in the Arrhenius-type equation for the shift factor 𝑎𝑇for Al–Cu–Mg alloys. Temperature region 𝐸𝐴in Arrhenius-type equation for 𝑎𝑇 From 𝐸󸀠data [eV/atom] From 𝐸󸀠󸀠 data [eV/atom] Low temperature (AA 2024-T3) 0.30 No master curve was obtaineda High temperature (AA 2024-T3) 0.80 0.80 Low temperature (AA 2024) 0.30 No master curve was obtaineda High temperature (AA 2024) 0.80 0.80 aOnly values of activation energy below 0.01 eV/atom (with little physical significance) resulted in an acceptable degree of superpositionof 𝐸󸀠󸀠 curves. Therefore, this result was disregarded. Advances in Materials Science and Engineering 7 30000 40000 50000 60000 70000 80000 90000 0 0.5 1 1.5 2 Storage modulus E(MPa) log(f) −1−1.5 −0.5−2 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C 390∘C 405∘C 425∘C (a) 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 log(f) Loss modulus E (MPa) 0 0.5 1 1.5 2 −1−1.5 −0.5−2 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 185∘C 215∘C 245∘C 275∘C 290∘C 305∘C 320∘C 335∘C 350∘C 365∘C 375∘C 390∘C 405∘C 425∘C (b) Figure 4: Isothermal storage modulus 𝐸󸀠(a) and loss modulus 𝐸󸀠󸀠 (b) versus logarithm of the frequency 𝑓from DMA tests on solution treated AA 2024 at frequencies ranging from 0.01 to 150 Hz and from 50 to 425∘C. Table 5: Characteristic parameters of the Debye peaks in Figure 9(a), associated with a hypothetical mechanical relaxation. The values of 𝐸𝑢 and 𝐸𝑟were fitted empirically. Only 𝜏at 375∘C was fitted empirically, while 𝜏at 350 and 365∘C was obtained with an Arrhenius equation for 1/𝜏((4)) with activation energy 0.60 eV/at. Temperature 𝜏[s] 𝐸𝑢[GPa] 𝐸𝑟[GPa] 𝐽𝑢[GPa−1]𝛿𝐽[GPa−1] 375∘C 0.50 61.5 53.6 1.63×10−2 2.40×10−3 365∘C 0.59 64.5 56.9 1.55×10−2 2.07×10−3 350∘C 0.77 65.3 57.9 1.53×10−2 1.95×10−3 results it appears that phase transformations have also an influence on 𝐸󸀠and 𝐸󸀠󸀠,assuggestedpreviously[1,7,29]. 3.3. Analysis of the Measured 𝐸󸀠󸀠 Peaks. For AA 7075-T6, the isothermal 𝐸󸀠󸀠 curves at 350, 365, and 375∘Cshowa peakataround3Hz,asshowninFigure1(b).ForAA7075, this peak shifts from 1 to 10 Hz as temperature rises from 320 to 425∘C as shown in Figure 2(b). Figure 9(a) shows Debye peaks superimposed to the isothermal 𝐸󸀠󸀠 curves of AA 7075-T6 at 350, 365, and 375∘C. The Debye peaks correspond to a hypothetical mechanical relaxation with characteristic parameters as given in Table 5. The values of 𝐸𝑢and 𝐸𝑟in Table 5 were fitted empirically bearing in mind that (1) from the results in Figure 1(b), 𝐸𝑢and 𝐸𝑟should be around 60 and 50 GPa, respectively; (2) 𝐸𝑢>𝐸 𝑟,since materials exhibit a stiffer response at higher frequencies; and (3) 𝐸𝑢and 𝐸𝑟decrease with temperature, as usual for elastic moduli of metals [25–27]. Since the values of 𝜏at the various temperatures are linked by (4) (the activation energy being 0.60 eV/atom, from the results in Table 3), only the value of 𝜏at 375∘C was fitted empirically. The purpose of this rough fitting was not to compute the best fit values for those parameters (then this procedure would not be the most appropriate) but simply to show that since the measured 𝐸󸀠󸀠 8 Advances in Materials Science and Engineering Solution treated AA 7075 AA 7075-T6 95∘C 110∘C 125∘C 155∘C 72500 70000 67500 65000 62500 60000 log(f·aT) Storage modulus E(MPa) 302520151050 50∘C 65∘C 80∘C Figure 5: Master curve showing the storage modulus 𝐸󸀠versus the logarithm of the frequency 𝑓times the shift factor 𝑎𝑇.These curves resulted from horizontal shifting of the isothermal 𝐸󸀠curves from DMA tests on AA 7075-T6 and solution treated AA 7075, at temperatures from 50 to 155∘C. peaks are much broader than the Debye peak, they do not correspond to a single relaxation, but to several overlapping or coupled relaxations or to a single mechanism with a distribution of relaxation times. The results in Figure 9(b) also suggest that the measured viscoelastic response may be associated with more than one mechanical relaxation process. Particularly, the value of 𝛿𝐸,asobtainedfromtheDebye equation corresponding to a single relaxation, is smaller than the magnitude of the relaxation of 𝐸󸀠that can be inferred from 𝐸󸀠results in the limited test frequency range. For one test on AA 2024-T3, an inflexion is observed centeredaround1Hzfor365to425 ∘C, and peaks are observed at around 100 Hz for 100 to 160∘C. However, for most tests on AA 2024-T3 and 2024, no 𝐸󸀠󸀠 peak can be observed in the studied frequency and temperature ranges. For example, the Zener peak often present in solution treated and quenched Al–Cu alloys at 173–175∘C and 1 Hz [1] is not observed in our results. If for AA 2024 (which was solution treated and quenched) the peak is not present, this may be due to (1) presence of GP-II zones, (2) the effect of the other elements present in this alloy aside from Cu, (3) the formation of phase 𝜃󸀠, causing a decline of the Zener peak, which may eventually vanish when 𝜃󸀠phase formation is complete [1], or,(4)mostlikely,theCucontentinthestudiedalloysbeing so low that there are not enough Cu atomic pairs for this Zener peak to become visible (for instance, Golovin et al. [15] did not distinguish a Zener peak in Al–Mg alloys due to reorientation of Mg–Mg atom pairs in the aluminum solid solution for alloys with Mg content below 5 wt.%). In most of Storage modulus E(MPa) log(f·aT) Loss modulus E (MPa) 4500 4000 3500 3000 2500 2000 1500 1000 500 0 −3−4−5 72500 70000 67500 65000 62500 60000 57500 55000 52500 50000 E,375 ∘C E,365 ∘C E,350 ∘C E,335 ∘C E,320 ∘C E, 350∘C E, 365∘C E, 375∘C E, 335∘C E,320 ∘C (a) −12 −10 −8−6−4 log(f·aT) Storage modulus E(MPa) Loss modulus E (MPa) 6000 5000 4000 3000 2000 1000 0 60000 55000 50000 45000 40000 35000 30000 25000 E, 375∘C E,365 ∘C E, 350∘C E, 335∘C E,320 ∘C E,350 ∘C E,365 ∘C E,375 ∘C E,335 ∘C E,320 ∘C (b) Figure 6: Master curves showing the storage modulus 𝐸󸀠(hollow symbols) and the loss modulus 𝐸󸀠󸀠 (solid-fill symbols) versus the logarithm of the frequency 𝑓times the shift factor 𝑎𝑇. These curves resulted from horizontal shifting of the isothermal curves of 𝐸󸀠and 𝐸󸀠󸀠 from DMA tests on AA 7075-T6 (a) and solution treated AA 7075 (b), from 320 to 375∘C. Advances in Materials Science and Engineering 9 AA 2024-T3 Solution treated AA 2024 50∘C 65∘C 80∘C 95∘C 110∘C 125∘C 155∘C 302520151050 72500 70000 67500 65000 62500 60000 57500 log(f·aT) Storage modulus E(MPa) Figure 7: Master curve showing the storage modulus 𝐸󸀠versus the logarithm of the frequency 𝑓times the shift factor 𝑎𝑇.These curves resulted from horizontal shifting of the isothermal 𝐸󸀠curves from DMA tests on AA 2024-T3 and solution treated AA 2024, at temperatures from 50 to 155∘C. our results (see Figure 2 in [7]), we can hardly distinguish the broad relaxation peak (three times broader than the Debye peak) at 135∘C, associated with 𝜃󸀠phase, with magnitude proportional to 𝜃󸀠phase fraction [1]. For the studied alloys, the results show monotonical growth of 𝐸󸀠󸀠 with temperature above 200∘C (see, for instance, Figure 2 in [6] and Figure 2 in [7], resp.). This is usually explained by presence of coupled or overlapping relaxations [1, 15] and by the high-temperature internal friction background, which is roughly exponential with temperature for most materials [15, 16]. This background is linked to thermally activated viscous deformation, mainly associated with diffusion-controlled dislocation motion. A relaxation probably contributing to this growth of 𝐸󸀠󸀠 is the internal friction peak typically exhibited by polycrystalline aluminum centered at ≈300∘C.Thispeakistraditionallyassociatedwith grain boundary sliding (GBS) [1, 15, 16] and has been observed also in Al–Mg, Al–Zn–Mg, and Al–Cu–Mg alloys [12, 13]. For AA 2024-T3 and 2024, a relaxation peak associated with 𝜃phase[10]isalsoprobablycontributingtothemonotonical growth of 𝐸󸀠󸀠. Finally, there is also a contribution by the linear thermoelastic background, which is proportional to temperature,butitshouldnotbeverysignificantinthestudied frequency range [16]. For the studied alloys in the considered ranges, it is difficult to ascertain whether any 𝐸󸀠󸀠 peak is fully captured (i.e., whether the results capture both tails of the peak). Hence, itisnotconvenienttousetheHNmodelorCDmodelto analyze the results, because it is difficult to establish whether thepeaksareasymmetricornot.Analternativetoobtain information from the relaxation process (or processes) responsible for those peaks is to analyze the observed tails of the peaks [17, 18]. For a given value of Ω𝜏, it can be derived from (1) and (3) that the value of 𝐸󸀠󸀠 isthesamenomatter the frequency. Hence, for each of the curves corresponding to a different frequency in Figure 2 in [7], we obtained the temperatureforasamevalueof𝐸󸀠󸀠 (i.e., a same value of Ω𝜏). With these data, an Arrhenius plot can be obtained following (4), showing the logarithm of the frequency versus the reciprocal of the temperature, where ln(Ω)=ln(Ω𝜏)−ln(𝜏0)− 𝐸𝐴/𝑘𝐵𝑇(see Figure 10). Since the value of Ω𝜏is known, from the Arrhenius plot it is possible to estimate 𝐸𝐴and 𝜏0. Using this procedure for four different values of 𝐸󸀠󸀠 for each alloy (1980, 2000, 2020, and 2040 MPa for AA 7075- T6 and 1280, 1300, 1320, and 1340 MPa for AA 2024-T3), the apparent activation parameters were obtained for the monotonical growth region of 𝐸󸀠󸀠. The differences in the computed activation energy for the four cases with respect to the average are below 1.5% for AA 7075 and 4.3% for AA 2024. This proves the robustness of the method and the precision and high-confidence of the activation energy results. Particularly, the apparent activation parameters, which may correspond to superposed tails of coupled or overlapped peaks, are 𝐸𝐴= 1.50±0.01eV/atom and 𝜏0= (1.44±0.40)×10−15 sforAA 7075-T6 and 𝐸𝐴= 1.83±0.05eV/atom and 𝜏0= (1.45± 1.40) × 10−17 s for AA 2024-T3. These values suggest that the relaxations may correspond effectively to GBS [1, 15]. For example, the relaxation times are far from those typical of relaxations caused by dislocations (for which 𝜏0usually ranges from 10−10 to 10–13) [15], while the activation energies are close to 1.48 eV/atom, as reported for GBS [1, 15]. The higher value of 𝐸𝐴for AA 2024-T3 indicates that, for given conditions, it is more difficult and less frequent for GBS to proceed in this alloy compared to AA 7075-T6. Since GBS has a significant influence on creep at high stress values, for example, in the beginning of the tertiary creep acceleration [30], this would be consistent with the creep response of Al–Cu–Mg alloys, which is superior to Al–Zn–Mg alloys in the tertiary stage; see, for instance, a comparison of the creep behavior of AA 2419 at 340 MPa and 100∘C[31]withthatof AA7010 at 350 MPa and 100∘C [30]. AA 7010 has higher minimum creep strain rate and has lower time to fracture (718000 versus 838000 s) and shows higher creep strain at fracture. Moreover, since fatigue is a consequence of microstructural changes that must have also an effect on the viscoelastic response [2], if GBS contributes also to fatigue problems, the higher value of 𝐸𝐴for the relaxation associated with GBS in Al–Cu–Mg alloys is also consistent with the better fatigue response of these alloys compared to Al–Zn–Mg alloys. 4. Conclusions The main conclusions of this investigation are summarized as follows: (1) The TTS principle, which has not been applied to crystalline materials before, was successfully applied totheviscoelasticresponseofcommercialAA7075