ISSN 0031-918X, Physics of Metals and Metallography, 2017, Vol. 118, No. 12, pp. 1255–1261. © Pleiades Publishing, Ltd., 2017. Original Russian Text © L.S. Metlov, I.G. Brodova, V.M. Tkachenko, A.N. Petrova, I.G. Shirinkina, 2017, published in Fizika Metallov i Metallovedenie, 2017, Vol. 118, No. 12, pp. 1331–1337.
STRUCTURE, PHASE TRANSFORMATIONS, AND DIFFUSION
Bimodal Structures of Solids Obtained under Megaplastic Strain L. S. Metlova, b, I. G. Brodovac, *, V. M. Tkachenkoa, A. N. Petrovac, and I. G. Shirinkinac a
Galkin Donetsk Physicotechnical Institute, ul. Rozy Luksemburg 72, Donetsk, 83114 Ukraine bDonetsk National University, ul. Shestisotletiya 21, Vinnitsa, 21021 Ukraine c Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, ul. Sof’I Kovalevskoi 18, Ekaterinburg, 620137 Russia *e-mail:
[email protected] Received March 14, 2017; in final form, June 26, 2017
Abstract—The evolution of the defect structure of aluminum alloys of different compositions under megaplastic strain at a high quasi-hydrostatic pressure has been studied. The theoretical analysis and numerical calculation of the evolution of the defect structure of solids (metals) under this impact have been performed within the three-defect model of nonequilibrium evolution thermodynamics. The calculation of defect kinetics shows that the presence of coarse grains submerged into a matrix of fine grains at specified parameters and coefficients leads to the additional generation of dislocations and, as a consequence, to the greater dislocation strengthening of a material. Keywords: megaplastic strain, nonequilibrium thermodynamics, kinetics, structure, defects, dislocations, grain boundaries, electron microscopy, fragmentation, dynamic recrystallization DOI: 10.1134/S0031918X17120109
INTRODUCTION Both the generation of structural defects and their annihilation (recrystallization) occur under megaplastic strain (MPS). If these processes have a linear effect on each other, the attainment of a steady state is monotonical in time. However, the process may have a nonlinear character in real practice, and these mutually countercurrent processes may be quicker or slower to form a cyclic behavior over time and space [1–4]. In particular, the processes of recrystallization may invariably lead to the local formation of coarse grains, which restores the plasticity of the material. These grains will further be fragmented again, which provides high strength. Hence, the bimodal distribution of the grain sizes appears in material subjected to MPS, and this distribution will be in dynamic equilibrium. In real practice, the existence of a bimodal distribution of grain sizes was observed in pure metals [5–8] and alloys [8–15] subjected to MPS treatment. The simultaneous presence of coarse and fine grains provides a combination of high strength and relatively good plasticity even for pure copper [5] or nickel [6]. In the case of pure metals, coarse and fine grains have an identical crystal structure. In the case of highentropy alloys with a great number of components, coarse and fine grains may additionally differ from each other in their component composition and phase state.
A number of models and mechanisms, but no unitary theoretical description have been recently proposed for the formation of a bimodal distribution of grain sizes in the MPS process. The approach based on nonequilibrium evolution thermodynamics (NET) [17, 18] was also proposed for describing the kinetics of structural defects under MPS [17, 18] and tested on a number of certain examples, including the description of a time-circular character of evolution [4]. The main objective of this work was to ascertain the applicability of NET for the modeling of the effect of a bimodal distribution of grain sizes on the evolution of the structure and properties and verify the theoretical model in comparison with experimental data [9, 14–16]. EXPERIMENTAL METHODS OF STUDY AND MATERIALS The structural evolution of industrial aluminum alloys of different compositions, such as AMts, V95, and A2024, under shear MPS at a high quasi-hydrostatic pressure was studied. Torsion in Bridgman anvils at a rate of 1 rpm at a pressure P = 4 GPa and room temperature was used as an MPS treatment method. Disk-shaped ingots cut from the central part of a thermally treated rod with a diameter of 10 mm and a thickness of 0.75 mm were used. The number of anvil rotations was n = 1, 5, 10, and 15 and, according to cal-
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RESULTS AND DISCUSSION
(a)
Experimental results on the formation of a bimodal distribution of grain sizes under MPS were obtained on several aluminum alloys belonging to different doping systems. They represent thermally hardenable alloys of the Al–Mg–Cu (A2024) [16] and Al–Zn– Mg–Cu (B95) [14] systems, and a thermally nonhardenable alloy of the Al-Mn system (AMts) [15].
12 n, %
10 8 6 4 2 0
100
200
300 D, nm
400
500
600
(b)
n, %
40 30
A more abundant variety of structures was revealed for alloy AMts after MPS. A mixed structure, which predominantly consists of grains with high-angle boundaries (HABs) and low-angle boundaries (LABs) (400 nm) and less defective recrystallized grains, is observed at a true strain е = 3.9. Then, the recrystallized structure becomes predominant within a range of true strains е = 4.1–5.5. The mixed structure, the major volume of which is occupied by secondary fragmented grains with a size of 150 nm and a developed inner substructure, appears again at е = 6.4–6.9. Isolated recrystallized grains free from dislocations appear again at triple junctions of these crystallites. Hence, a cyclic character of the formation of submicrocrystalline (SMC) states in alloy AMts under quasi-static strain was revealed.
20 10 0
100
300
500 700 D, nm
900
(c)
n, %
30 20 10
0
100
300
500 D, nm
700
The histograms of the size distribution of grains– subgrains depending on the accumulated strain in alloys of different compositions are plotted in Fig. 1. The mixed bimodal structure in A2024 alloy after MPS consists of a great number of fine grains with a size of 100–200 nm and isolated coarse grains with sizes above 500 nm. The ratio between these modes varies depending on the degree of accumulated strain and is most pronounced at е ≈ 5.5 (Fig. 1a).
900
Fig. 1. Grain–subgrain size distribution histograms for alloys (a) A 2024, е = 5.5, (b) V95, е = 6.4, and (c) V95, е = 6.9.
culation, corresponded to the true strain e = 4.0, 5.5, 6.5, and 6.9 at half-radius points of a specimen. In the structural studies, the sizes of matrix fragments in the strained materials were calculated from dark-field electron microscopy images, which were taken on Philips СМ-30 and JEM -200 CX transmission electron microscopes and processed by the SIAMS Photolab automatic method. To eliminate the effect of strain nonuniformity along the radius of a specimen in this strain method, all of the structural characteristics were determined at the half radius of a specimen.
The nature of physical processes in alloy B95 under MPS also variates depending on the regime of strain. The predominant mechanism of the relaxation of inner strains in this alloy within a broad range of true strains е ≤ 6.4 is fragmentation, the structure is disintegrated to a nanolevel, and the average grain size at е = 6.4 is 55 nm. At е ≥ 6.9, another elastic energy relaxation channel, namely, low-temperature dynamic recrystallization becomes active in the alloy, and a mixed structure with a bimodal distribution of grain sizes appears (Fig. 1c). In particular, coarse grains with maximum sizes of 600 nm and average sizes increases to 115 nm appear alongside with the areas of nanograins. Electron microscopic studies have demonstrated the evolution of the fine structure of aluminum alloys under MPS and made it possible to reveal the main regularities of its development with an increase in the accumulated strain. Independently of the composition of the alloy, the formation of a strained structure occurs due to two competitive processes, i.e., fragmentation and dynamic recrystallization.
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In the first process, new low-angle boundaries are first formed inside initial subgrains via the arrangement of lattice dislocations into walls, and the further development of rotation strain modes leads to the appearance of high-angle grain misorientations and the formation of an SMC structure. The observed nonuniform contrast inside crystals indicates a high level of elastic stresses and a high density of dislocations. This stage of formation of SMC states is described by the unimodal grain size distribution histogram. A mixed structure of grains with different sizes is formed with an increase in the accumulated strain. An elastically stressed state persists in the region of fine grains that is exhibited in the light-field images as unsteady high-angle boundaries and a nonuniform contrast inside crystallites. The boundaries inside coarse grains are more regular, and the number of dislocations inside coarse grains is much lower. The appearance of a small number of coarse grains and the bimodal distribution of the average size of the entire ensemble indicates the beginning of abnormal grain growth in the process of dynamic recrystallization. The ratios between the volumes of two SMC structures formed by different mechanisms in the mixed structure depend not only on the accumulated strain, but also on the degree of matrix doping. At the same shear strain, a fragmented structure predominates in complexly doped alloys, and a recrystallized structure predominates in lowly doped alloys. In other words, a more steady-state bimodal structure component appears in alloys with a less doped matrix at a smaller accumulated strain. Despite the above described distinctions in the behavior of aluminum alloys during MPS, it is important to identify the main general regularities in the formation of their bimodal structure at this stage of study. These include the close sizes of fine and coarse grains, i.e., 100 and 500 nm. According to the presented histograms, the volume fractions of coarse grains and their density are also close. In particular, the density of coarse grains hG is 2.0 × 1017 m–3 in the structure of alloy A 2024 and 4.0 × 1017 m–3 in alloy V95. The considered metal structure consisting of coarse and fine grains is schematically reproduced in Fig. 2. It is assumed that grains with different sizes represent two different defects, each of which should be described with a particular kinetic equation. Thus, fine grains have shared boundaries with each other and collectively form a continuous medium and are characterized by the density of grain boundaries. Coarse grains are isolated from one another, submerged into the matrix of fine grains, and can be described as bulk defects of mesolevel. Their density is equal to the number of grains per unit volume. If dislocations are added to these defects, and the density of PHYSICS OF METALS AND METALLOGRAPHY
Fig. 2. Schematic representation of a metal structure with a bimodal grain size distribution.
the effective internal energy u with allowance for the triple contribution will have the form [17, 18]
u = u0 +
∑
m = b,D,G
(ϕ
0m hm
− 1 ϕ1mhm 2 2
)
+ 1 ϕ 2mhm3 − 1 ϕ3mhm4 3 4 + ϕ bG hb hG + ϕ bDhb hD + ϕGDhG hD,
(1)
where u0 is the internal energy of a defectless material; hb , hG , and hD are the number density of grain boundaries for fine grains, the density of coarse grains, and the density of dislocations, respectively; ϕ0m, ϕ1m, ϕ2m, and ϕ3m are the model parameters for each type of defect apart; and ϕ bG , ϕ bD , and ϕGD are the model parameters taking into account the pairwise interactions between different types of defects. The summands of Eq. (1) in the parentheses describe the contributions from each individual defective subsystem, i.e., from dislocations (subscript D), boundaries of fine grains (subscript b), and coarse grains (subscript G). The other summands in Eq. (1) describe the interaction between defective subsystems. The coefficients depending on the elastic strain as a controlling parameter are
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u0 = 1 λ ( ε ii ) + μ ( ε ij ) , 2 2
(
2
)
2 2 ϕ 0m = ϕ*0m + g mε ii + 1 λ m ( ε ii ) + μ m ( ε ij ) , 2
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where ε ii and ( ε ij ) = ε ji ε ij are the first and second invariants of the tensor of elastic strains, μ and λ are the Lame elastic moduli, μ m and λ m are the deficit of these moduli due to the effect of m-type defects, and ϕ*0m, ϕ1*m, еm, and gm are the other expansion coefficients [3]. In the case of grain boundaries, the description of the kinetics requires a higher degree of approximation for the internal energy (Eq. (1)) due to the general trend of the formation of bimodal grain size distributions. In the case of dislocations and coarse grains, it is sufficient to restrict consideration to the contributions of only the second order for the number density of defects, setting ϕ2D = ϕ3D = 0 and ϕ2G = ϕGD = 0 in order to decrease the total number of free parameters in the problem. The alternation of signs in parentheses in Eq. (1) reflects the le Chatelier principle, according to which a thermodynamic process of higher level is oriented to the compensation of effects from thermodynamic processes of lower level. The elastic strains are controlling problem parameters and, according to the Hooke law, can be expressed through the stresses as 2
λ (3) ε ij = 1 σ ij − σ ii δ ij . 2μ 2μ (3λ + 2μ ) In this model, it is assumed that the process can be controlled by specifying the acting stresses. In the case of slow loadings (at a strain rate below 10–1 s–1) implemented under MPS, it is possible to assume that the acting stresses coincide with the ultimate plastic yield independently of the loading rate and automatically grow with increasing material strength. At a dislocation level, the strengthening law is specified by the ultimate plastic yield τ as τ = μ ( S hD − whb ) ,
(4)
where μ is the shear modulus, S is the dislocation strengthening coefficient, and w is the coefficeint of weakening due to sliding along grain boundaries. In this certain case, the kinetic equations can be written in the explicit form
∂ hb = γ b ∂ u = γ b (ϕ 0b − ϕ1b hb ∂t ∂ hb 2 3 + ϕ 2b hb − ϕ3b hb + ϕ bG hG + ϕ bDhD ), ∂ hD = γ D ∂u ∂t ∂ hD = γ D(ϕ 0D − ϕ1DhD + ϕ bDhb + ϕGDhG ).
absolute values. In fact, multiplying all of the coefficients by the same value only leads to the reestimation of the kinetic coefficients in Eq. (5). In connection with this, the main coefficients in the internal energy expansion with respect to the density of grain boundaries must be ϕ0b and ϕ1b at the initial stage of MPS process and ϕ2b and ϕ3b at its final stage, as their ratios determine the steady-state eigenvalues of the density of grain boundaries without consideration for the effect of defects of other types. The same conclusion is true for the coefficients ϕ0D and ϕ1D for dislocations and ϕ0G and ϕ1G for coarse grains. The effect of elastic stresses on these coefficients is taken into account separately and specified in the general form by Eqs. (2) and (3). The coefficients ϕbG, ϕbD, and ϕGD are not major, but still very important in the general kinetics of defects. For example, the coefficient ϕbG immediately characterizes the interaction between fine grain boundaries and coarse grains. We shall assume that the initial specimen consists of coarse and uniformly distributed grains, and the grain system uniformly evolves as a result of MPS, i.e., the only grain size mode (fine grains) is formed until the system attains a steady-state regime. In experiment, this process may correspond to the initial structure fragmentation regime. Let us further artificially introduce coarse grains to see what effect will be produced by their presence on the attained parameters of the material. If we make such a simplification (assumption), than the system is only described by the first two equations in set (5) within the two-defect model, which takes into account only fine grain boundaries (b) and dislocations (D). The third defect (G, coarse grains) is only added after the system attains a steady-state regime at a time moment t = 0.6 s. In experiment, these coarse grains appear as a result of relaxation processes, such as dynamic recrystallization, and can be fragmented again under further strain treatment. Hence, the generation and annihilation of coarse grains with the attainment of steady-state values occur in the material. The calculations are given for the effective equation coefficients. Let us estimate the coefficeint ϕ0G for coarse grains. The effective internal energy of a solid is
u = u − μ eq hG, (5)
Specifying the initial problem conditions, it is possible to calculate the number densities of all defects at any time moment. It follows from Eqs. (5) that the kinetics of defects depends on the ratio between the free energy expansion coefficients, rather then of their
(6)
where u is the internal solid’s energy determined by Eq. (1) and μ eq = ∂ u is the chemical potential or ∂ hG energy of a single defect (coarse grain). If the chemical potential is assumed to be the excess energy of a coarse grain, it is equal to the half energy of its outer boundaries. Then, the estimate obtained for the coefficient of a cubic grain is ϕ 0G ≈ 2μ eq = α GS G = 6α GdG2 ,
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0D
* = 0.6867 × 10–24 J/m, g = 1 × 10 −8 J/m, ϕ1D D −2 μ D = 0.1 × 10 J/m, eD = 6 × 10 −23 J/m, γD = 3 × 1025 J—1 m–1 s–1, ϕ = 1.0 × 10 −16 J, ϕ* = 0.4 J/m2, bD
0.8 hG × 1017, m–3
where αG is the specific boundary surface energy, αG = 0.5–1 J/m2 for metals and alloys [19], SG is the boundary surface area, and dG is the coarse grain size. Let us consider the following set of the model parameters and coefficients (J/m) for calculations [4]: λ = 2.08 × 1010 Pa, μ = 2.08 × 1010 Pa, ϕ* = 5 × 10 −9 J/m,
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0.2
0.5
1.0
1.5
2.0
2.5
hD × 1015, m–2
80 hG = 0, m–3 h0G = 8 × 1014, m–3 h0G = 8 × 1015, m–3 h0G = 8 × 1016, m–3
60 40 20
(b) 0
0.5
1.0
1.5
2.0
2.5
hb × 106, m–1
40 30 20
hG = 0, m–3 h0G = 8 × 1014, m–3 h0G = 8 × 1015, m–3 h0G = 8 × 1016, m–3
10
(c) 0
0.5
1.0
1.5
2.0
2.5
0.8 0.6 εij, %
According to the applied technique, the density of coarse grains abruptly changes at a time moment t = 0.5 s by a specified value and further slowly decreases, sustaining the response from the continuum of fine grains and dislocations (Fig. 3a). The density of fine grain boundaries (Fig. 3c) and the density of dislocations (Fig. 3b) at the initial stage grow before the appearance of coarse grains, tending to 26.8 × 106 m–1 and 26.6 × 1015 m–2, respectively. This density of fine grain boundaries corresponds to the average size slightly higher than 100 nm. These characteristics attain a new asymptotic limit upon introduction of coarse grains, and its value grows with increasing density of coarse grains to reach 35 × 106 m–1 at a density of coarse grains of 8 × 1016 m–3. This corresponds to the average size of fine grains of less than 100 nm. On the whole, the average size of fine grains is comparable with its experimentally observed values by the order of magnitude. On one hand, coarse grains submerged into a matrix of fine grains act as barriers for mechanisms of interactions between fine grains, which results in the generation of dislocation and the strengthening of a material. On the other hand, the generation, sliding, in coarse grains, the mutual interaction of dislocations result in a decrease in the number of coarse grains due to fragmentation and an increase in the density of fine grain boundaries, which also improves the strength properties of a solid. An abrupt increase in the density of dislocations is observed when the density of coarse
hG = 0, m–3 h0G = 8 × 1014, m–3 h0G = 8 × 1015, m–3 h0G = 8 × 1016, m–3
0.4
0
* = 0.08 × 10–6 J/m, ϕ = 0.56 × 10 −12 J, ϕ1b 2b ϕ3b = 0.3 × 10 −19 J m, gb = 3 J/m2, λ b = 2.5 × 10 5 J/m2,
104 J–1 m–3 s–1, ϕ bG = 1.0 × 10 −16 J m, and dG = 500 × 10–9 m. The calculation of the system kinetics at these parameters and coefficients is given in Fig. 3. The cases when there are no coarse grains (hG = 0) throughout the entire period of calculations, and the cases with a different initial number of coarse grains (h0G) were considered.
0.6
(a)
0b
J/m, J/m2, μ b = 3.12 × 10 8 e b = 3.6 × 10 −4 6 − 16 –1 –2 s–1 , ϕGD = 1.0 × 10 J m2, γ b = 1 × 10 J m − 6 ϕ0G = 0.96 × 10–12 J, ϕ1G = 1.65 × 10 J m3, γG = 1 ×
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hG = 0, m–3 h0G = 8 × 1014, m–3 h0G = 8 × 1015, m–3 h0G = 8 × 1016, m–3
0.2
(d) 0
0.5
1.0
1.5
2.0
2.5
t, s
Fig. 3. Change in the density of (a) coarse grains hG, (b) dislocations hD, and (c) fine grain boundaries hb, and (d) elastic strains versus calculation time. No. 12
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grains grows to h0G = 8 × 1016 (Fig. 3b). In this case, the maximum elastic shear strain also increases (Fig. 3d) and, according to Eq. (4), the ultimate yield also grows. Hence, the model results show that the attainment of a limit asymptotic value by all the parameters of a material occurs by a standard scheme. Plastic strain processes predominate at the initial stage, leading to the fragmentation of grains, the monotonical formation of a defective structure, and an increase in the density of dislocations and the volumetric density of the overall surface area of fine grain boundaries. In a certain time period, this elastic energy relaxation channel ceases to be efficient, and the defective structure evolves due to the activation of another relaxation process, namely, dynamic recrystallization. When the rates of these processes become equal to each other, the system attains asymptotic (limit or steady-state) values of all its parameters. The formed balance is destroyed again upon the introduction of coarse grains, and new steady-state values of the same parameters are formed over time. The experimental results given in the first part of this paper prove that the formation of bimodal grain size distributions is a real problem of material science, and the experimental and theoretical sizes and volumetric densities of structural components are comparable with each other by the order of magnitude. At the same time, we should point to the existence of some objective difficulties in a more detailed comparison of theoretical results and experimental data [20]. Analyzing the obtained model results, it is possible to suggest that the effect of additional strengthening in the structure of alloys upon introduction of coarse grains is similar to the effect of dispersion strengthening due to precipitates of new phases. CONCLUSIONS (1) The studies of aluminum alloys subjected to megaplastic shear strain at a high quasi-hydrostatic pressure have disclosed the mechanisms of the formation of a strained submicrocrystalline structure due to fragmentation and dynamic recrystallization. The existence of two elastic energy relaxation channels leads to the formation of a mixed structure characterized by a bimodal grain size distribution. (2) A mathematical model that describes the structure of a material with a bimodal distribution of structural elements with consideration for the evolution of dislocations and grain boundaries has been proposed. It has been shown that the presence of coarse grains submerged into a matrix of fine grains promotes the generation of dislocations and the elongation of grain boundaries, which leads to the additional strengthening of the material due to the activation of the dislocation mode.
ACKNOWLEDGMENTS This work was performed within the state task of the Federal Agency for Scientific Organizations of Russia, program “Structure” no. 01201463331. Electron microscopic studies were performed in the Department of Electron Microscopy of the Shared Facilities Center of the Mikheev Institute of Metal Physics (Ural Branch, Russian Academy of Sciences) “Experimental Center of Nanotechnologies and Advanced Materials.” REFERENCES 1. V. N. Varyukhin, E. G. Pashinskaya, L. S. Metlov, A. F. Morozov, A. S. Domareva, S. G. Cynkov, V. G. Cynkov, and T. P. Zaika, “Application of hydroextrusion with torsion for production of massive metallic samples with submicroscopic structure,” Fiz. Tekhn. Vys. Davl. 12 (1), 29–41 (2002). 2. A. M. Glezer, “On the nature of ultrahigh plastic (megaplastic) strain,” Bull. Russ. Acad. Sci.: Phys. 71, 1722– 1730 (2007). 3. A. M. Glezer and L. S. Metlov, “Physics of megaplastic (severe) deformation in solids,” Phys. Solid State 52, 1162–1169 (2010). 4. L. S. Metlov, A. M. Glezer, and V. N. Varyukhin, “Cyclic character of the evolution of the defect structure and the properties of metallic materials during megaplastic deformation,” Russ. Metall. (Metally) 2015, 269–273 (2015). 5. Q.-W. Jiang and X.-W. Li, “Effect of pre-annealing treatment on the compressive deformation and damage behavior of ultrafine-grained copper,” Mater. Sci. Eng., A 546, 59–67 (2012). 6. T. R. Lee, C. P. Chang, and P. W. Kao, “The tensile behavior and deformation microstructure of cryorolled and annealed pure nickel,” Mater. Sci. Eng., A 408, 131–135 (2005). 7. F. S. J. Poggiali, E. B. Figueiredo, M. T. P. Aguilar, and P. R. Cetlin, “Grain refinement of commercial purity magnesium processed by ECAP (Equal channel angular pressing),” Mater. Res. 15, 312–316 (2012). 8. O. Sedivy, V. Benes, P. Ponizil, P. Kral, and V. Sklenicka, “Quantitative characterization of microstructure of pure copper processed by ECAP,” Image Anal. Stereol. 32, 65–75 (2013). 9. I. G. Brodova, A. N. Petrova, and I. G. Shirinkina, “Comparing specific features of the structural formation of aluminum alloys during severe and intense plastic deformation,” Bull. Russ. Acad. Sci.: Phys. 76, 1233–1237 (2012). 10. M. Haouaoui, I. Karaman, H. J. Maier, and K. T. Hartwig, “Microstructure evolution and mechanical behavior of bulk copper obtained by consolidation of micro- and nanopowders using equal-channel angular extrusion,” Metall. Mater. Trans., A 35, 2935–2949 (2004). 11. K. S. Raju, V. S. Sarma, A. Kauffmann, Z. Hegedus, J. Gubicza, M. Peterlechner, J. Freudenberger, and G. Wilde, “High strength and ductile ultrafine-grained Cu–Ag alloy through bimodal grain size, dislocation
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Translated by E. Glushachenkova
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