In conventional coarse-grained (CG) metallic materials, the deformation is mainly controlled by the motion of full lattice dislocations in the grain interiors. Grain boundaries act as obstacles against dislocation motion, therefore the dislocations formed at Frank–Read (FR) sources inside the grains are accumulated in pileups, e.g., at grain boundaries. The dislocations in pileups apply repulsive forces to successive dislocations emitted by the sources. As the grain size decreases, the sources are located closer to the grain boundaries; therefore the emission of dislocations from the FR sources becomes more difficult. Thus, for very small grain sizes (10–20 nm) dislocation pileups do not form and the dislocation sources are rather located in the grain boundaries. In this case, the dislocations emitted from a boundary travel throughout the grain and annihilated in the opposite boundary. Experiments and theoretical modeling revealed that when the grain size decreases below ∼10 nm, rather partial dislocations are emitted from grain boundaries instead of perfect lattice dislocations [1–7]. Nanostructured metals produced by severe plastic deformation (SPD) techniques often have nonequilibrium grain boundaries that contain excess dislocations (referred to as extrinsic or nongeometrically necessary or statistically stored dislocations) in addition to the geometrically necessary ones. Some dislocations may dissociate into pairs of Shockley partials, which could move away from the grain boundary under a stress. Such grain boundaries thus act as partial dislocation sources. However, partial dislocations can also be emitted from grain boundaries by atomic reshuffling [2]. The effect of grain size on the emission of partial dislocations from grain boundaries and the related deformation twinning has been studied for face-centered cubic (fcc) nanostructures on the model depicted in Figs. 7.1–7.3 [8,9]. The model assumes a grain with a square (111) slip plane, as shown in Fig. 7.1. Under an external shear stress τ, a leading Shockley partial dislocation with the Burgers vector b₁ is emitted from the grain boundary XY. This leading partial dislocation consists of the three segments XE, EF, and FY in Fig. 7.1. τ is oriented at an angle β with line EF. The ends of this partial dislocation are pinned at the triple junctions X and Y, so that two segments of partial dislocation lines XE and FY are deposited on grain boundaries. Similarly, a trailing Shockley partial dislocation with the Burgers vector b₂ is emitted from the grain boundary XY. This partial consists of the segments XC, CD, and DY. The two partial segments EF and CD are separated by a stacking fault (SF) ribbon. The two partials react to form two perfect edge dislocation segments XC and YD with the Burgers vector b =  b₁  +  b₂ at grain boundaries. The partial segments EF and CD would form a perfect screw dislocation with the Burgers vector b if they reacted. For the formation of a single leading partial, τ has to perform the work needed to increase both the area of the SF behind the partial and the length of the dislocation segments XE and FY. Using the formulas for the forces acting on the leading partial dislocation segment (see Chapter 4), the critical stress, τ_P for moving the leading partial XEFY can be derived as [9]

where G is the shear modulus, a is the lattice parameter, ν is the Poisson's ratio, γ is the stacking fault energy (SFE), and d is the grain size.

Considering the grain boundary strengthening in polycrystalline CG metallic materials, the yield strength is usually correlated to the grain size by applying the Hall–Petch equation [24–26]. In a grain of a polycrystal, the plastic deformation sets in when the resolved shear stress in one or more slip systems reaches the critical value necessary for the initiation of dislocation glide. In a polycrystalline material, the grains have different crystallographic orientations relative to the direction of the external load, therefore the resolved shear stresses acting on the various slip systems also change from grain to grain. In the grains with favorable orientations (i.e., having slip systems with large Schmid factors), the resolved shear stresses are high and the plastic deformation starts by the activation of one or more slip systems. In unfavorably oriented grains, the shear stresses are not enough to initiate dislocation glide, therefore these grains deform only elastically. In the plastically deformed grains, dislocations are formed at the (FR) sources, and these dislocations are accumulated in pileups at the grain boundaries as depicted for grain A in Fig. 7.6. The elastic stresses of dislocations in the pileups are added to the shear stresses caused by the external load, thereby assisting the initiation of plastic deformation in the adjacent, elastically deformed grain having unfavorable orientation relative to the applied force (grain B in Fig. 7.6). The macroscopic plastic deformation sets in when all grains start to deform plastically. As the repulsive force between dislocations in pileups is inversely proportional to dislocation spacing, the smaller the grain size, the lower the number of dislocations in the pile-ups. As a consequence, when the grain size is reduced, the contribution of the stress field of dislocations to the initiation of plasticity in the adjacent grain is smaller; therefore larger applied stress is necessary for yielding the material, i.e., the yield strength will be higher. Thus, there is an inverse relationship between grain size and yield strength, as demonstrated by the Hall–Petch equation [24–26]:

where σ_Y is the yield strength, σ₀ is the friction stress, k_Y is the strengthening coefficient (a constant unique to each material), and d is the average grain diameter. For instance, the values of σ₀ and k_Y are ∼25 MPa and ∼5000 MPa nm^1/2, respectively, for Cu [26] as illustrated in Fig. 7.7.

where b is the magnitude of the Burgers vector and G is the shear modulus [33]. τ as a function of L^−1/2 for Cu is plotted in Fig. 7.8. When the length of the dislocation segment is large, the stress needed for the operation of a source increases more or less linearly with L^−1/2. However, when the length of an FR source is reduced below 10–15 nm, the slope of τ versus L^−1/2 plot decreases. Moreover, if L is smaller than 4 nm in Cu, the stress required for the operation of an FR source decreases with the reduction of its length. In materials with grain sizes between 20 and 100 nm, a considerable fraction of FR sources may have smaller lengths than 10–15 nm that can explain the deviation from the conventional Hall–Petch behavior. Additionally, the strong reduction of the numbers of dislocations in pileups with decreasing grain size yields a decrease of the exponent of d in Eq. (7.6) from −1/2 to −1 which may cause the observed reduction of k_Y in the Hall–Petch plot [16]. The value of −1 for the Hall–Petch exponent is also predicted by the model in which two dislocations locate within one grain with the distance of d/3 between them, where d is the grain diameter [34].

where k is the Boltzmann's constant, T is the absolute temperature, Ω is the atomic volume, δ is the grain boundary width, and D_GB is the grain boundary diffusion coefficient. Assuming Ω = 1.3 × 10⁻²⁹ m³, δ = 1 nm, D_GB = 4.7 × 10⁻²⁰ m²/s at RT, and ε˙=10−4 for Cu with d = 20 nm, the stress required for Coble creep is 112 MPa as obtained from Eq. (7.8), which is much less than the calculated stress needed for dislocation glide (∼800–900 MPa). The easy occurrence of Coble creep suggests its significant contribution to plasticity at RT for grain sizes less than 20 nm. Moreover, Eq. (7.8) predicts that when the grain boundary sliding becomes the main deformation mechanism, the yield strength will decrease with decreasing grain size, which corresponds to a considerable fraction of experimental results. However, Coble creep cannot be the sole deformation mechanism in the inverse Hall–Petch behavior as the experimental yield strength values do not fall so steeply with decreasing grain size as the trend, σ ∼ d³, predicted by Eq. (7.8) [36]. For instance, for inert gas–condensed Cu the inverse Hall–Petch effect was described as

where A ≈ 1290 MPa and k′ ≈ 2170 MPa nm^1/2 [27]. Most probably, intragranular dislocation motion as a complementary deformation mechanism also occurs to sustain the integrity of the material as the sliding of grains without changing their shape would yield voids between them. It should be also noted that the voids, micro- or nanocracks, and poorly bonded interfaces in nanomaterials processed by powder consolidation (e.g., inert gas condensation) may also yield a similar strength degradation as the inverse Hall–Petch behavior [23]. Although, the experimental results obtained on different nanomaterials do not prove convincingly the general existence of the inverse Hall–Petch effect, molecular dynamic simulations predicts this behavior for very small grain sizes [37]. It is noted, however, that the results of molecular dynamic simulations are valid only for very high strain rates (10⁸–10⁹ s⁻¹). Moreover, the change of the deformation mechanism inside the grains when their sizes decrease below ∼20 nm may also contribute to the inverse Hall–Petch behavior. This is illustrated in Fig. 7.9, where the shear stresses required for the emission of full lattice (τ_L), leading partial (τ_P), trailing partial (τ_trail), twinning partial (τ_twin), and shrinking partial (τ_shrink) dislocations from grain boundaries are plotted as a function of grain size in Cu. These stresses are calculated according to Eqs. (7.1)–(7.5) with assuming G = 47 GPa, a = 0.361 nm, ν = 0.34, γ = 62 mJ/m², and β = 30 degrees. It seems that below ∼30 nm, twinning becomes a considerable deformation mechanism. For twin nucleation, first an initial SF should be formed at the stress corresponding to point N. Then, twinning occurs at the stress corresponding to point M that is much lower than that at point N. When the grain size falls below ∼20 nm (point L in Fig. 7.9), partial dislocations are emitted from the grain boundaries instead of full dislocations since the former mechanism requires smaller stress. In nanocrystalline materials, the grains have a size distribution. By decreasing the average grain size, more and more grains in the distribution become so small that their interiors deform rather by twinning which requires smaller stress than for dislocation glide in the large nanograins. Therefore, the macroscopic yield strength decreases with decreasing the average grain size in accordance with the inverse Hall–Petch behavior.

For UFG fcc metals processed by ECAP the dislocation density and the yield strength show similar developments as a function of strain and both quantities saturate approximately at the same strain [16,42]. This suggests that it is worthwhile studying the relationship between these quantities. The dislocation density determined by X-ray line profile analysis and the corresponding yield strength measured by mechanical tests for different fcc metals and alloys are listed in Table 7.1. The relationship between the dislocation density (ρ) and yield strength (σ_Y) for plastically deformed metals is generally characterized by the Taylor equation:

where σ₀ is the friction stress, α is a constant depending on the arrangement of dislocations, and M^T is the Taylor factor. Since the value of αM^T is of the order of 1, it is possible to check the validity of Eq. (7.10) by plotting the value of (σ_Y − σ₀) against the product of Gbρ^1/2 for different fcc metals and solid solution alloys processed by SPD, as shown in Fig. 7.11. The good correlation between these two quantities in Fig. 7.11 demonstrates that in UFG fcc metals and solid solutions processed by SPD the yield strength is essentially determined by the interactions between dislocations. This observation can be explained by the fact that in materials processed by SPD at low homologous temperatures, the grain refinement occurs mainly by the arrangement of dislocations into subgrain boundaries and/or dislocation walls. Consequently, large fraction of boundaries consists of dislocations and the effect of subgrain/grain boundaries on dislocation motion can be regarded as the interaction between dislocations [43]. Concerning the solid solution strengthening, the solute atoms may increase the strength in two ways: (1) directly by pinning the dislocations and impeding their motion in a solute–dislocation interaction and (2) indirectly by hindering the annihilation of dislocations during deformation, leading to an increase in the dislocation density in a dislocation–dislocation interaction. The direct and indirect effects of solute atoms are taken into account in the friction stress, σ₀, and the dislocation term in the Taylor equation. For severely deformed metals where the dislocation density is relatively high, the latter term is dominant so that the hardening effect of the solute atoms is manifested basically in the higher dislocation density.

where x is the average size of dispersoids and L is the average distance between them. For instance, in supersaturated Al–4.8Zn–1.2Mg–0.14Zr and Al–5.7Zn–1.9Mg–0.35Cu (wt.%) alloys processed by eight passes of ECAP at 200 ​°C, stable incoherent MgZn₂ precipitates (η-phase particles) form [48]. The different contributions to yield strength are calculated from Eq. (7.11) using the values of L and x obtained from transmission electron microscopy (TEM) images and the dislocation density determined by X-ray line profile analysis. The average size of the precipitates in the Al–4.8Zn–1.2Mg–0.14Zr and Al–5.7Zn–1.9Mg–0.35Cu alloys are 30 and 20 nm, respectively [48]. The average distance, L, between the precipitates is approximately 120 nm in the Al–4.8Zn–1.2Mg–0.14Zr alloy and 80 nm in the Al–5.7Zn–1.9Mg–0.35Cu alloy. The value of σ₀ was taken as 20 MPa measured for pure Al [52] because the lattice parameter of the matrix obtained from the X-ray diffractograms agrees with the value for pure Al within the experimental error. The strength contributions originating from the dislocation density given by the second term in Eq. (7.11) are 133 and 137 MPa for Al–4.8Zn–1.2Mg–0.14Zr and Al–5.7Zn–1.9Mg–0.35Cu, respectively, as the dislocation densities in the two matrices are very close to each other. The third term in Eq. (7.11) related to the dislocation–precipitation interaction gives 156 and 214 MPa for Al–4.8Zn–1.2Mg–0.14Zr and Al–5.7Zn–1.9Mg–0.35Cu, respectively, where the difference between the two values is due to the different size and dispersion of precipitates [48]. Thus, the smaller particles with a more dense distribution lead to a higher strength for the Al–5.7Zn–1.9Mg–0.35Cu alloy. Finally, the sum of the three components leads to estimated total strengths of 309 and 371 MPa, which are in good agreement with the values of 290 and 350 MPa determined by mechanical testing for the Al–4.8Zn–1.2Mg–0.14Zr alloy and the Al–5.7Zn–1.9Mg–0.35Cu alloy, respectively (see Table 7.1).

Beside the yield strength, the ductility of UFG and nanomaterials is also an important parameter describing the mechanical behavior. The ductility is usually characterized by the maximum elongation or the maximum true strain (ε_max) achieved until inhomogeneous deformation (necking) sets in during tension. After necking, the deformation is localized mainly to the neck, and due to the decrease of the cross section at the neck, the force required for further straining decreases, i.e., the deformation becomes unstable (it does not stop immediately when the force is reduced). As a consequence of inhomogeneous deformation after necking, the true stress (σ) and true strain (ε) vary along the longitudinal axis of the specimen; therefore the following usual formulas for the calculation of σ and ε can only be applied before necking:

where l and l₀ are the actual and initial lengths of the sample, respectively, F is the applied force, A and A₀ are the actual and initial cross sections of the sample, respectively. Eq. (7.12b) reflects the unchanged volume during plastic deformation. The deformation is stable only in the homogeneous regime, i.e., when the first derivative of the force over the sample length, dFdl, is larger than zero. Using Eqs. (7.12a–c), this criterion can be expressed by the true stress and true strain as

where m is the strain rate sensitivity parameter defined as

where k is the Boltzmann constant and T is the absolute temperature of deformation. The activation volume decreases with decreasing grain size in fcc metals [57,58]. The rate-controlling mechanism for conventional polycrystalline fcc metals is the glide movement of a dislocation through the dislocation forest. The activation area is equal to the spacing of forest dislocations multiplied by b and the activation volume is therefore equal to 100–1000 × b³ for dislocation mechanism. For UFG fcc materials, the increase of the dislocation density yields a decrease in the average dislocation spacing, but this trend is almost completely compensated by the increase of flow stress, thereby m only slightly increases. In the nanocrystalline regime, V∗ decreases drastically to ∼10 × b³ as a consequence of the change in the rate-controlling mechanism [35]. Dislocation-boundary interactions or grain boundary sliding [59] are two possible mechanisms operating in nanomaterials which may cause the reduction in V∗ and the corresponding increase in m (see Fig. 7.16a).

In Chapter 5, the defect structures in UFG Ni samples sintered from NPs under different consolidation conditions are reviewed. In this section, the mechanical behaviors of these materials are compared. The samples are consolidated from NPs with the particle size of 100 or 50 nm. The sintering is carried out by hot isostatic pressing (HIP) or spark plasma sintering (SPS) method. The details of the consolidation procedures are given in Chapter 5 and also listed in Table 7.3. Both the NPs and the consolidated materials contain NiO. The amount of NiO is characterized by the integrated intensity ratio of the NiO and Ni peaks on the X-ray diffractograms at 2Θ = 37.4 and 44.6 degrees, respectively. The intensity ratios for the initial powders having particle sizes of 100 and 50 nm are 0.5% and 1.2%, respectively, which does not change during HIP processing as a result of the careful isolation of the sample from air. On the other hand, in SPS processing, the capsule containing the powder was broken in air that yields an increase of NiO content from 0.5% to 0.9% and from 1.2% to 1.5% for the powders with particle sizes of 100 and 50 nm, respectively [83]. When the consolidation by SPS is performed in Ar, the amount of NiO does not increase significantly (see Table 7.3). The grain size and the defect densities are listed in Table 5.2 in Chapter 5 for the samples sintered by SPS in air and by HIP in Ar.

In Chapter 5, the grain size distribution and the defect structure were studied in Ni materials consolidated from blends of NP and coarse powder (CP). In this section, the mechanical properties of these materials are studied. In addition to the bulk samples processed from 100% NP and 100% CP powders (referred to as samples UFG and CG, respectively), two additional specimens with different volume fractions of powders NP and CP were prepared. The volume fractions of CG and UFG components are 61% and 39% for sample A, respectively, while these values are 39% and 61% for sample B [87]. Fig. 7.26a shows the resulting true stress–true strain curves for samples CG, A, B, and UFG. The yield strength was plotted as a function of the UFG volume fraction in Fig. 7.26b. It can be seen that the yield strength increases with increasing the amount of the UFG content of the samples. However, the yield strength values determined for samples A and B can be found under the straight line connecting the datum points of the fully CG and UFG samples. This means that the yield strength of the composite-like materials cannot be obtained by a linear interpolation from the strength values of the fully CG and UFG specimens [87].

The annihilation of dislocations during SPD is controlled by thermally activated mechanisms, such as vacancy migration, therefore the decrease of processing temperature and the increase of strain rate are expected to yield a higher dislocation density, thereby enhancing the strength of plastically deformed metals. The combined influence of these two effects is usually measured by the Zener–Hollomon parameter (Z), given as Z=ε˙eQkT, where ε˙ is the strain rate given in the unit s⁻¹, Q is the activation energy of the mechanism controlling the rate of deformation, k is the Boltzmann constant, and T is the absolute temperature. Indeed, experiments showed that the grain size decreased and the dislocation density increased with increasing the value of Z, resulting in a higher yield strength and a reduced ductility (e.g., for Cu see Ref. [91]). As a consequence, high strain rate deformation methods are expected to be very effective in processing UFG metallic materials with very high strength. In addition, plastic deformation at high strain rates naturally occurs in projectile impact and explosion processes. Therefore, there is a demand for a better understanding of the deformation behavior of UFG and nanocrystalline materials at high strain rates. Different dynamic plastic deformation techniques, such as drop mass bench [92], air hammer [93], split Hopkinson pressure bar, and direct impact Hopkinson pressure bar (DIHPB) [94,95] tests, have been developed for studying the mechanical performance of materials at strain rates between 10³ and 10⁵ s⁻¹. It was found that these techniques are capable for refining the grain structure to the UFG regime in very short times (typically in a millisecond) [92,96].

where τ is the shear stress acting on the dislocation in the glide plane in the direction of the Burgers vector, b is the magnitude of the Burgers vector, and B is the drag coefficient [100]. The drag coefficient B can be given as [100]

where T is the absolute temperature of deformation and the coefficients B_e and B_p are associated with electron and phonon drags, respectively. The B_p is about one order of magnitude larger than B_e, therefore the former one gives the main contribution to B at RT. The average velocity of dislocations can be related to the strain rate as

where ρ_m is the mobile dislocation density and M is the Taylor factor. Viscous drag is the dominant deformation mechanism if the dislocation velocity is higher than 0.1 × c_s, where c_s is the transversal sound velocity (the upper limit of dislocation velocity which is 2370 m/s for Zn). For UFG-Zn the steep increase of the flow stress was not observed even at very high strain rates (see Fig. 7.30), suggesting that viscous drag effects are negligible for this material. The lack of dislocation drag phenomenon in UFG-Zn can be explained by the relatively large dislocation density (∼10¹⁴ m⁻² as shown in Fig. 7.31) and the small grain size (∼200 nm), as both dislocations and grain boundaries hinder the acceleration of dislocations to very high velocities which is necessary for the occurrence of viscous drag effects [98]. The velocity of dislocations in UFG-Zn is about 1 m/s (∼10⁻³ × c_s) for the highest applied strain rate (∼10⁴ s⁻¹), as can be estimated from Eq. (7.19) with the assumption that the mobile dislocation density has the same order of magnitude as the total dislocation density (∼10¹⁴ m⁻²). Due to the high dislocation density in UFG-Zn, the high strain rate can be achieved by a slower motion of dislocations. The observed difference between the stress and strain-rate characteristics of UFG-Zn and CG-Zn at very high strain rates is in accordance with the results obtained for an electrodeposited nanostructured copper [101]. It is noted that during quasistatic straining of CG-Zn at RT in addition to dislocation slip, twinning also operates as it has been shown in previous works [102,103]. For UFG-Zn, considerable twinning activity was not observed at either low or high strain rate [98].

where ρ_mass is the mass density and c_p is the specific heat. The plastic work can be determined from the stress–strain data, and using this value the calculated temperature rise is about 100–200 K at the highest applied strain rate (∼10⁴ s⁻¹). However, due to the high heat conductivity of the tungsten bars used as projectile and transmitted bars in the DIHPB set up, this temperature decreased very quickly to RT. Namely, this cooling period can be estimated as [98]:

where the heat reduction of ΔQ equals to 0.9 W_p, k is the thermal conductivity, x is the thickness, and A is the area of the sample surface contacted with the bars at the end of compression. The cooling period obtained from Eq. (7.21) is about 5 × 10⁻² s which is much larger than the duration of compression (∼10⁻⁴ s). Therefore, first the temperature increases by about 100–200 K during DIHPB test, then it decreases to RT very quickly. The significant increase of the sample temperature most probably yields recovery of the microstructure which may be the reason of the reduction of the dislocation density and the increase of the crystallite size with increasing strain rate above ∼10³ s⁻¹ (see Fig. 7.31). These observations are also valid for other UFG materials.