Hall-Petch analysis for temperature and strain rate dependent deformation of polycrystalline lead

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УДК 621. 891
Влияние температуры и скорости деформации поликристаллического свинца на параметры уравнения Холла-Петча
В. Е. Панин, R.W. Armstrong1
Институт физики прочности и материаловедения СО РАН, Томск, 634 055, Россия 1 Center for Engineering Concepts Development, Department of Mechanical Engineering, University of Maryland,
College Park, MD 20 742, USA
Проведен анализ экспериментальных значений параметра kz уравнения Холла-Петча, полученных при растяжении образцов поликристаллического свинца в широком интервале температур T с двумя скоростями нагружения. В среднем интервале T получено хорошее согласие теоретических и экспериментальных зависимостей k = f (T). Занижение экспериментальных значений ke по сравнению с рассчитанными по теории плоских скоплений дислокаций при очень низких T связывается с высокой кривизной границ зерен, которая вызывает очень высокую низкотемпературную пластичность поликристаллов и генерацию в них полосовых структур.
Ключевые слова: дислокационный расчет ke уравнения Холла-Петча, поликристаллический свитец, согласие экспериментальных и теоретических зависимостей k2 = f (T), отсутствие согласия при очень низких T, роль кривизны кристаллической решетки и развития полосовых структур
Hall-Petch analysis for temperature and strain rate dependent deformation
of polycrystalline lead
V.E. Panin and R.W. Armstrong1
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634 055, Russia 1 Center for Engineering Concepts Development, Department of Mechanical Engineering, University of Maryland,
College Park, MD 20 742, USA
A dislocation pile-up analysis of the Hall-Petch kz for the tensile deformation of polycrystalline lead over a wide range of temperature T and at two strain rates has been made. The predicted and experimental dependencies of the Hall-Petch ke = f (T) are in good agreement. Lower than predicted kE values at very low temperatures are attributed to the high curvature of grain boundaries which experience high localized plasticity and consequent shear banding.
Keywords: dislocation pile-up analysis of the Hall-Petch ke for polycrystalline lead, agreement between theoretical and experimental dependencies ke = f (T), the agreement lack at very low T, the role of crystal lattice curvature and shear bands development
1. Introduction
A dislocation pile-up analysis has been given for the Hall-Petch (H-P) kE of essentially pure face-centered cubic (fcc) metals-copper, nickel, and aluminum-being controlled by cross-slip [1]. Agreement was obtained for the three metals by substitution at low proof strains of the cross-slip shear stress Tm for the concentrated shear stress Tc in the H-P kf relationship [2]
In Eq. (1), mT and mS are the Taylor and Sachs orientation factors, respectively, G is the shear modulus, b is the dislocation Burgers vector, and a ~ 0.8 is an average screw or edge dislocation factor. The recent results on polycrys-talline lead reported by Panin, Moiseenko and Elsukova [3] for k0. 002 and its temperature dependence are shown here to agree with the same model description.
2. Experimental measurements
Panin et al. [3] measured for polycrystal lead the temperature dependence at two strain rates of both H-P con-
© Panin V.E., Armstrong R.W., 2015
Table 1
Comparison of predicted and experimental values of Hall-Petch ke
Metal Shear modulus, GPa Burgers vector, nm Tjjj, MPa S '- Theoretical Experimental
Al 25.0 0. 290 4.5 1. 20 1. 40 [9]
Ni 62.0 0. 250 17.0 3. 30 4. 90 [10]
Cu 30.5 0. 256 29.0 3. 10 5. 00 [11]
Pb 4.8 0. 350 1.6 0. 33 0. 43 [3]
stants, g08 and k8, in the reciprocal square root of grain size l1/2 relationship
G?=Go?+ k? l-½. (2)
The measurements covered initial true proof strains of 8 = 0. 002 and continuing onward to 8 = 0. 05. Temperatures in the range between 78 and 435 K were included in separate measurements at strain rates of 0. 0017 or 0. 035 s-1. Metallographic evidence was presented of grain-boundary shearing associated with the measurements that were interpreted to provide evidence of curvature and '-rotational wave flows'- near to both low and high angle grain boundaries in the polycrystal microstructures. At the lower strain rate and at larger strains, the H-P k8 values were shown to decrease (with increasing strain), very much more so from higher values measured at the lower temperatures. At higher strain rate, the k8 measurements remained relatively constant except for a small decrease at the lowest temperatures of testing.
3. Hall-Petch evaluation of k8
The suggestion that the initial proof strain value of k8 for aluminum was controlled by cross-slip was invoked to explain its relatively low value [1]. Later connection with cross-slip stresses was employed to explain essentially the same value of k8 being obtained for nickel and copper materials despite the shear modulus of nickel being approximately two times larger than that for copper [4]. The cross-slip shear stress for nickel is approximately one-half of that for copper and so the product of shear modulus and substituted cross-slip shear stress in Eq. (1) is about the same value for the two metals. The ratio of cross-slip shear stresses for nickel and copper matches the reverse ratio of stacking fault energies [5].
Table 1 provides evaluations of k8 at ambient temperature for all three metals. The shear moduli were determined to apply for the {111} slip plane from the measurements reported by Kelly and Groves [6]. The Tm measurements are average values determined for ~300 K test results from the compilation reported by Bell [7]. Higher measurements of Tm from later nickel and copper single crystal deformation reports were employed in a more recent comparison of the k8 values thus bringing the theoretical values closer to
the experimental ones [4]. Alternatively, lower experimental values of k8 have been reported in a recent review also, at least for nickel [8]. The same sources of G and Tm have been employed to produce the favorable comparison of k8 values that are listed in Table 1 for lead.
4. The temperature dependence of k8
In accordance with Eq. (1), the temperature and strain rate dependencies of k^ for copper [1, 12] have been matched also with the same dependencies observed for xm expressed in the form
tiii =- exp (-?t)
In Eq. (3), B and P are experimental constants that are accounted for on a thermal activation strain rate analysis [12].
Figure 1 shows the same comparison for the recent measurements reported for lead at the two strain rates. Such strain rate dependence enters through the parameter P in the relationship [12]
? = ?o-?iln
Fig. 1. Temperature dependence of k8 for lead from the reported measurements [3]. 8 = 0. 0017 (1), 0. 035 s-1 (2)
Fig. 2. Laser pumping mechanism of a dislocation generation in a 3D grain by ion cluster AB in a grain boundary [20]
In Eq. (4), the experimental constants P0 and P1 taken together with T in Eq. (3) demonstrate a coupled dependence for thermally-activated dislocation mobility in the thermal activation strain rate analysis. From the values of P for the two straight line dependencies drawn in Fig. 1, a value of P0 = 0. 0026 K-1 and P1 = 0. 0024 K-1 are obtained. The P0 value compares very well, for example, with value of P0 = = 0. 0032 K-1 for the temperature dependence of ke2 for copper [13] while the value of P1 for lead is much greater than the copper strain rate dependent value of P1 = 0. 0002 K-1 determined previously for the similar dependencies both of ke2 and Tjjj. Extrapolation of the linear dependencies in Fig. 1 to T = 0 leads to a same relatively high value of ke ~ ~ 6 MPa -mm½ at T = 0 K. The lower experimental values of k at the lower temperatures fall significantly below pre-
diction and, hence, raise an issue of possible deformation-induced localized heating effect.
5. Plastic instability at very low and high temperatures
Many papers have been focused on the simulation of the Hall-Petch equation using the dislocation-based theory [13]. Table 1 shows good agreement between the predicted and experimental values of Hall-Petch ke. At very low and rather high temperatures as well as under plastic deformation of micropillars, the Hall-Petch equation has the changed form g = g0e+ke1[3, 14−17]. Recent description has been given of the combined effect of both micropillar and material grain size influences relative to Hall-Petch influence for nanopolycrystal copper material [18]. A theoretical transition to inverse grain size rather than inverse square root of grain size has been predicted for nanopolycrystalline materials with relatively small ke values [19].
The form of the Hall-Petch equation depends on the state of grain boundaries where flows of structural transformations locate [3]. Providing these flows induce dislocations in 3D grains, which are responsible for self-consistency of deformation of misoriented grains in a polycrys-talline aggregate, the Hall-Petch equation acquires the form a = aoe + ke1Figure 2 illustrates the scheme of dislocation formation [20], which is based on the very important result obtained for a one-dimensional crystal [21]. According to [21], with a unique minimum of the pair interaction potential of atoms, an increase in local interatomic spacing gives rise to bifurcation minima of the potential of the particle system. This point is of fundamental importance for the validation of the mechanism of formation of deformation defects in a deformable solid. It is known that the interface between heterogeneous/dissimilar media demonstrates a periodic modulation in tensile and compressive normal stresses [22]. Flows of structural transformations at grain boundaries form clusters 1 (Fig. 2) of positive ions in zones of tensile normal stresses. An excess positive charge
Fig. 3. Extrusion, x210 (a) and delamination, X1850 (b) of the material in near-boundary regions of the lead polycrystal in tension at T = = 300 K, e = 15%, v = 2.1%/min [3]
73 173 273 373 473 T, K
Fig. 4. Temperature dependence of elongation 5 at v2 = 2.1%/min for tension of polycrystals Pb (1), Pb + 0. 01% As (2), Pb + 0. 24% Sb (3), Pb + 0. 03% Te (4), Pb + 0. 03% Cu (5), and Pb + 1.9% Sn (6)
in grain-boundary clusters should be screened by electron gas of layers adjacent to 3D crystalline grains. The interatomic spacing in zones 2 given in Fig. 2 increases, resulting in the formation of interstitial bifurcation vacancies. The number of degrees of freedom grows sharply in near-boundary regions of the polycrystal, which alters completely the structural state of grain boundaries. This is apparent from the formation of nonequilibrium interstitial bifurcation vacancies, widening of grain boundaries (Fig. 3, a), breakdown of the crystal structure (Fig. 3, b), a quasi-viscous plastic flow, and increase in low-temperature plasticity (Fig. 4).
It is reasonable that the Hall-Petch k8 is considerably reduced at a new structural state of grain boundaries (Fig. 1). The modulation frequency of tensile normal stresses decreases at grain boundaries, and there appear large clusters of positive ions in zones of tensile normal stresses, which cause high curvature in near-boundary regions of 3D grains, presenting sites of shear band nucleation.
Thus, the scheme of self-consistency of plastic deformation of misoriented grains in a polycrystalline aggregate should become qualitative different due to high lattice curvature arising at grain boundaries and near-boundary regions. Under these conditions, the collective self-consistency of plastic deformation of misoriented grains through dislocations in adjacent grains disappears. Consequently, the Hall-Petch equation g = g08 + k81& quot-12 is replaced by the equation a = a08+k8l1. The exponent of l1 can even be much less than one [15].
A similar effect is observed at rather high temperatures (Fig. 1) when grain-boundary sliding develops intensively and induces high lattice curvature in near-boundary regions as well as grain fragmentation (Fig. 5). Under such conditions, plastic flow of each anisotropic grain proceeds autonomously and is accommodated by its own intragranular mechanisms of deformation. This makes the dislocation pile-
Fig. 5. Autonomous plastic deformation and severe fragmentation of separate polycrystalline grains of Pb + 1.9% Sn alloy at T= 493 K, 8 = 10%, X200
up analysis of the Hall-Petch k8 inadequate for tensile deformation of such polycrystals.
Introduction of slightly soluble impurities at grain boundaries results generally in an increase in k8 under normal conditions. Figure 6 presents temperature dependences of k8 = f (T) for Pb and Pb + 1.9% Sn alloy, which approaches the solubility limit of tin in lead [3]. The dependences are obtained at 8 = 0.2% and high strain rate in tension v3 = 54%/min. With reference to Fig. 6 it is seen that at T & gt- 150 K the k8 value is higher for the alloy than for lead, which seems quite natural due to strengthening of lead by alloying elements. However, at T & lt- 150 K there is a rapid decrease in k8, even much below the lead value. It is of interest that under these tension conditions the alloy plasticity 5 is abnormally high (above 50%) and far exceeds the lead plasticity 5 (Fig. 4). The same effect has been previously observed in association with thermal instability [23]- and, is in line also with prediction of shear band suscepti-
0. 9-
& lt-N
1 °-6& quot-
& lt-3
0. 3-
o. o-.-.-.-. -
100 200 300 400
T, K
Fig. 6. Temperature dependence of the k8 parameter for Pb (1) and Pb + 1.9% Sn alloy (2) at 8 = 0.2% and v = 54%/min
bility being gauged by the ratio of ke and K, the thermal conductivity [24], that is kjK*. Undoubtedly, the low temperature value of K is appreciably higher for the Pb + + 1. 9% Sn alloy.
The considered anomalies of the low-temperature behavior of the Hall-Petch ke are indicative of the importance of the state of polycrystalline grain boundaries and processes in them, which should be studied in combination with the analysis of the dislocation structure of intragranular slip.
6. Discussion
The multilevel approach of physical mesomechanics to describing physical plasticity and strength of polycrystals takes a solid as consisting of two subsystems, namely, a 3D crystalline grain subsystem and a 2D planar subsystem (grain boundaries and surface layers). The planar subsystem has no translational invariance and is characterized by a low Gibbs thermodynamic potential- first flows of structural transformations in a loaded solid develop in its planar subsystem [25, 26]. The model represented in Fig. 2 is universal for deformation defects of all types: dislocations, discli-nations, shear bands, and cracks. This model operates by the mechanism of laser pumping. Flows of structural transformations in grain boundaries form clusters of positive ions in zones of tensile normal stresses, which cause local lattice curvature in layers adjacent to 3D grains. An excess positive charge of an ion cluster is screened by electron gas from zone 2 shown in Fig. 2- increased interatomic spac-ings and allowed bifurcation states in interstices are observed in this zone [20, 21]. Positive ions of clusters pass to bifurcation structural states of interstices 2, thus forming a dislocation nucleus. In the case of a large ion cluster and numerous interstitial bifurcation states in this zone, discli-nations, shear bands, and viscous plastic flow may occur in near-boundary regions of the material with high concentration of interstitial bifurcation vacancies. Under these conditions the Hall-Petch ke decreases and low-temperature polycrystalline plasticity increases abnormally.
In alloys with grain boundaries highly strengthened by slightly soluble elements, high lattice curvature arising in near-boundary regions leads to an increased concentration of athermal interstitial bifurcation vacancies. The solubility of alloying elements in near-boundary regions increases while their concentration at grain boundaries decreases. Under these conditions the alloy ke reduces, being even below the pure lead value (Fig. 6)
The multilevel approach of physical mesomechanics provides a satisfactory explanation of the low-temperature anomaly of ke in the studied lead-based materials, which enhances with decreasing strain rate and increasing strain [3]. A higher strain rate of the polycrystal affects an anomalous decrease in ke at very low temperatures [3]. Deformation defects generate slowly in zones of high curvature of
grain boundaries, thus keeping high lattice curvature in these zones. This increases the concentration of interstitial bifurcation vacancies responsible for an anomalous reduction in ke at low deformation temperatures. An increase in local lattice curvature with growing strain of the polycrystal is a common knowledge [25].
The given results are in good agreement with the known investigations of adiabatic shear bands formed in dynamically loaded materials [24, 26]. Dynamic loading causes the formation of numerous zones of local lattice curvature- dislocations cannot move, thus giving way to intensive propagation of adiabatic shear bands.
7. Conclusion
A dislocation pile-up analysis of the Hall-Petch ke measured for the tensile deformation of polycrystalline lead over a wide range of temperatures and at two strain rates has been made. The average values of the predicted Hall-Petch ke and experimental one are in good agreement.
Temperature dependences of experimental ke2 for lead at two strain rates are linear for the middle temperature range. However, the experimental values of ke2 turned out to be anomalously low especially at 77 K and near 400 K. This is evident from their high deviation from the linear temperature dependence k2 = f (T).
Low experimental values of the Hall-Petch ke for lead polycrystals and lead-based alloys with slightly soluble impurities at very low and very high temperatures are explained using the mesomechanics concepts [20, 25] that are related to interstitial bifurcation structural states appearing in zones of local lattice curvature at grain boundaries, which induce shear bands in grains as well as provide viscous grain-boundary sliding and high low-temperature plasticity of the material.
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Поступила в редакцию 24. 04. 2015 г.
CeedeHun 06 aemopax
Victor E. Panin, Dr. Sc. (Phys. -Math.), Prof., Head of Laboratory, ISPMS SB RAS, paninve@ispms. tsc. ru Ron Armstrong, Prof., University of Maryland, USA, rona@umd. edu

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