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The effect of scale levels of rotational plastic deformation modes on the strain resistance of polycrystals
T.F. Elsukova and V.E. Panin / Physical Mesomechanics 13 12 (2010) 62–69
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The effect of scale levels of rotational plastic deformation modes on the strain resistance of polycrystals T.F. Elsukova and V.E. Panin* Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia Experiments on polycrystalline lead alloys with different states of grain volumes and boundaries under tension in a wide range of test temperatures have been performed. The qualitative and quantitative characteristics of the grain size dependence of the strain resistance of the examined materials are shown to be substantially affected by the development of rotational plastic deformation modes at different scale levels. This effect must be accounted for in the interpretation of the parameters of the HallPetch equation in the framework of the multiscale approach employed in physical mesomechanics. Keywords: scale levels of rotational deformation modes of polycrystals, parameters of the HallPetch equation
1. Introduction Physical mesomechanics treats grain boundaries in a deformed polycrystal as a separate mesoscopic structural level of deformation that plays an important functional role in the propagation of plastic shear in a heterogeneous medium [1, 2]. The shear develops in a grain to give rise to a rotational deformation mode at a higher structural level due to grain-boundary defect fluxes along grain boundaries. This is evident as grain-boundary sliding [3]. The constrained rotations of grains as a unity are responsible for induced stress concentrators at grain boundaries and for accommodation deformation regions near grain boundaries [4]. The role of the rotations can be determined by the HallPetch equation used in physical mesomechanics to describe the accommodation of adjacent grains in a polycrystalline aggregate. Using the concepts of physical mesomechanics [1, 2], an analysis of the HallPetch equation, V V0 Kd 1 2, was performed for high-purity lead polycrystals subjected to tension in a wide range of test temperatures [5]. The experimental conditions enabled qualitative changes in the strain mechanisms operative at mesoscopic structural levels to be produced. The special features of the temperature dependence of plasticity and the accommodation grain-
* Corresponding author Prof. Victor E. Panin, e-mail: [email protected] Copyright © 2010 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2010.03.008
boundary mechanisms were found to be closely correlated. The mesoscopic strain mechanisms grain-boundary sliding and migration, mesoscale fragmentation of grains, formation of severe local curvature regions, etc. are shown to significantly affect the shape of the strain resistance grain size curve. Among these mechanisms, grain-boundary sliding is of special importance because it determines the form of the HallPetch equation and affects the qualitative characteristics of the dependence of its parameters on applied loading. The present paper deals with further analysis of the Hall Petch equation in terms of structural-scale levels of deformation of solids. The effect of scale levels of rotational plastic deformation modes on the strain resistance of polycrystals and parameters of the HallPetch equation is estimated in lead alloys with essentially different grain volumes and boundaries as an example. 2. Test materials and experimental technique The test materials were binary alloys based on highpurity lead S000 (Russian classification). To provide a purposeful change in the states of grain volumes and boundaries, use was made of two types of elements slightly soluble in lead: horophylic additions (As and Sn) significantly segregated at grain boundaries and a horophobic addition (Te). The latter is not prone to segregation at grain boundaries and is concentrated on lattice defects. The concentration of the additions was taken to be within solid so-
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lution, including the room-temperature solubility limit (1.9, 0.01 and 0.03 wt. % for Sn, As and Te, respectively). By increasing the test temperature the degree of departure from the solubility limit can be varied and in doing so the state of the grain boundaries in the alloys will be changed. To exclude an uncontrollable effect of natural impurities, 99.999% pure lead and alloying elements were employed. Flat specimens with cross-sectional area of 4.8 mm2 and gage length of 20 mm were subjected to tension at a rate of 54.0 %/min at 77548 K, which corresponds to (0.10.9) Tm for the examined materials. The stress-strain curves were plotted by averaging the data for 57 specimens. The resulting stress-strain curves were used to make plots of applied stress V versus d 1 2 for different degrees of strain (H = 0.2, 1, 2, 3 and 5 %) at different test temperatures Tt . The ability of the polycrystals to transfer strain from one grain to another, K, and the strain resistance in a grain volume, V0 , were estimated by the least-squares technique, extrapolating the straight lines for V = f (d ) to d 1 2 = 0 and d 1 0. The specimen surface was electropolished and examined by optical, interference and scanning electron microscopy followed by a quantitative assessment of grain-boundary migration, fragmentation, and single and multiple slip.
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3. Results and discussion Consider the effect of As and Sn additions on the behavior of the parameters V 0 and K as a function of the degree of strain and test temperature. The additions are horophylic relative to Pb and are substantially segregated at grain boundaries. Figure 1 shows the temperature dependence of V 0 and K for a Pb 0.01 % As alloy. When these plots are compared with those constructed for pure lead (Fig. 2), it is apparent that for H d 2 %, the curves V 0 (T ) are identical, whereas for higher degrees of strain, the curves for the lead alloy lie somewhat higher. The As addition gives rise to a number of special features of the K(T ) plot, however. First, this dependence is more pronounced for the examined alloy than for the pure metal: the absolute values of K for PbAs increase by a factor of 310 at low test temperatures and show a modest increase at elevated temperatures. Second, the peak in the K(T ) curve for lead is observed at low temperatures only for the yield stress and does not occur in the alloy with H = = 0.2 %. Yet the dependence for PbAs does exhibit a peak at higher H. The higher is H, the more pronounced is the peak. Finally, at low and moderate temperatures, the temperature dependence of K vanishes in Pb for H = 5 %, whereas in PbAs, the dependence is as strong as for H < 5 %. A comparison of the K(T ) curves for the pure metal and alloy revealed other striking features of importance for our discussion relating to the effect of small alloying addi
a
a
b b
Fig. 1. Effect of the test temperature on the parameters V 0 (a) and K (b) of a Pb 0.01 % As alloy for H = 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
Fig. 2. Effect of the test temperature on the parameters V0 (a) and K (b) of pure lead. The degree of strain is 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
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tions on the strain resistance of polycrystals. As was shown in [6], addition of 0.01 % of As caused a dramatic increase in the yield stress for Pb at low and moderate temperatures. At high test temperatures, on the contrary, the yield stress was decreased. Inspection of Figs. 1 and 2 discloses that it is the parameter K which is responsible for the increase in the yield stress in this case, as the As addition makes no contribution to V0 for H d 2 %, with K being increased by four- or fivefold. The As effect appears to be due to the fact that As demonstrates high segregation ability relative to Pb grain boundaries [7] and in consequence is almost entirely concentrated at grain boundaries. This makes grain-boundary deformation difficult and favors a large increase in VGB during intracrystalline shear strains. Another horophylic addition used was tin for which the solubility in lead is 200 times higher than that of arsenic. The examined alloys contained 1.9 (the solubility limit at 300 K) or 0.4 % of Sn. The temperature dependence of V0 and K for the alloys under review is presented in Figs. 3 and 4. It is evident from the curves that for V0 at 77300 K, the strain resistance exhibits an anomalous temperature dependence evident as local maxima in the V0 (T ) plots at 300 K. For K(T ) in the low-temperature range, the effect of the degree of strain is reversed in comparison with Pb and PbAs: the higher is H, the higher is K. As is the case with
PbAs, the K(T ) curves show peaks. Notably, the lower is H, the more pronounced is the peak for K. At high test temperatures, the temperature variations of V 0 and K exhibit the same qualitative relationships as pure lead does. The aforementioned unusual temperature dependence of V0 for the PbSn alloys is responsible for an anomalous concentration dependence of the strain resistance of PbSn solid solutions. This is evident in the fact that an increase in the tin concentration from 0 to 1.9 % causes V to increase, then to decrease drastically and, ultimately, to increase again in a gradual manner (Fig. (5) [8]). The manifestation of the anomalous behavior of V0 (T ) essentially depends on the degree of strain and loading conditions. The anomaly is at its maximum at low test temperatures, high strain rates and low degrees of strain. The anomalous temperature dependence of V0 found in our experiments suggests that the unusual low-temperature effects observed in PbSn are due to the special features of the intracrystalline shear strains involved. The anomalies are superficially similar to those found for copper alloys in [9] where plots of V and V0 were built as nonmonotonic functions of concentration and temperature (Figs. 6 8). In copper alloys, however, the anomalies were not found at low degrees of strains, but they did occur where the strain was increased. The effect was traced to constrained multiple slip associated with the special features of the resulting dislocation structure [9].
a a
b
Fig. 3. Effect of the test temperature on the parameters V 0 (a) and K (b) of a Pb 0.4 % Sn alloy. The degree of strain is 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
b
Fig. 4. Effect of the test temperature on the parameters V 0 (a) and K (b) of a Pb 1.9 % Sn alloy. The degree of strain is 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
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c
b
Fig. 5. Strain resistance versus concentration of tin at Tt
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169 (a), 20 (b) and 180 qÑ: H = 0.2 (1), 1 (2), 3 (3) and 10 % (4)
In PbSn alloys, the anomaly is at its maximum for the yield stress. As H is increased, the anomalous dependence is smoothed out and then vanishes. Microscopic examination has revealed that the most distinct unusual mechanical behavior of PbSn is due to the preferred intracrystalline single slip, localization of accommodation processes (like quasi-viscous extrusion of the material) at grain boundaries (Fig. 9) and distortion of square cells of the reference grid, including grain boundaries, with relatively small alteration of the shape of the cells inside grains (Fig. 10). The latter circumstance is evidence of rotations of grains as a unity. The lack of grain-boundary sliding plays an important role in this case.
The data under consideration are demonstration that the anomalous temperature dependence of V 0 for PbSn is also associated with constrained multiple slip. In these conditions, single slip is provided by easier quasi-viscous flow in the grain-boundary regions of constrained deformation due to reduced shear stability of the crystal lattice caused by the alloying effect. This is attributable to a number of factors. First, in solid solutions, especially in the vicinity of the solubility limit, atomic displacements are so large that
a
b
Fig. 6. Strain resistance versus concentration of CuAl alloys under tension: H = 9 % and the grain size d = 0.21 (1), 0.05 (2), 0.025 (3) and 0.01 mm (4)
Fig. 7. Dependence of the parameters V 0 (a) and K (b) in the HallPetch equation on the concentration of solid solution of CuAl alloys under tension: H = 0.2 (1), 1 (2), 3 (3) and 7 % (4)
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a *
)
b * )
Fig. 8. Temperature dependence of the strain resistance of Cu + 17.3 at. % Al (1, 2) and Cu + 17.3 at. % Al + 0.5 at. % Fe alloys (3): the grain size d = 55 µm and H = 3 (1) and 7 % (2, 3)
Fig. 9. Single slip and extrusion along a region at the grain boundary AB in alloy Pb 1.9 % Sn: Tt 196 K and H = 15 %, without interference (a) and with interference (b), u 200
shear instability of the lattice may arise [10]. Second, since alloys with 0.4 1.9 % Sn correspond to the solubility limit above 77 K, they appear to be oversaturated in varying de-
grees at 77 K. This causes added instability of the lattice. Third, the degree of nonequilibrium of the material may be further increased by the segregation at grain boundaries.
*
)
Fig. 10. Distortion of square cells of the reference grid at the grain boundaries AB in Pb 1.9 % Sn: Tt
196 K and H = 15 %, u 300
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67 a
Fig. 11. Grain-boundary sliding and migration in Pb 1.9 % Sn: Tt = 300 K and H = 15 %, u 240
b
At strain temperatures above 250 K, the PbAs and Pb Sn alloys are solid solutions, and oversaturation of grain boundaries with alloying elements vanishes. Accordingly, the role of grain-boundary sliding and migration grows in importance at deformation (Fig. 11). In these conditions, the accommodation rotational modes multiple slip, severe curvature of the grain surface, fragmentation of the material in grain volumes are realized inside the grains (Fig. 12). The coefficient K in the HallPetch equation above 250 K decreases drastically, whereas the parameter V0 exhibits an anomalous increase (Fig. 4). Quite similar effects were observed in CuAl alloys for V0 V0 (T ) and K = = K (T ) [9]. The association of the coefficient K in the Hall Petch equation with rotational deformation modes and hence an important role of scale levels of accommodation rotations were overlooked, however. An addition horophobic relative to lead was tellurium. The temperature dependence of the parameters K and V0 of the HallPetch equation for the Pb 0.03 % Te alloy is shown in Fig. 13. It follows from a comparison of the plots
Fig. 13. Effect of the test temperature on the parameters V 0 (a) and K (b) of alloy Pb 0.03 % Te for H = 0.2 (1), 1 (2), 2 (3) and 5 % (5)
for Pb 0.03 % Te and those constructed for pure lead (Fig. 2) that the Te addition causes a dramatic increase in K and V0 . Unlike lead and eutectic alloys, the low-temperature peaks in the K (T) plots are missing in PbTe. The strong effect of the horophobic tellurium addition on the coefficient K in the HallPetch equation is similar in a qualitative sense to that of the horophylic arsenic addition and does a good job of suppressing grain-boundary sliding (Fig. 14), for it is not segregated at lead grain boundaries. Microscopic studies have revealed that the tellurium effect on grain-boundary sliding and coefficient K is associated with the fact that the accommodation rotational processes involved in the realization of grain-boundary sliding are constrained in the grain-boundary regions with strong chemical bonds. 4. On a physical interpretation of the parameters of the HallPetch equation The well-known relationship between the strain resistance and the grain size in a deformed polycrystal has been found in [11, 12] and is of the following form:
Fig. 12. Multiple slip and bending of the grain surface in Pb 1.9 % Sn: Tt 300 K and H = 30 %, u 180
V V0 K1d 1 K 2 d 1 2, (1) where V 0 , K1 and K 2 are the parameters of the material and d is the average grain size. It is standard practice to use
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Fig. 14. Microstructure of Pb 0.03 % Te alloy: Tt = 30 %, u 110
300 K and H =
only the summands V0 and Kd 1 2 in Eq. (1), which is the case in this work. Here V 0 is related to the shear resistance in a grain volume, whereas K is linked with the resistance to strain transfer from grain to grain. A rich variety of models has been suggested to explain Eq. (1) with different interpretations of the parameters V0 and K. Most of them are based on the single-scale approach of the dislocation theory. Strictly speaking, conventional interpretations of the parameters V0 and K account for the translational mechanisms of plastic shear strains alone. Physical mesomechanics employs a multiscale approach relying on a hierarchically self-organized plastic flow that follows the shear + rotation pattern. The plastic deformation propagates as an autowave process [2]. Shear is generated as local structural transformation in the vicinity of a stress concentrator of a certain scale and develops as a relaxation process in the field of maximum tangential stresses Wmax . The direction of Wmax and that of the loading axis do not coincide, which causes rotational deformation modes to give rise to other stress concentrators that provide for the propagation of plastic shear as a relaxation process. Plastic shear in a grain volume of a deformed polycrystal generates constrained rotations at the boundaries with adjacent grains. The stress fields of the constrained rotations may be relaxed by the following mechanisms: strain-induced defect fluxes at grain boundaries with consequent grain-boundary sliding, multiple slip in a grain volume with the resultant dispersion of rotational modes at lower-lying mesoscale levels inside grains, generation of localized deformation bands of an accommodation nature at grain boundaries, fragmentation of the material in a grain volume, rotations of grains as a unity,
migration of grain boundaries and development of extrusion (intrusion) of local regions in the material, pore and crack nucleation. Different relaxation mechanisms of the field of rotational moments are at work depending on the material, loading type and conditions. Clearly, these mechanisms must be accounted for in interpreting the parameters W0 and K in the HallPetch equation, which has been demonstrated convincingly by the results obtained in this work and in [9]. Summarizing briefly the examined effects, we may conclude that (i) a material is characterized by low values of V0 and high values of K if constrained rotational modes are localized within and near grain boundaries and (ii) a material exhibits a high level of V 0 and a low level of K if the rotational modes are dispersed in a grain volume. The multiscale interpretation of the parameters V0 and K has a simple physical meaning. Condition (1) is fulfilled in the case of single slip in polycrystalline grains. The attendant rotations of the material give rise to a counterpropagating field of accommodation rotations by the Ashby mechanism in a region near grain boundaries [13]. This effect is responsible for the low level of V 0 and high values of K. In coarse-grained materials with multiple slip in the base metal and switch to single slip due to introduction of alloying elements, the mechanical characteristics are adversely affected by alloying (Figs. 5 and 12). An increase in the test temperature causes grain-boundary sliding and intracrystalline multiple slip to enhance the strain resistance (Figs. 3, 4 and 8). In fine-grained materials, no anomaly is observed owing to high values of K and extended grain boundaries [9]. The high values of V and V0 in the case of multiple slip are associated with the development of dispersed rotational modes in the entire grain volume. It is not surprising that the coefficient K becomes very low. An important point to remember in connection with this is that in polycrystals with cellular dislocation substructure, the HallPetch equation is satisfied not for the grain size but for that of dislocation cells [14]. As is shown theoretically in [15], about 40 % of strain hardening of the material is associated with rotational deformation modes. In actual fact, the percentage is even higher when all kinds of rotational deformation modes are considered. The multiscale approach taken in physical mesomechanics to an analysis of the parameters V0 and K in the HallPetch equation provides an insight into the mechanisms involved in the plastic deformation of polycrystals under different loading conditions. 5. Summary We have performed systematic studies on the role of the scale levels of rotational deformation modes in the strain resistance of polycrystals with purposeful changes in the
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states of grain volumes and boundaries. The results obtained enabled us to reveal a number of hitherto unknown mechanisms and effects of critical importance for the problem under consideration. The materials under review are Pb As, Pb Sn and Pb Te. Evidence has been found for the decisive influence of the rotational deformation modes on the degree and special features of the temperature dependence of the polycrystals at different scale levels of the plastic flow. It has transpired that horophylic additions segregated at grain boundaries of polycrystals are stimulatory to localization of rotational deformation modes at a higher-lying scale level and are responsible for reduced strain resistance. Horophobic additions dissolved in the crystal lattice of grain volumes give rise to rotational modes at a lower-lying scale level in the grain volume and cause a dramatic increase in the strain resistance. A quantitative assessment has shown that grain-boundary sliding in an alloy with a horophobic addition (Pb Te) that forms a chemical compound with the base metal is suppressed due to constrained rotational accommodation processes in grain-boundary regions with strong chemical bonds. Both copper and lead alloys have been found to exhibit an anomalous temperature dependence of the shear resistance in grain volumes. The effect is attributed to constrained multiple slip in grain volumes at low test temperatures and to localization of rotational deformation modes in the grainboundary regions of polycrystals. In these conditions, the ability of polycrystals to transfer strain from grain to grain is impaired and hence parameter K is increased dramatically. The main factors responsible for the effect of scale levels of rotational deformation modes on the parameters V 0 and K in the HallPetch equation have been revealed.
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References [1] Physical Mesomechanics of Heterogeneous Media and ComputerAided Design of Materials, Ed. by V.E. Panin, Cambridge Interscience Publishing, Cambridge, 1998. [2] V.E. Panin, Synergetic principles of physical mesomechanics, Phys. Mesomech., 3, No. 6 (2000) 5. [3] O.A. Kaibyshev and R.Z. Valiev, Grain Boundaries and Properties of Metals, Metallurgiya, Moscow, 1987 (in Russian). [4] Yu.V. Grinyaev and V.E. Panin, Stress state in elastically loaded polycrystal, Sov. Phys. J., 21, No. 12 (1978) 1598. [5] V.E. Panin, T.F. Elsukova, and Yu.V. Grinyaev, The mechanism of grain size influence on strain resistance of polycrystals in the context of structural deformation levels of solids. Part I. Consideration of the mesoscopic structural levels of deformation in analyzing the Hall Petch equation, Phys. Mesomech., 6, No. 12 (2003) 91. [6] T.F. Elsukova and K.P. Zhukova, Effect of Temperature and Alloying on Plastisity and Macroscopic Deformation of Lead Alloys, in Substructure and Mechanical Properties of Metals and Alloys, Izd-vo TPI, Tomsk (1988) 77 (in Russian). [7] N.I. Afanasiev and T.F. Elsukova, Influence of impurities on discontinuous decomposition rate of PbSn alloys, Fiz. Met. Metalloved., 57, No. 1 (1984) 96 (in Russian). [8] T.F. Elsukova, K.P. Zhukova and V.E. Panin, Concentration dependence of flow stress of Pb-Sn solid solutions under deformation, Phys. Met. Metallogr., 64, No. 6 (1987) 1158. [9] V.E. Panin, E.F. Dudarev and L.S. Bushnev, Structure and Mechanical Properties of Substitution Solid Solutions, Metallurgiya, Moscow, 1971 (in Russian). [10] G.L. Krasco, Static displacements and shear instability of solid solution crystal lattice, Phys. Lett. A, 55, No. 4 (1975) 260. [11] B.O. Hall, Deformation and ageing of mild steel, Proc. Phys. Soc., 64, No. 1 (1951) 747. [12] N.J. Petch, The cleavage strength of polycrystals, J. Iron Steel Inst., 174 (1953) 25. [13] M.F. Ashby, The deformation of plastically non-homogeneous materials, Phil. Mag., 21, No. 170 (1970) 399. [14] V.I. Trefilov, Yu.V. Milman and S.A. Firstov, Physical Principles of Strength of Refractory Metals, Naukova Dumka, Kiev, 1975 (in Russian). [15] I.Yu. Smolin, P.V. Makarov and R.A. Bakeev, Generalized model of elastic-plastic medium with independent plastic rotation, Fiz. Mezomekh., 7, Spec. Iss., P. 1 (2004) 89 (in Russian).
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The effect of scale levels of rotational plastic deformation modes on the strain resistance of polycrystals T.F. Elsukova and V.E. Panin* Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia Experiments on polycrystalline lead alloys with different states of grain volumes and boundaries under tension in a wide range of test temperatures have been performed. The qualitative and quantitative characteristics of the grain size dependence of the strain resistance of the examined materials are shown to be substantially affected by the development of rotational plastic deformation modes at different scale levels. This effect must be accounted for in the interpretation of the parameters of the HallPetch equation in the framework of the multiscale approach employed in physical mesomechanics. Keywords: scale levels of rotational deformation modes of polycrystals, parameters of the HallPetch equation
1. Introduction Physical mesomechanics treats grain boundaries in a deformed polycrystal as a separate mesoscopic structural level of deformation that plays an important functional role in the propagation of plastic shear in a heterogeneous medium [1, 2]. The shear develops in a grain to give rise to a rotational deformation mode at a higher structural level due to grain-boundary defect fluxes along grain boundaries. This is evident as grain-boundary sliding [3]. The constrained rotations of grains as a unity are responsible for induced stress concentrators at grain boundaries and for accommodation deformation regions near grain boundaries [4]. The role of the rotations can be determined by the HallPetch equation used in physical mesomechanics to describe the accommodation of adjacent grains in a polycrystalline aggregate. Using the concepts of physical mesomechanics [1, 2], an analysis of the HallPetch equation, V V0 Kd 1 2, was performed for high-purity lead polycrystals subjected to tension in a wide range of test temperatures [5]. The experimental conditions enabled qualitative changes in the strain mechanisms operative at mesoscopic structural levels to be produced. The special features of the temperature dependence of plasticity and the accommodation grain-
* Corresponding author Prof. Victor E. Panin, e-mail: [email protected] Copyright © 2010 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2010.03.008
boundary mechanisms were found to be closely correlated. The mesoscopic strain mechanisms grain-boundary sliding and migration, mesoscale fragmentation of grains, formation of severe local curvature regions, etc. are shown to significantly affect the shape of the strain resistance grain size curve. Among these mechanisms, grain-boundary sliding is of special importance because it determines the form of the HallPetch equation and affects the qualitative characteristics of the dependence of its parameters on applied loading. The present paper deals with further analysis of the Hall Petch equation in terms of structural-scale levels of deformation of solids. The effect of scale levels of rotational plastic deformation modes on the strain resistance of polycrystals and parameters of the HallPetch equation is estimated in lead alloys with essentially different grain volumes and boundaries as an example. 2. Test materials and experimental technique The test materials were binary alloys based on highpurity lead S000 (Russian classification). To provide a purposeful change in the states of grain volumes and boundaries, use was made of two types of elements slightly soluble in lead: horophylic additions (As and Sn) significantly segregated at grain boundaries and a horophobic addition (Te). The latter is not prone to segregation at grain boundaries and is concentrated on lattice defects. The concentration of the additions was taken to be within solid so-
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lution, including the room-temperature solubility limit (1.9, 0.01 and 0.03 wt. % for Sn, As and Te, respectively). By increasing the test temperature the degree of departure from the solubility limit can be varied and in doing so the state of the grain boundaries in the alloys will be changed. To exclude an uncontrollable effect of natural impurities, 99.999% pure lead and alloying elements were employed. Flat specimens with cross-sectional area of 4.8 mm2 and gage length of 20 mm were subjected to tension at a rate of 54.0 %/min at 77548 K, which corresponds to (0.10.9) Tm for the examined materials. The stress-strain curves were plotted by averaging the data for 57 specimens. The resulting stress-strain curves were used to make plots of applied stress V versus d 1 2 for different degrees of strain (H = 0.2, 1, 2, 3 and 5 %) at different test temperatures Tt . The ability of the polycrystals to transfer strain from one grain to another, K, and the strain resistance in a grain volume, V0 , were estimated by the least-squares technique, extrapolating the straight lines for V = f (d ) to d 1 2 = 0 and d 1 0. The specimen surface was electropolished and examined by optical, interference and scanning electron microscopy followed by a quantitative assessment of grain-boundary migration, fragmentation, and single and multiple slip.
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3. Results and discussion Consider the effect of As and Sn additions on the behavior of the parameters V 0 and K as a function of the degree of strain and test temperature. The additions are horophylic relative to Pb and are substantially segregated at grain boundaries. Figure 1 shows the temperature dependence of V 0 and K for a Pb 0.01 % As alloy. When these plots are compared with those constructed for pure lead (Fig. 2), it is apparent that for H d 2 %, the curves V 0 (T ) are identical, whereas for higher degrees of strain, the curves for the lead alloy lie somewhat higher. The As addition gives rise to a number of special features of the K(T ) plot, however. First, this dependence is more pronounced for the examined alloy than for the pure metal: the absolute values of K for PbAs increase by a factor of 310 at low test temperatures and show a modest increase at elevated temperatures. Second, the peak in the K(T ) curve for lead is observed at low temperatures only for the yield stress and does not occur in the alloy with H = = 0.2 %. Yet the dependence for PbAs does exhibit a peak at higher H. The higher is H, the more pronounced is the peak. Finally, at low and moderate temperatures, the temperature dependence of K vanishes in Pb for H = 5 %, whereas in PbAs, the dependence is as strong as for H < 5 %. A comparison of the K(T ) curves for the pure metal and alloy revealed other striking features of importance for our discussion relating to the effect of small alloying addi
a
a
b b
Fig. 1. Effect of the test temperature on the parameters V 0 (a) and K (b) of a Pb 0.01 % As alloy for H = 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
Fig. 2. Effect of the test temperature on the parameters V0 (a) and K (b) of pure lead. The degree of strain is 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
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tions on the strain resistance of polycrystals. As was shown in [6], addition of 0.01 % of As caused a dramatic increase in the yield stress for Pb at low and moderate temperatures. At high test temperatures, on the contrary, the yield stress was decreased. Inspection of Figs. 1 and 2 discloses that it is the parameter K which is responsible for the increase in the yield stress in this case, as the As addition makes no contribution to V0 for H d 2 %, with K being increased by four- or fivefold. The As effect appears to be due to the fact that As demonstrates high segregation ability relative to Pb grain boundaries [7] and in consequence is almost entirely concentrated at grain boundaries. This makes grain-boundary deformation difficult and favors a large increase in VGB during intracrystalline shear strains. Another horophylic addition used was tin for which the solubility in lead is 200 times higher than that of arsenic. The examined alloys contained 1.9 (the solubility limit at 300 K) or 0.4 % of Sn. The temperature dependence of V0 and K for the alloys under review is presented in Figs. 3 and 4. It is evident from the curves that for V0 at 77300 K, the strain resistance exhibits an anomalous temperature dependence evident as local maxima in the V0 (T ) plots at 300 K. For K(T ) in the low-temperature range, the effect of the degree of strain is reversed in comparison with Pb and PbAs: the higher is H, the higher is K. As is the case with
PbAs, the K(T ) curves show peaks. Notably, the lower is H, the more pronounced is the peak for K. At high test temperatures, the temperature variations of V 0 and K exhibit the same qualitative relationships as pure lead does. The aforementioned unusual temperature dependence of V0 for the PbSn alloys is responsible for an anomalous concentration dependence of the strain resistance of PbSn solid solutions. This is evident in the fact that an increase in the tin concentration from 0 to 1.9 % causes V to increase, then to decrease drastically and, ultimately, to increase again in a gradual manner (Fig. (5) [8]). The manifestation of the anomalous behavior of V0 (T ) essentially depends on the degree of strain and loading conditions. The anomaly is at its maximum at low test temperatures, high strain rates and low degrees of strain. The anomalous temperature dependence of V0 found in our experiments suggests that the unusual low-temperature effects observed in PbSn are due to the special features of the intracrystalline shear strains involved. The anomalies are superficially similar to those found for copper alloys in [9] where plots of V and V0 were built as nonmonotonic functions of concentration and temperature (Figs. 6 8). In copper alloys, however, the anomalies were not found at low degrees of strains, but they did occur where the strain was increased. The effect was traced to constrained multiple slip associated with the special features of the resulting dislocation structure [9].
a a
b
Fig. 3. Effect of the test temperature on the parameters V 0 (a) and K (b) of a Pb 0.4 % Sn alloy. The degree of strain is 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
b
Fig. 4. Effect of the test temperature on the parameters V 0 (a) and K (b) of a Pb 1.9 % Sn alloy. The degree of strain is 0.2 (1), 1 (2), 2 (3), 3 (4) and 5 % (5)
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c
b
Fig. 5. Strain resistance versus concentration of tin at Tt
65
169 (a), 20 (b) and 180 qÑ: H = 0.2 (1), 1 (2), 3 (3) and 10 % (4)
In PbSn alloys, the anomaly is at its maximum for the yield stress. As H is increased, the anomalous dependence is smoothed out and then vanishes. Microscopic examination has revealed that the most distinct unusual mechanical behavior of PbSn is due to the preferred intracrystalline single slip, localization of accommodation processes (like quasi-viscous extrusion of the material) at grain boundaries (Fig. 9) and distortion of square cells of the reference grid, including grain boundaries, with relatively small alteration of the shape of the cells inside grains (Fig. 10). The latter circumstance is evidence of rotations of grains as a unity. The lack of grain-boundary sliding plays an important role in this case.
The data under consideration are demonstration that the anomalous temperature dependence of V 0 for PbSn is also associated with constrained multiple slip. In these conditions, single slip is provided by easier quasi-viscous flow in the grain-boundary regions of constrained deformation due to reduced shear stability of the crystal lattice caused by the alloying effect. This is attributable to a number of factors. First, in solid solutions, especially in the vicinity of the solubility limit, atomic displacements are so large that
a
b
Fig. 6. Strain resistance versus concentration of CuAl alloys under tension: H = 9 % and the grain size d = 0.21 (1), 0.05 (2), 0.025 (3) and 0.01 mm (4)
Fig. 7. Dependence of the parameters V 0 (a) and K (b) in the HallPetch equation on the concentration of solid solution of CuAl alloys under tension: H = 0.2 (1), 1 (2), 3 (3) and 7 % (4)
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66
a *
)
b * )
Fig. 8. Temperature dependence of the strain resistance of Cu + 17.3 at. % Al (1, 2) and Cu + 17.3 at. % Al + 0.5 at. % Fe alloys (3): the grain size d = 55 µm and H = 3 (1) and 7 % (2, 3)
Fig. 9. Single slip and extrusion along a region at the grain boundary AB in alloy Pb 1.9 % Sn: Tt 196 K and H = 15 %, without interference (a) and with interference (b), u 200
shear instability of the lattice may arise [10]. Second, since alloys with 0.4 1.9 % Sn correspond to the solubility limit above 77 K, they appear to be oversaturated in varying de-
grees at 77 K. This causes added instability of the lattice. Third, the degree of nonequilibrium of the material may be further increased by the segregation at grain boundaries.
*
)
Fig. 10. Distortion of square cells of the reference grid at the grain boundaries AB in Pb 1.9 % Sn: Tt
196 K and H = 15 %, u 300
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67 a
Fig. 11. Grain-boundary sliding and migration in Pb 1.9 % Sn: Tt = 300 K and H = 15 %, u 240
b
At strain temperatures above 250 K, the PbAs and Pb Sn alloys are solid solutions, and oversaturation of grain boundaries with alloying elements vanishes. Accordingly, the role of grain-boundary sliding and migration grows in importance at deformation (Fig. 11). In these conditions, the accommodation rotational modes multiple slip, severe curvature of the grain surface, fragmentation of the material in grain volumes are realized inside the grains (Fig. 12). The coefficient K in the HallPetch equation above 250 K decreases drastically, whereas the parameter V0 exhibits an anomalous increase (Fig. 4). Quite similar effects were observed in CuAl alloys for V0 V0 (T ) and K = = K (T ) [9]. The association of the coefficient K in the Hall Petch equation with rotational deformation modes and hence an important role of scale levels of accommodation rotations were overlooked, however. An addition horophobic relative to lead was tellurium. The temperature dependence of the parameters K and V0 of the HallPetch equation for the Pb 0.03 % Te alloy is shown in Fig. 13. It follows from a comparison of the plots
Fig. 13. Effect of the test temperature on the parameters V 0 (a) and K (b) of alloy Pb 0.03 % Te for H = 0.2 (1), 1 (2), 2 (3) and 5 % (5)
for Pb 0.03 % Te and those constructed for pure lead (Fig. 2) that the Te addition causes a dramatic increase in K and V0 . Unlike lead and eutectic alloys, the low-temperature peaks in the K (T) plots are missing in PbTe. The strong effect of the horophobic tellurium addition on the coefficient K in the HallPetch equation is similar in a qualitative sense to that of the horophylic arsenic addition and does a good job of suppressing grain-boundary sliding (Fig. 14), for it is not segregated at lead grain boundaries. Microscopic studies have revealed that the tellurium effect on grain-boundary sliding and coefficient K is associated with the fact that the accommodation rotational processes involved in the realization of grain-boundary sliding are constrained in the grain-boundary regions with strong chemical bonds. 4. On a physical interpretation of the parameters of the HallPetch equation The well-known relationship between the strain resistance and the grain size in a deformed polycrystal has been found in [11, 12] and is of the following form:
Fig. 12. Multiple slip and bending of the grain surface in Pb 1.9 % Sn: Tt 300 K and H = 30 %, u 180
V V0 K1d 1 K 2 d 1 2, (1) where V 0 , K1 and K 2 are the parameters of the material and d is the average grain size. It is standard practice to use
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Fig. 14. Microstructure of Pb 0.03 % Te alloy: Tt = 30 %, u 110
300 K and H =
only the summands V0 and Kd 1 2 in Eq. (1), which is the case in this work. Here V 0 is related to the shear resistance in a grain volume, whereas K is linked with the resistance to strain transfer from grain to grain. A rich variety of models has been suggested to explain Eq. (1) with different interpretations of the parameters V0 and K. Most of them are based on the single-scale approach of the dislocation theory. Strictly speaking, conventional interpretations of the parameters V0 and K account for the translational mechanisms of plastic shear strains alone. Physical mesomechanics employs a multiscale approach relying on a hierarchically self-organized plastic flow that follows the shear + rotation pattern. The plastic deformation propagates as an autowave process [2]. Shear is generated as local structural transformation in the vicinity of a stress concentrator of a certain scale and develops as a relaxation process in the field of maximum tangential stresses Wmax . The direction of Wmax and that of the loading axis do not coincide, which causes rotational deformation modes to give rise to other stress concentrators that provide for the propagation of plastic shear as a relaxation process. Plastic shear in a grain volume of a deformed polycrystal generates constrained rotations at the boundaries with adjacent grains. The stress fields of the constrained rotations may be relaxed by the following mechanisms: strain-induced defect fluxes at grain boundaries with consequent grain-boundary sliding, multiple slip in a grain volume with the resultant dispersion of rotational modes at lower-lying mesoscale levels inside grains, generation of localized deformation bands of an accommodation nature at grain boundaries, fragmentation of the material in a grain volume, rotations of grains as a unity,
migration of grain boundaries and development of extrusion (intrusion) of local regions in the material, pore and crack nucleation. Different relaxation mechanisms of the field of rotational moments are at work depending on the material, loading type and conditions. Clearly, these mechanisms must be accounted for in interpreting the parameters W0 and K in the HallPetch equation, which has been demonstrated convincingly by the results obtained in this work and in [9]. Summarizing briefly the examined effects, we may conclude that (i) a material is characterized by low values of V0 and high values of K if constrained rotational modes are localized within and near grain boundaries and (ii) a material exhibits a high level of V 0 and a low level of K if the rotational modes are dispersed in a grain volume. The multiscale interpretation of the parameters V0 and K has a simple physical meaning. Condition (1) is fulfilled in the case of single slip in polycrystalline grains. The attendant rotations of the material give rise to a counterpropagating field of accommodation rotations by the Ashby mechanism in a region near grain boundaries [13]. This effect is responsible for the low level of V 0 and high values of K. In coarse-grained materials with multiple slip in the base metal and switch to single slip due to introduction of alloying elements, the mechanical characteristics are adversely affected by alloying (Figs. 5 and 12). An increase in the test temperature causes grain-boundary sliding and intracrystalline multiple slip to enhance the strain resistance (Figs. 3, 4 and 8). In fine-grained materials, no anomaly is observed owing to high values of K and extended grain boundaries [9]. The high values of V and V0 in the case of multiple slip are associated with the development of dispersed rotational modes in the entire grain volume. It is not surprising that the coefficient K becomes very low. An important point to remember in connection with this is that in polycrystals with cellular dislocation substructure, the HallPetch equation is satisfied not for the grain size but for that of dislocation cells [14]. As is shown theoretically in [15], about 40 % of strain hardening of the material is associated with rotational deformation modes. In actual fact, the percentage is even higher when all kinds of rotational deformation modes are considered. The multiscale approach taken in physical mesomechanics to an analysis of the parameters V0 and K in the HallPetch equation provides an insight into the mechanisms involved in the plastic deformation of polycrystals under different loading conditions. 5. Summary We have performed systematic studies on the role of the scale levels of rotational deformation modes in the strain resistance of polycrystals with purposeful changes in the
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states of grain volumes and boundaries. The results obtained enabled us to reveal a number of hitherto unknown mechanisms and effects of critical importance for the problem under consideration. The materials under review are Pb As, Pb Sn and Pb Te. Evidence has been found for the decisive influence of the rotational deformation modes on the degree and special features of the temperature dependence of the polycrystals at different scale levels of the plastic flow. It has transpired that horophylic additions segregated at grain boundaries of polycrystals are stimulatory to localization of rotational deformation modes at a higher-lying scale level and are responsible for reduced strain resistance. Horophobic additions dissolved in the crystal lattice of grain volumes give rise to rotational modes at a lower-lying scale level in the grain volume and cause a dramatic increase in the strain resistance. A quantitative assessment has shown that grain-boundary sliding in an alloy with a horophobic addition (Pb Te) that forms a chemical compound with the base metal is suppressed due to constrained rotational accommodation processes in grain-boundary regions with strong chemical bonds. Both copper and lead alloys have been found to exhibit an anomalous temperature dependence of the shear resistance in grain volumes. The effect is attributed to constrained multiple slip in grain volumes at low test temperatures and to localization of rotational deformation modes in the grainboundary regions of polycrystals. In these conditions, the ability of polycrystals to transfer strain from grain to grain is impaired and hence parameter K is increased dramatically. The main factors responsible for the effect of scale levels of rotational deformation modes on the parameters V 0 and K in the HallPetch equation have been revealed.
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