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Localization of plastic deformation in single-crystal and polycrystalline materials with bcc, hcp and tetragonal lattice
V.I. Danilov, S.A. Barannikova, K.V. Gonchikov, M.A. Kunavina, and L.B. Zuev
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia
The macro-localization patterns of plastic flow observed at the linear stage of the deformation curve have been investigated using silicon iron (bcc), white tin (bct) and single-phase magnesium-base alloy (hcp). It is shown that in all the materials investigated the evolution of deformation localization occurs in the form of running autowaves whose velocity are inversely proportional to the reduced coefficient of work hardening.
1. Introduction
Macrolocalization of plastic flow may be treated in the framework of the self-organization concept involving open non-equilibrium systems (e.g. loaded object-loading device). In so doing, the most striking features of this phenomenon are revealed as the strain distribution pattern evolves in the form of running phase autowave, that is equidistant flow nuclei propagate at a constant rate along the loaded specimen. The above localization type was first discovered in 1994 for the single copper-base alloy crystals in a state of oversaturated solid solution [1, 2]. Later on the generation of phase autowaves of macrolocalization at the stage of linear work hardening was observed for single crystals of chro-mium-nickel and high manganese fcc iron-base alloys. Moreover, it was found that autowave velocities are related to the reduced coefficient ofwork hardening, i.e. Vaw = EG/9, where G is the shear modulus, 9 is the coefficient of work hardening and E is the constant having the units of velocity [3, 4].
At the same time similar investigations were also performed for polycrystalline metals and alloys [4-6]. It was found that during the deformation at the stage of linear plastic hardening at a constant non-zero value of 9, in the deforming material there travel equidistant nuclei of localized deformation. The rates of flow nuclei were measured for various materials at different values of 9. Of particular significance is the fact that introduction of the measured rates into the above dependence does not affect the constant E and correlation factor [3]. Thus in [4] the same dependence was constructed using data obtained for eight different materials va-
rying with respect to microscopic mechanisms involved in plastic flow, extension axis orientation of single crystals, interstitial impurity content, grain size of polycrystalline specimens and disperse precipitate content. The only feature the materials had in common, with the exception of zirconium-base alloy, was hcp lattice. In view of the above, the question arises as to whether localization of deformation in metals and alloys having other types of crystal lattice also evolves in the form of running autowave.
2. Materials and experimental methods
Localization of deformation in polycrystalline materials has been investigated and the results obtained are presented herein. The data on material structure and composition are listed in Table 1. All the test specimens were flat dumbbell ones. They were prepared from rolled sheets by stamping. The structure of silicon iron specimens as determined by the standard procedure used for transformer steel manufacture revealed large grains of a-solid solution of Si and C in Fe with {110}<001> texture. The specimens of white tin (P-Sn) were prepared in the form of bands by rolling at room temperature; hence most of these also had {110}<001> texture. Prior to testing, the specimens of commercial single-phase alloy (MA8) were subjected to recrystallization by annealing in vacuum for 4 h at 620 K. Also listed in Table 1 are the data obtained for single silicon iron crystals grown by the Bridgman method in an atmosphere of inert gas. These have the same composition as polycrystalline silicon iron. The single-crystal silicon iron specimens also were dumbbell-shaped. They were cut out
© VI. Danilov, S.A. Barannikova, K.V Gonchikov, M.A. Kunavina, and L.B. Zuev, 2004
Table 1
Material Structure Grain size, um Fe, wt. % Mg, wt. % Sn, wt. % Si, wt. % Mn, wt. % C, wt. % Ce, wt. % As, wt. % Al, wt. % Zn, wt. % The rest
Silicon iron bcc (4.5 ± 3) • 103 >96.1 — — 2.8-3.8 — <0.02 — — — <0.08
MA8 commercial alloy hcp 12.5 <0.05 >97.0 — 0.1 1.3-2.2 0.15-0.35 — 0.1 0.3 <0.3
Tin P-Sn bct 1.2 • 103 < 10-4 — 99.999 — — — — 10-4 < 3 • 10-4 3 • 10-5 <5 • 10-4
of single-crystal ingots in such a way that their longitudinal axis is oriented in the direction [143] and the work surface has index (168).
All the specimens were tested in tension at a constant rate at room temperature. Localization patterns were obtained by recording the displacement fields for individual points on the deforming specimen surface with the aid of double-exposure speckle interferometry, with an increment in the total conventional deformation, 8 tot, being 0.2 %. The method is described in detail elsewhere [7].
During the tensile loading of the single-crystal Fe-Si specimens oriented in the direction [143], the deformation at the yield point involves two slip systems. Therefore, the deformation curve shows no easy glide stage; the viscoelastic transition is followed by the stage of linear plastic hardening (stage II), which can be readily distinguished on the flow curve by numeric differentiation.
An analysis of the multistage behavior of the deformation curve was performed for the polycrystalline specimens in the coordinates (s - e), where s = g(1 + 8) is the true stress and e = ln(1 + 8) is the true strain (Fig. 1). Evidently, the diagrams constructed for the specimens of bcc Fe-Si and P-Sn with tetragonal crystal lattice (bct) (Fig. 1, curves 1 and 3, respectively) belong to the most commonly occurring type of curve, i.e. one that shows a very short transition stage and has an increasing work hardening coefficient.
The diagrams obtained for the Mg-Mn-Ce specimens, have, on the contrary, a lengthy transition stage and are pa-
s, MPa
0.0 0.1 0.2 0.3 e
Fig. 1. Deformation curves: Fe-3% Si (1), Mg-Mn-Ce (1), P-Sn (3)
rabolic in shape. For this reason, the linear work hardening stage was determined for the former specimens by replotting the curves in the co-ordinates (s - e1/2). The same stage on the resulting curve has the form of quadratic parabola and is much more pronounced among the remaining, almost rectilinear, portions of the curve [8]. Using the above procedure, it was found that stage II on the curves of Fe-Si and P-Sn is in the intervals 0.008 < e n <0.038 and 0.01 < e n < 0.07, respectively, and the coefficient of work hardening, 9, is 1750 MPa and 33 MPa, respectively.
In the case of Mg-Mn-Ce, however, the above procedure failed; therefore, the Honeykombe representation [9] was invoked, according to which the deformation curve of a polycrystal is represented by the following formula
s = s 0 + Ken, (1)
where s0 ~ G02 and n < 1. To the various portions of the plot there correspond different values of the exponent n, with n varying in either a continuous or a discrete manner. In the latter case, at n = 1 the linear stage II is clearly revealed with Kjj = 9 and stage III falls into several substages. Using the above procedure, the linear stage is distinguished on the deformation curve obtained for MA8 alloy in the interval 0.02 < en < 0.04 and the value of the work hardening coefficient 9 obtained for the same stage is 875 MPa.
3. Results and discussion
Although the evolution of localization patterns has been registered for the entire flow curve, we shall restrict our consideration to the results obtained for the stage of linear plastic hardening. Thus, at a constant value of the work hardening coefficient, periodic space-time arrangements of local elongations are found to form and propagate in the specimens of silicon iron, P-tin and magnesium alloy, i.e. autowaves are generated. The wave parameters have been determined for the above materials (Fig. 2) by plotting the flow nuclei positions against time in the fashion described in Ref. [4]. As is seen from Fig. 2, at stage II the localized deformation nuclei traverse the specimen length several times. Therefore, the autowaves observed fall into the phase autowave type, since switching or excitation autowaves would only traverse once the entire space of the specimen [10]. Straight-line approximation has been performed and the distance between the straight lines has been measured along the vertical and horizontal scales to yield wavelength and wave period, respectively. The inclination of the straight
Fig. 2. The kinetics of localized deformation nuclei at the stage of linear plastic hardening: Fe-3 % Si (a), P-Sn (b), Mg-Mn-Ce (c)
lines to the 7-axis determines autowave velocity. The autowave parameters are listed in Table 2.
The data presented in Table 2 were treated using the dependence Vaw - 9/G [4]. It has been found that the propagation rates of localization zones derived for single-crystal and polycrystalline silicon iron and Mg-Mn-Ce alloy are described satisfactorily by the above dependence (Fig. 3, line 2). The constant E varies from 6.61 • 10-7 m/s to 6.59 • 10-7 m/s.
Table 2
Material G, GPa [22] 0, MPa 0/G A, mm t, s Vaw • 105, m/s
Fe-Si, polycrystal 86 1750 0.02 5.5 ± 0.5 325 1.7
Fe-Si, single crystal 86 900 0.01 3.5 ± 0.5 60 5.8
Mg-Mn-Ce 17 875 0.05 3.0 ± 0.5 100 3.3
P-Sn 17 33 0.002 4.3 ±0.5 59 7.3
Correlation coefficient p also varies insignificantly (0.94 and 0.92). The inverse relationship between Vaw and the reduced coefficient of work hardening is discussed in [4]. It is postulated that the variation in the nuclei motion rate is determined by the changes in the length of the glide path at the linear stage and the value of 9 is determined by the formation of planar pileups A and by the condensation of dislocations in the same [11].
In this instance
dVaw = O-^-d9. (2)
819 2
Thus, apparently, Vaw ~ 9-1. Here O is a dimensional constant that has the meaning of density of energy flux. The physical sense of the parameter A can be determined by analysis of the types of dislocation structures that form in the material. Thus at stage II, network dislocation structure would form from individual dislocation pileups [12, 13]. In so doing, the length of the glide path is naturally determined by the characteristic size of nets, i.e. parameter A. The same holds for metallic systems with fcc, hcp and bcc crystal lattice [13].
In the case of P-Sn, one is faced with a more complex situation. As is seen from Fig. 3, the autowave velocity falls into a set of data, which was obtained by straight-line extra-
V, m/s
0.00020
0.00015 0.00010
0.00005 0.00000
0 200 400 600 G/0
Fig. 3. Autowave velocity against reduced values of the work hardening coefficient
polation using line 2 instead of line 1. This includes the velocity values that have been derived at the easy glide stage and on the yield plateau for the deforming specimens. In this instance, the value of the plastic hardening coefficient 9 is determined by the formation of point defects, by the impact of long-range fields of individual dislocations and by lattice friction [14], i.e.
9 = 9def +Tlat =9def + D / Y p, (3)
where 9 def is the deformation due to the impact of lattice defects; Tlat = D / y p is the lattice friction; y p is the shear deformation at the easy glide stage and D is the material constant that is inversely proportional to the yield point. The main factor that inhibits the deformation at stage II is the contact interaction of dislocations from the different slip systems. At the stages considered herein, however, the latter factor is inoperative; therefore, the hardening attained is insignificant or close to zero. It can be shown that such is indeed the case when dislocation glide occurs in tin at room temperature. It is known [15] that as early as on the yield plateau, the deformation of single-crystal and polycrystalline P-Sn involves three active slip systems, i.e. {110}<001>, {100}<001> and {101}<101>, with the first of these being slightly predominant. Therefore, after the transition stage, barriers are expected to appear in the material due to crossing of dislocations from the different systems and stage II of linear work hardening is expected to begin. In the case of P-Sn, TL = 505 K and the recrystallization threshold, TR = 200 K. Consequently, the deformation at room temperature (T ~ 300 K) evolves in the conditions of intense dynamic recrystallization. The above conditions favor, at low loading rates (6.67 • 10-5 s-1), disintegration of barriers, which might have otherwise inhibited the evolution of deformation.
4. Conclusions
Macrolocalization of plastic flow in materials with bcc, hcp and tetragonal crystal lattice has been investigated. The results obtained reveal that at the stage of linear plastic hardening, the evolution of macrolocalization occurs in the form of mobile equidistant deformation zones, which travel at a constant rate, thereby generating autowaves of plastic
flow. The autowaves observed in silicon iron and magnesium-base alloy propagate at a rate, which is inversely proportional to the reduced coefficient of work hardening and is described by the same dependence that holds also for single-crystal and polycrystalline materials with fcc crystal lattice. In the case of P-Sn, the deformation evolves in dynamic recrystallization conditions and 9<< 10-2 G; therefore, the autowave velocity is described by the generalized dependence Vaw - 9 I / G obtained for the fcc single crystals tested.
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