Low-temperature work-hardening of cadmium single crystals

LOW-TEMPERATURE WORK-HARDENING OF CADMIUM SINGLE CRYSTALS B. WIELKE.

W. TIKVIC

II. Physikalisches Institut der Universitlt

Wien, Austria

and A SVOBODOVA Department

of

and P. LUKAC

Metal Physics. Charles university, Prague. Czechosiovakia

(Received 9 June 1976: in revised form 15 Februnry 1977) Abstract-Stress-strain curves of Cd-single crystals were measured at low temperatures down to 4.2 K. The resolved critical shear stress ro/p decreases with increasing temperature, showing a slope three to four lines larger in the interval 4.2-30 K than in the interval 30-300 K. The work-hardening coefficient shows a distinct temperature dependence in the easy-glide stage A as well as in stage B. The activation volume obtained from strain rate change experiments decreases during deformation by more than one order of magnitude. The experimental results are compared to the behaviour of f.c.c. metals. The temperature-dependence of the reduced work-hardening coefficients 6Jfi and 6,:~ cannot be explained by means of the commonly used work-hardening theories. R&sum&-Dn a enregistre ies courbes contmint~d~formation de mon~ris~ux de cadmium a basse temperature, jusqu’a 4,2 K. La cission critique reduite re/_u diminue lorsqu’on augmente la temperature et presente une pente trois a quatre fois plus grande entre 4.2 et 30 K qu’entre 30 et 3OOK. Le coefficient d%crouissage depend de Ia temperature au stade A (glissement facile) et au stade B. Le volume d’activation, obtenu au tours d’essais de sauts de la vitesse de deformation, diminue de plus dun ordre de grandeur au tours de la deformation. On compare ces resultats experimentaux au comportement des metaux c.f.c. On ne peut pas expliquer la variation en fonction de la temperature des coefficients d’ecrouissage ritduits 0,Jp et 0a:p par les theories usuelles de I’dcrouissage. Zusammenfassung-An Cd-Einkristallen wurden Verfestigungskurven bei niedrigen Temperaturen bis zu 4.2 K hinunter gemessen. Mit zunehmender Temperatur sinkt die kritische RieBspannung ~‘1 ab mit einer Steigung, die im Temperaturintervall von 4,2 bis 30 K drei bis viermal gr6Ber als im Interval1 30 bis 300 K ist. Der Verfestigungskoeffizient weist eine deutliche Temperaturabhlngigkeit im Einfachgleitbereich A und im Bereich B auf. Das Aktivierungsvolumen wurde mit Dehngeschwindigkeitswechsem bestimmt; es wird w&end der Verformung urn fiber eine Gr~Benordnung kleiner. Die ex~~menteli~ Ergebnisse werden mit dem Verhalten von kfz. Metallen verglichen. Die Tem~~tur~b~ngigkeit der reduzierten Verf~tigun~sk~ffizient~ @,/p und t?Bf~(kann mit den iiblichen Ve~estigungsth~~en nicht e&l&t werden. 1. Ii\iRODUCTION

Experimental investigations of the temperature dependence of flow stress and the rate of work-hardening of hexagonal metals were until now limited mainly to temperatures higher than 77 K [l-j]. In this temperature range, the experimental results seem to obey the predictions of a model proposed by Seeger [I, 6,7], from which it follows that the workhardening rate of the well-develo~d easy-glide range at low temperatures should be constant. Al~ou~ some anomalies were observed on Zn occasionally cl, 2,8,9] they were limited to some points around 77 K. Only a few measurements were made at 4.2 K [9-l 11. Recently, Wielke [ 123 has measured the work-hardening rate in easy-glide range of Zn and Cd single crystals deformed at low temperatures. It was shown that the work-hardening rate varies with temperature in an unexpected way. In the present work, the temperature dependence of the parameters of the work-hardening curve and the stress depen-

dence of the activation volume of cadmium single crystals is investigated also at very low temperatures.

2. EXPERIMENTAL

PROCEDLRES

Single crystals of 4 mm dia. were grown in a glass tube at the rate of 4 cm h-’ using a modiied Bridgman method. Specimens about 45 mm long were obtained by cutting from longer single crystals, Orientations were deter~ned by the rotating-crystal method and are shown in Fig. 1. They were chosen near @001) orientation to avoid deformation twinning. Before the deformation, the samples were polished in a solution of 240 g CrOs and 20g Na2S04 in IOOOccm H20. Tensile tests were carried out on a hydraulic Instron machine model 1253 with an initial strain rate of 8 x lo- J s- ‘. A gas spray cryostat [13] was used for r> 20 K, at T c 20 K the experiments were performed with a spray and/or bath cryostat. Differential 1071

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a Fig. f. ResoIved shear stress vs shear strain curves of cadmium single crystals deformed between 4.2 and 295 K. The specimen orientations are shown in the unit triangle. strain rate change experiments were carried out by alternating the speed of the hyd~ulic driven piston. 3. REWXTS Resolved shear-stress-strain curves are calculated from load-elongation curves under assumption of single glide [14]. Figure 1 shows the shear stressshear strain curves obtained at various temperatures. In the literature, the work-hardening curves are usually divided into three stages. As can be seen from Fig. 1, the third stage C in the low-temperature range is only poorly developed and is missed completely at T ,<48 K. Figure 2 shows a typica1 example of the

strain dependence of the work-hardening rate 0 for T= 77K A large region with a relatively low value of the work-hardening rate, corresponding to easy glide stage A, is followed by an increase of the workhardening rate, which reaches a maximum value {stage 3) and drops again (stage C). Comparing the strain dependence 8 (a) of a typical f.c.c. metal with our results of Cd, one can see that the length of easyglide range differs by an order of magnitude. Also, the transition from easy-glide to stage $3 or II is only wefi defined in the case of f.c.c. metals. In Fig. 3, the temperature dependence of the reduced critical resotved shear stress so:‘!6is shown. One gets the value of the critical shear stress by extra-

I

Fig 2. Strain dependence of the work-hardenj~g rate of Cd compared with Ag [XJ at 77 K. Definition of & t&and

stage length aA are shown.

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s- I 0 0

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Fig. 3. Temperature dependence of the reduced critical resolved shear stress of cadmium single crystals. The insert shows the determination of T,,. Straight lines indicate directly measured Ar,,jA?:

o* 0

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200 T

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polating the stress-strain curve to a = 0 using the slope of stage A, @.+.Taking another generally used

Fig. 5. Temperature dependence of the length of stage A, aA, and the shear strain o, = a (&,,L

definition. the stress at a certain amount of plastic strain, no quantitative difference in the temperature dependence is obtained. Data of the shear modulus fi = C,, are taken from Garland and Silverman [15]. It shoutd be pointed out that the slope in ~o(~~/~ at temperatures lower than 30 K is three or four times higher than that at higher temperatures. Similar behaviour was also observed for Zn single crystals by Wielke [16]. One gets the same shape of the s,(T) curve by extrapolating the values of Ahs/ATmeasured by temperature change experiments [17] to r = ro. The points connected by straight lines in Fig. 3 show such directly measured values Ar/ATof the same crystal. In Fig. 4, the reduced work-hardening rate in stage A, fJ+.,/pis plotted against temperature. It shows a distinct tem~mture dependence also at very low temperatures. The same qualitative behaviour is observed on Zn single crystals [12]. The well-known decrease of BA/p above 170 K is governed by recovery (3,183. The extension of stage A, called aA, as a function

of temperature is shown in Fig. 5. The definition for aA = ~(0 = 2BA)(see Fig. 2} is different from the usual definition where the first deviation from the linear stage A is taken [3]. The latter one seems not to be unambiguous. Though uA decreases with decreasing temperature, it maintains a high value of about 70% even at 4.2 K. The temperature dependence of ant, the shear strain at the maximum slope of the work-hardening curve 0,,,, is also shown in Fig. 5. The temperature dependence of ~$,,JIL the maximum value of reduced work-hardening rate in stage B, is given in Fig 6. Since no stage C exists at very low temperatures, the slope of the stress-strain curve at fracture was taken as value for @_in these cases. In spite of the scatter of the measured values, there is a distinct decrease of @,,,with increasing temperature, which up to now has not been reported in the Iiterature and is different to the behaviour of f.c.c. metals, where &ill is nearly independent of temperature [7].

I.2

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Fig. 4. The reduced work-hardening rate 0,./p in the easyglide range as a function of temperature.

T

IKI

Fig. 6. Temperature dependence of the maximum workhardening rate @,,% in stage B.

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Fig. 7. Stress increments due to strain rate changes 1: 10 vs shear stress at three different temperatures. The change of the experimental activation volume, defined by V, = kT (dtnd;Arf [ 193, with deformation was also measured for T= 4.2, 30 and 77 K. Figure 7 shows that the stress increment AT at a strain rate change 1: 10 increases with increasing stress. The upward changes were evaluated. The increment Ar was defined by the ins~n~n~us stress response at the yield point. The slope.of Arfr) depends on temperature. No range of constant Ar(r) is observed. Further measurements are described in Ref. [17]. 4. DISCCSSION

Now the measured data shall be compared with the data in the literature and the differences to typical f.c.c. metals will be worked out. In the literature [7,20], there was emphasized a certain analogy between the work-hardening behaviour of h.c.p. and f.c.c. metals which is not valid for the low-temperature defo~tion of Cd. The temperature dependence of the critical resolved shear stress of Cd can be interpreted by means of thermally activated glide. Ono [21] discussed the temperature dependence of Q for various given interaction potentials (foroedistance curves), between dislocation and obstacle. For all realistic models, one obtains a function r,(T) with a slope dr,/dTincreasing with decreasing temperature. A detailed discussion can also be found in ref. [22]. Our measurements agree with the idea that the temperature dependence of so/g is controlled by thermally activated glide of dislocations. The measured temperature dependence of the work-hardening coefficients 8, and S,,, differs dis-

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tinctly from the data in the literature [3,7]. All authors thought the work-hardening in analogy to f.c.c. metals to be constant in the low-temperature region below 15O-200 K (T= 0.3 . ..0.5 T,, T, = melting point). This error obviously results from the fact that not enough measurements were performed below 200 K. The main differences between the deformation behaviour of h.c.p. Cd and typical f.c.c. metals (Ag, Au, Al, Cu. Ni, etc.) are the following: (i) the length of easy-glide is much larger for Cd than for f.c.c. metals (typically q r 0.05). Moreover, the length of stage A increases with increasing temperature (see Fig. 5), while it decreases for f.c.c. metals [Tj. {ii) In the low-temperature region, the work-hardening coefficient 6,/p for Cd decreases rapidly with increasing temperature, while e,/hr is increasing [7]. (iii) The difference between Cd and f.c.c. metals seems to be especially striking for the maximum work-hardening coefficients ~~~~~ and t$t/~. For f.c.c. metals, t?tJp is not only temperature independent but it also has practically the same value (4, -r p/300) for all metals. This does not hold for Cd (and Mg), as can be seen from Fig. 8. In the examined temperature interval, 6,,, decreases rapidly with increasing temperature. This temperature dependence couid arise from a dislocation process involving thermally activated cross-slip. The main results of the strain-rate-change experiments are the absolute values of the activation volume u, = kT Aln&/Ar and its decrease from the beginning of deformation. There exists no region of constant t’, which means that the density of obstacles which are overcome by thermal activation increases from the beginning The obstacle density can be estimated from the so-called thermodynamic activation volume u with the well-known relation v = dAyb, where i. is the mean obstacle distance along the dislocation line, AX is the mean obstacle diameter (which must be of the order of b, the Burgers vector). Besides the problems of quantitative thermal activation analysis (constancy of the pre-exponential factor, absolute value of Ax, the Friedel relation [25-J). the thermodynamic activation volume L’and the experimental activation volume u, are related by c = cc, (c being a factor of the order 1) [17,193. The mean distance 1 between obstacles is 1 c i, so the obstacle density p = 1-2 can be estimated [17] by p > (Ayb/u,)‘. At the beginning of deformation (Fig. 7) it is r, = 2 x lo4 b3 and with Ax z 2b the density p becomes p > 10’ cm- ‘. A detailed q~nti~tive analysis is given in a further paper [ 173. Since the grown-in forest density is about 103-104cm-‘. it can be said that the forest dislocations @ or E + Z-dislocations) cannot explain the high obstacle density of >107cm-2 at the beginning of deformation. Keeping in mind all the experimental results, it seems impossible to understand the work-hardening behaviour of Cd by means of one of the commonly

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Fig. 8. Comparison between the temperature dependence (r, = melting point) of the maximum workhardening coefficient of f.c.c. and h.c.p. metals. used simple models. Some discussion of the easy-glide work-hardening of Cd and Zn is given in ref. Cl’]. The author comes to the conclusion that there is no single process known which can explain the experimental results. Without any further information. that means especially systematic electron microscopy. no further conclusions can be drawn. Therefore the temperature dependence of the work-hardening behaviour of Cd cannot be understood. Acknowledgements-The authors are grateful to Prof. G. Schiick for valuable critical discussions to this work, and to Prof. Kratochvil and to Dr. B. Sprusil for reading this manuscript and for their valuable comments.

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