The flow stress and dislocation structure of nickel deformed at very high pressure

Materia& Science and Engineer&g American Society for Metals, Metals Park, Ohio, and Elsevier Sequoia S.A., Lausanne-Printed in the Netherlands

111

The Flow Stress and Dislocation Structure of Nickel Deformed at Very High Pressure w. A. JESSER and D. K U H L M A N N - W I L S D O R F Department o[ Materials Science, University 0[' Virginia, Charlottesville, Va. 22901 (U.S.A.) (Received September 8, 1971)

Summary * The arrangement, density and average link-lengths of the dislocations in nickel specimens, severely deformed in shear under high normal pressures, were Jbund to be largely independent of pressure, even while Riecker, Towle and Rooney had observed their flow stress to rise strongly with pressure. Moreover, the dislocations were not arranged into well-defined cell walls. Both of these findings strongly suggest that the j?ictional stress acting on the dislocations dominates the flow stress at high pressures and is responsibleJbr its pressure dependence. A quantitative analysis of the measurements indicates a (near)proportionality between the frictional stress and pressure. When this effect is ascribed to the volume changes of the dislocation core in its motion over Peierls hills and valleys, the magnitude of the said volume changes is found to be near one-hundredth of an atomic volume per lattice plane, which appears to be a reasonable number when compared with a total volume expansion of about one atomic volume per dislocation and lattice plane.

INTR ODUCTI ON

In the past, several investigators 1- 3 have studied the mechanical deformation of nickel specimens at pressures of over 5 kilobars. The most recent such work by Riecker, Towle and Rooney 3 indicated that the shear strength of nickel, grossly deformed by shear in an opposed anvil apparatus at pressures of from 10 to 150 kilobars, is consistent with the "unified" theory ofworkhardening4. Since according to that theory the flow stress is determined by the average free link lengths in the dislocation network, while an investigation of other grossly deformed materials by Towle and Riecker 5 had indicated that the Peierls-Nabarro force as derived previously6 * For French and German translations of the Summary, see p. 116.

Mater. Sci. Eng., 9 {1972)

could well be the factor controlling the flow stress in metals under high pressures, it seemed instructive to investigate the dislocation structures and average free dislocation link lengths of the above-mentioned grossly deformed specimens, in an attempt to decide whether the free dislocation link length or the Peierls stress controlled their flow stress. The requisite grossly deformed nickel wafers, the same that were investigated by Riecker et al. 3, were kindly provided by these authors. EXPERIMENTAL

The specimens employed in the present experiment were 0.010 in. thick, 99.995~ pure Ni wafers that had been deformed at 27°C between a pair of tungsten carbide anvils with circular faces of 0.25 in. diameter, as described by Riecker et al. 3. In this, the upper anvil was rotated at a constant speed of nine revolutions per hour while the torque necessary to prevent rotation of the lower anvil was measured, thus allowing a determination of the shear strength of the wafer as a function of pressure. For examination in the transmission electron microscope, the wafers were electrolytically thinned in a 50~o by volume sulfuric acid solution at a current density of about 15 amp/cm 2. In order to facilitate the thinning, which proved to be rather more difficult than had been anticipated, a solid state switching device was constructed by means of which the potential was applied in pulses of about 0.1 sec duration, at a frequency of four pulses per second. In this way, the gas bubbles evolved during any one pulse convect away from the specimen before the next pulse. In addition, a magnetic stirrer was employed during the thinning process. The thinned regions were carefully separated from the main part of the wafer by a rotating cut with a sharp razor blade, and then mounted between two microscope grids for observation in either the Siemens Elmiskop 1A or the RCA 500 kV electron microscope.

! !2

W.A. JESSER, D. K U H L M A N N - W I L S D O R F

Fig. 1. Transmission electron micrograph at 100 kV of a thinned nickel wafer deformed at 40 kilobars. Note the imperfect dislocation structure and relatively large grain size compared with Fig. 2. M = 57000 x.

RESULTS

Figures 1 and 2 are electron micrographs of typical structures found in the thinned nickel specimens. As seen, they contain a high density of dislocations in a structure composed of unusually small crystallites. However, no well-defined dislocation cells such as are typical of f.c.c, metals after creep or straining beyond, say, the middle of stage II of the stress-strain curve, were found. Indeed, recognizable but still only ill-defined dislocation cells were observed in only one nickel specimen. This latter specimen had been deformed at 10 kilobars, the lowest pressure employed in the experiment. The dislocation density, p, was measured by counting the number, N, of dislocation intersections with arbitrary straight lines of length L, superimposed on the micrographs, and substituting into the equation 7 p = 2N/Lt.

(1)

Here t is the film thickness, which was roughly 1500 ~. The average link length, i, of the dislocations was found by measuring the free dislocation lengths between pinning points, as seen projected into the film plane, on the micrographs. The actual link lengths were assumed to be twice t4ae measured lengths, rather than 4/re larger, the factor appropriate to random orientations. This was done in an attempt to correct for the tendency of dislocations in thin films to rotate toward orientations steeply inclined Mater. Sci. Eng., 9 (1972)

to the film plane and thereby to shorten their projected image length in the micrograph. A necessity for this correction was in fact indicated by an apparent inconsistency between the values of i and

Fig. 2. Transmission electron micrograph at 400 kV of a thinned nickel wafer deformed at 100 kilobars. The irregular dislocation structure and relatively small grain size are to be compared with Fig. 1. Note that well developed dislocation cells are virtually absent from both. M = 57000 x.

1 13

F L O W STRESS A N D DISLOCATION STRUCTURE OF N I C K E L TABLE 1

Relevant experimentally determined data : p is the hydrostatic pressure imposed during the shear deformation at the flow stress "c; the mean cell diameter is d, and the mean free dislocation length between adjacent nodes is l, while p is the measured dislocation density and G the shear modulus at pressure p as given by Bridgman 1°. Parameters derived from the experimental data are m r = 12p (eqn. (3)), and values for e obtained by the use of eqns. (5a) and (5b) employing n = 3, v =0.31 and b = 2.51 ~.. Finally included are computed values tor (r - % ) , where the preponderant contribution to r 0 is believed to be the Peierls-Nabarro stress. (r-"co)* is obtained from using the empirical relationship eqn. (4) with e computed from eqn. (5b), while (z-"c0)** is derived from the average linklength, l, as the stress required to bow out the longest links according to the meshlength theory, employing eqn. (2) with n=3, b=2.51 A, and v=0.31.

Specimen No.

p [kbar]

r [kbar]

G [kbar]

(l

p

[A]

[101° cm - z]

1 [A~]

mf

c~

(r - "co )*

("c- "co)**

[kbar]

[kbar]

2.7 2.8 2.6 3.4 2.1

3.14 4.19 2.36 3.66 1.70

f r o mfrom Ni Ni Ni Ni Ni

1-1 1 6 1 1l 1 12 1-14

10 40 90 100 120

1.8 6.0 9.4 9.8 10.5

815 861 937 950 980

5500 470 610 300 940

12.7 11.6 8.0 15.3 4.4

240 180 400 240 620

measured dislocation densities but, in any event, none of the dislocation measurements is sufficiently accurate for a reliance beyond, say, a 30~o to 5 0 ~ margin to be placed on them, particularly since some rearrangements of dislocations almost certainly occurred once the pressure was released, as discussed below. According to workhardening theory 4, ] is related to the flow stress, ~, as Gb(1 - v / 2 ) , - T° = 2~nni(1 - ~ ) m (ni/b)

(2)

where ro is the frictional stress on the dislocations including the Peierls-Nabarro stress, G is the modulus of rigidity, v is Poisson's ratio, b is the magnitude of the Burgers vector, and n-~3 is a geometrical parameter. Further, p and i are related by p = mf/l z

(3)

where f, typically about 1/5 in f.c.c, metals deformed at low or vanishing pressure, is the fraction of volume occupied by the dislocation cell walls, and m is a geometrical parameter found to be near 5 for pure f.c.c, metals 4'8, rendering mf~-1. Since p~_ 1/l 2 was recently found also to be true for heavily drawn iron wire 9, eqn. (3) with m f ~- 1 is believed to have in fact a rather wide applicability. From eqns. (2) and (3) follows the empirically well-known relationship between the flow stress and the dislocation density r - Z o = ~Gb p ~ ~- Gb p~ / 3 with Mater. Sci. Eng., 9 (1972)

(4)

0.75 0.38 1.28 0.88 1.69

_

eqn. (5b)

eqn. (5a)

0.378 0.382 0.393 0.324 0.413

0.368 0.349 0.401 0.368 0.429

(l-v/2)ln(ni/b) 2~n (1 - v)

(5a)

or

(1

- ~(1Z~

ln(3/p~b)

(Sb)

according to refs. 4 and 9, provided that m.f~ - 1, i.e. p~_ 1/l z as explained, and that n_~3.

Measurements of p and i, along with the average crystallite diameter, d, obtained from the nickel samples, are given in Table 1. The values of the flow stress, r, listed in the table are those experimentally determined by Riecker et al. 3 while the values of the shear modulus, G, were obtained from work reported by Bridgman 1°. Also listed are values for m f a n d obtained from the experimental data on the basis of eqns. (3) and (5), as well as (z - % ) values that would be deduced from eqn. (2), and from eqn. (4)in conjunction with eqn. (5b), respectively, on the basis of the measured values of ~,i and p. In Fig. 3, the experimental as well as some theoretical results are displayed. The topmost curve is an interpolation of the measurements of the flow stress z, according to ref. 3, which are indicated as solid dots. As will be seen, a problem exists at pressures below 40 kilobars, or more specifically, with the measured flow stress at 10 kilobars pressure. Namely, the most natural curve taking account of all data points would pass through the origin as is indicated by the dashed branch of the curve. However, strongly deformed nickel samples exhibit substantial flow stresses also when they have been strained at atmospheric pressure, and thus the curve could not possibly pass through the origin. For

114

w . A . JESSER, D. K U H L M A N N - W I L S D O R F

example, the polycrystalline nickel specimens tested at atmospheric pressure by Bridgman 1 and Haasen and Lawson 2 exhibited final tensile stresses referred to the original cross section of 5.4 and 3.6 kilobars, respectively. Since the strains concerned were well below those employed in the present samples, the actual flow stress value at vanishing pressure could hardly be less than 2 kilobars even taking into consideration the somewhat higher purity of the wafers and the fact that the Schmid factor for the shear type test performed in the opposed anvil apparatus could have been perhaps only half as large as in the tensile tests of refs. 2 and 3. Indeed, Riecker and Towle 11 have pointed out the said difficulty in their technique stating: "Valid shear strength data can be obtained only above a threshold pressure which may be quite high for materials which are intrinsically strong and which workharden extensively. This threshold.., is typically in the range of 10-50 kilobars". For this reason, then, the measured point at p = 10 kilobars was disregarded as being almost certainly in error. The solid curve drawn is that resulting from as smooth an interpolation of all other points as possible.

axis at the same point as the flow stress curve, for the reason that the frictional stress, %, is known to be negligibly small at atmospheric pressure. Essentially, this curve represents the flow stress that would be determined for the samples at atmospheric pressure, after they had been predeformed to saturation at the respective high pressures. Its initial rise is consistent with Haasen and Lawson's observation: that specimens deformed at 5 kilobars pressure showed a slightly higher flow stress on retesting at normal pressure than did samples predeformed to the same degree at atmospheric pressure. In the case of aluminum single crystals deformed to high strains this effect amounted to 4 ~ , in the case of moderately deformed nickel polycrystals to between 1 and 2 ~o. This is in quite good agreement with the value of 4 ~o that would be deduced for 5 kilobars pressure for the highly deformed nickel wafers from the initial slope of the (~-To)i-curve of Fig. 3. The downward slope of (z-Zo)i at high pressures, signifying a reduction of dislocation density, is consistent with the fact that, because of their associated volume expansion, the formation energy of dislocations rises by roughly one-fifth as the pressure is raised from atmospheric to 100 kilobars.

12-

DISCUSSION

10-

t o B

4

xj

/ /

__

l, 20

40

--Pressure

SO

80

100

120

(kilobar)

Fig. 3. Graphic representation of quantities listed in Table 1. The flow stress, z, experimentally determined by Riecker et al. 3, is given as solid circles interpolated by the topmost solid curve, while the lower curve indicates (z-%)i, the interpolated values of e~Gb~/p=(~-~o)*, (given as open circles) and of etGb/[= (x-v0)** (given as crosses). The probable value of the PeierlsNabarro stress, %, is found as the difference between the named two curves and is the trace in the middle.

The open circles shown in Fig. 3 are the computed values for (T-Zo)* and the crosses for (z-%)**. These data points are interpolated by the curve labelled (z-Zo)i, drawn so as to intersect the stress Mater. Sci. Eng., 9 (1972)

The most important direct result of the investigation embodied in Table 1 and Fig. 3 is that neither the experimentally determined values of the dislocation density, nor the mean free link lengths, nor even the overall appearance of the dislocation structure beyond 10 kilobars, show any dependence on pressure sufficiently strong to account for the rapid rise of flow stress with pressure. It is therefore concluded that the major source of the pressure dependence of the flow stress must be a strong rise of the frictional stress, %, with pressure. Since of all contributions to ~o only the Peierls-Nabarro stress could be significantly dependent on pressure as far as simple theory is concerned, it is, therefore, proposed that the large observed rise of flow stress under pressure originates in the pressure dependence of the Peierls-Nabarro stress. The magnitude of this stress as a function of pressure as it would result from the present measurements is found as the difference between the flow-stress curve and the ('c - Zo)i-curve. This has been entered in Fig. 3 and is the middle straight line passing through the origin. Admittedly, both interpolation curves on which

FLOW STRESS AND DISLOCATION STRUCTURE OF NICKEL

the deduced value of the frictional stress is based are somewhat adjustable and have partly been chosen so as to yield the indicated proportionality between To and pressure, where in fact there could be, and probably is, a mild curvature as well as a small positive value at p = 0 . Except for such minor variations, we believe the z(p)-dependence to be much as represented in Fig. 3. In this, the most disturbing aspect is the fact that the results ofrefs. 1 and 2 are substantially at variance with those of ref, 3, indicating a very much smaller pressure dependence of the flow stress which, because of the satisfactory agreement with respect to the contribution due to workhardening, means almost no pressure dependence of r o according to refs. 1 and 2. At this point the resolution of this conflict is impossible, especially since additional disagreements exist among the various experimental findings of high-pressure strength data and some results by Russian scientists, particularly with regard to Bridgman's work, as was noted by Towle 12. Pending more experimental work, we, therefore, believe the work of ref. 3 to be the most reliable. That the frictional stress dominates the flow stress at high pressures, as concluded above, is strongly indicated also by the observed dislocation arrangements. If the frictional stress represents the greater part of the total flow stress, the formation of cell walls must be inhibited because the stresses which arise from dislocation interactions and provide the driving force for cell wall formation 4' 13- ~5 are then opposed by the, on the average, larger frictional stress. This very situation is indicated by the already quoted results that at the higher pressures cell walls are very imperfectly formed, as illustrated in the micrographs, and that the best, but still imperfectly developed, cell structure was found at the lowest pressure. It follows also that some dislocation rearrangement must have taken place after completion of the deformation. Namely, at the high pressures, the dislocations could not have moved except in response to stresses above the heightened frictional stress. Undoubtedly, therefore, internal stresses up to that level must initially have existed, and must have been reduced to about, or below, the level of (~-~o) as given in Fig. 3 when the pressure was removed and the now more easily mobile dislocations rearranged. Presumably, the said rearrangements were not very large, since, at least for deformation pressures above 10 kb, they did not lead to dislocation cells or any other structures that would be Mater. Sci. Eng., 9 (1972)

| 15

clearly recognizable as having a low energy content. On the other hand, the comparatively large and erratic differences in the observed structures (compare Figs. 1 and 2) as well as the considerable scatter in the measured values of i and p (see Table 1 and Fig. 3) are quite likely the result of the discussed rearrangements, as they were influenced by the varying conditions of temperature and speed with which the pressure was removed. While we are thus uncertain about details of the dislocation structures that existed during the detormation, the conclusion that the frictional stress, To, controlled the deformation at high pressures seems not to be in doubt. The only apparent dependence of microstructure on pressure observed in the micrographs is that of the average crystallite diameter, d. Grain sizes in the specimen deformed at the lowest pressure were considerably larger than those in specimens deformed at high pressures. If this should be consistently true it could be of considerable importance in determining the workhardening rate (but not the flow stress 4'9) in stage II, but in the present samples the strains were very large, workhardening had ceased, and the deformation was more closely akin to creep than to normal low-temperature deformation. Hence, it is likely that the pressure dependence of d is the symptom of, rather than the cause of, restricted dislocation mobility at high pressures. Indeed, the total number of revolutions of one anvil during deformation varied slightly from specimen to specimen (between 0.48 and 1.12 revolutions), but there was no obvious correlation between d, i or p and the number of revolutions. A physical reason why the Peierls-Nabarro stress should rise with pressure is not hard to find. The extra volume that the atoms in the dislocation core occupy, compared with the same number of atoms in the undisturbed material, has been estimated and is in the order of one atomic volume per lattice plane (compare for example ref. 16). As the dislocation moves, this extra volume increases and decreases, going through one complete cycle while the dislocation moves through one Burgers vector distance, this being sometimes referred to as the "breathing" of the dislocation core. Assuming that the correlated volume change, AV per atomic plane of thickness b, varies sinusoidally with distance and has a maximum value of AVo, the excess energy per unit dislocation length under pressure p would be 1

P AV0 cos

2~z

x,

(6)

116

w . A . JESSER, D. KUHLMANN-WILSDORF

and the resulting stress on the dislocation, 1 dE

ZAv -

2 n p A V o . 2n

b dx -

b~

sin ~ - x

(7)

for a contribution to Zo of 2~zpAV° "CAVmax--

(8)

b3

With b -~ 2.5 A, the measured value of Zo --- 10 kilobar at p~- 150 kilobar would thus correspond to AVo/b a __ 10/(2rc x 150) --0.01, or roughly one-hundredth of an atomic volume change per atomic plane. This seems to be an entirely reasonable number. Also the nearly linear rise Of Zo with pressure, as seen in Fig. 3, is accounted for on the basis of eqn. (8).

strength with the shear strength calculated from the mesh-length theory of workhardening 4'8'9, on the one hand, and the empirical relationship between flow stress and dislocation density, on the other hand, it was concluded that the Peierls-Nabarro stress makes an increasing contribution to the flow stress, becoming dominant at pressures larger than about 50 kilobars. The volume change associated with the "breathing" of the core, as a dislocation passes over Peierls-Nabarro hills and valleys, that would semi-quantitatively account for the measurements, is in the order of 1~o of an atomic volume per lattice plane.

ACKNOWLEDGEMENTS CONCLUSIONS

Nickel wafers deformed in an opposed anvil apparatus 3 at pressures of from 10 to 120 kilobars were examined by transmission electron microscopy. The dislocation density, mean free link length and microstructure were observed to be substantially pressure independent. F r o m these observations, as well as from a comparison of the experimental shear

We are greatly indebted to Drs. Riecker, Towle and Rooney for supplying the samples used in this investigation. Further, sincere thanks are due to Dr. J. T. Moore, now of the Tennessee Eastman Company, Kingsport. Tennessee, who devised and constructed the switching device employed in the specimen preparation. The financial support of this research by the U. S. Atomic Energy Commission is gratefully acknowledged.

REFERENCES 1 P. W. Bridgman, J. Appl. Phys., 24 (1953) 560. 2 P. Haasen and A. W. Lawson, Jr., Z. Metallk., 49 (1958) 280. 3 R. E. Riecker, L. C. Towle and T. P. Rooney, Air Force Cambridge Res. Lab. Rept. 67-0475, 1967. 4 D. Kuhlmann-Wilsdorf, in J. P. Hirth and J. Weerman (eds.), Workhardenin9, Gordon and Breach, N.Y., 1968, p. 97. 5 L. C. Towle and R. E. Riecker, Appl. Phys. Letters, 13 (1968) 159. 6 D. Kuhlmann-Wilsdorf, Phys. Rev., 120 (1960) 773. 7 E. E. Underwood, in R. T. DeHoff and F. N. Rhines (eds.), Quantitative Microscopy, McGraw-Hill, 1968, p. 78.

Contrainte d'dcoulement et structure de dislocations du nickel ddformd sous de trks fortes pressions

On trouve que dans des 6prouvettes de nickel s6v~rement d&orm6es par cisaillement la r6partition et la densit6 des dislocations, ainsi que la longueur moyenne des segments de liaison, sont en grande partie ind6pendants de la pression exerc6e normalement au plan de cisaillement. Par contre, Riecker, Towle et Rooney avaient trouv6 que la contrainte d'6coulement de leurs 6prouvettes augMater. Sci. Eng., 9 (1972)

D. Kuhlmann-Wilsdorf, Trans. AIME, 224 (1962) 1047. D. Kuhlmann-Wilsdorf, Met. Trans., 1 (1970) 3173. P. W. Bridgman, The Physics of High Pressure, 1952, p. 386. R. E. Riecker and L. C. Towle, J. Appl. Phys., 38 (1964) 5189. L. C. Towle, J. Appl. Phys., 42 (1971) 2368. D. Kuhlmann-Wilsdorf, E. E. Laufer and H. Nine, J. Appl. Phys., 38 (1967) 896. 14 H. D. Nine and D. Kuhlmann-Wilsdorf, Can. J. Phys., 45 (1967) 865. 15 D. Kuhlmann-Wilsdorf and H. Nine, J. Appl. Phys., 38 (1967) 1683. 16 D. Kuhlmann-Wilsdorf, J. Appl. Phys., 36 (1965) 637. 8 9 10 11 12 13

mentait notablement avec la pression. D'autre part, les dislocations ne sont pas arrang6es en parois bien d6finies. Ces deux observations conduisent penser que la force de frottement qui s'exerce sur les dislocations a un effet pr6pond6rant sur la contrainte d'6coulement en pr6sence de fortes pressions. Cet effet est responsable de la variation de la contrainte d'6coulement avec la pression. L'analyse quantitative des mesures montre que la force de frottement est (sensiblement) proportionelle/~ la pression. Si l'on attribue cet effet/t une

FLOW STRESS AND DISLOCATION STRUCTURE OF NICKEL

117

modification de volume du coeur des dislocations lors de leur mouvement fi travers les collines et les vall6es de Peierls, on trouve que ce changement de volume est voisin d'un centi~me de volume ato-

mique par plan r6ticulaire. Ce chiffre parait raisonnable si on le compare/t 1'augmentation de volume totale qui est de l'ordre d'un volume atomique par dislocation et par plan r6ticulaire.

Die Flieflspannung und Versetzungsstruktur yon Nickel nach Verformung bei hohen Drucken

der Fliel3spannung ausmacht und somit ffir deren Druckabh~ingigkeit verantwortlich ist. Eine quantitative Analyse der Messungen ffihrt auf einen (nahezu) proportionalen Zusammenhang zwischen innerer Reibung und Druck. Wird dieser Effekt der Volumen~inderung des Versetzungskernes bei seiner Bewegung fiber Peierls-Berge und -T~iler zugeschrieben, so ergibt sich die Gr613e dieser Volumen~inderung zu etwa ein hunderstel Atomvolumen pro Gitterbene; dieser Wert erscheint im Vergleich mit der gesamten Volumenexpansion von etwa einem Atomvolumen pro Versetzung und Gitterebene sinnvoll.

Die Anordnung, Dichte und mittlere Bogenl~inge der Versetzungen in den unter hohem Druck stark gescherten Nickelproben sind im wesenlichen unabh~ngig vom Druck, obwohl Riecker, Towle und Rooney eine starke Zunahme der Fliel3spannung mit dem Druck beobachtet hatten. Die Versetzungen sind auBerdem nicht in wohldefinierten Zellw~inden angeordnet. Beide Ergebnisse sprechen daffir, dab die die Versetzungsbewegung behindernde Reibung bei hohen Drucken den gr6Beren Teil

Mater. Sci. Eng., 9 (1972)