Theory of the formation of lattice defects during plastic strain

THEORY

OF THE

FORMATION

OF LATTICE H.

G. VAN

DEFECTS

DURING

PLASTIC

STRAIN*

BIlERENt

A simple theory is presented by which a relation between the plastic strain and the concentration of vacancies, interstitials and dislocations is obtained. Dislocations are formed by sources under the action of an applied stress, the other defects by the movement of jogs in the dislocations. The action of a dislocation source under a varying stress is studied, for the case of static as well as of dynamic generation. After the first few per cent of strain, both methods of generation yield the same defect concentrations. By suitable elimination, the influence of workhardening can be left out of the theory. The results apply to zero temperature and plastic strains of 0.05 up to 1. They are compared with the observed critical shear stress, the elementary structure and the resistivity-strain relations in slightly deformed copper. THEORIE

DE

LA FORMATION DES DBFAUTS RETICULAIRES DEFORMATION PLASTIQUE

PENDANT

LA

11 est prLsent6 une theorie simple donnant une relation entre la deformation plastique et la concentration des lacunes, atomes interstitiels et dislocations. Les dislocations sont formCes par des sources sous l’action de la contrainte appliquee, tandis que les autres dCfauts r&ultent des “jogs” dans les dislocations. L’effet d’une contrainte variable sur une source de dislocations est Ctudie aussi bien pour la formation statique que dynamique. Apres les premiers pourcent de deformation, les deux modes de formation conduisent a la m&me concentration de defauts. Par un artifice appropri6, l’influence de 1’Ccrouissage peut Ctre CliminCe de la thtorie. Ces conclusions s’appliquent aux deformations B basses temperatures allant de 0,OS B 1 et elles sont compa&es au cisaillement critique, a la structure Clementaire et a la relation rCsistivitCdeformation dans le cuivre faiblement dCform6. EINE

THEORIE

DER

BILDUNG VON GITTERFEHLSTELLEN PLASTISCHEN VERFORMUNG

BE1 DER

Es wird eine einfache Theorie beschrieben, durch die die Zusammenhange zwischen der plastichen Verformung und der Konzentration von Leerstellen, Einlagerungen und Versetzungen gedeutet werden k6nnen. Unter der Einwirkung einer angelegten Spannung bilden sich aus den Quellen die Versetzungen; die anderen Fehlstellen entstehen durch die Bewegung der Treppenstufen (“jog-Sprung) in den Versetzungen. Fur den Fall der statischen, wie such fur den der dynamischen Wirkung, wird das Verhalten einer Versetzungsquelle bei unterschiedlicher, angelegter Spannung untersucht. Beide Wirkungsweisen ftihren nach den ersten wenigen Prozenten Verformung zu den gleichen (resultierenden) Fehlenstellenkonzentrationen. Durch eine geeignete Eliminierung kann der Einfluss der Verfestigung aus der Theorie eliminiert werden. Die Ableitungen mtissten bei niedriger Temperatur und Verformungen zwischen 0.05 und 1 gtiltig sein. Sie werden mit den beobachten Werten der kritichen Schubspannung, der Elementarstruktur und den Widerstands-Verformunts-Besiehungen in wenig verformtem Kupfer verglichen.

1. INTRODUCTION

strain. In section 4 a comparison is made between the theoretical deductions and the observed phenomena, notably the strain dependence of the electrical resistivity of copper.

It is generally accepted that plastic deformation of crystals, especially of metals, is governed by the action of dislocation sources under the influence of the applied stress. Another well conceived idea is the formation of vacancies and interstitials during deformation by some mechanism connected with the mutual crossing of dislocations. In this paper a theoretical discussion of the detailed processes occurring in deformed crystals is attempted and applied to the case of simple f.c.c. metals such as copper, deformed at very low temperature, where no thermal activation takes place. Section 2 deals with the behaviour of dislocation sources under a varying applied shear stress. It will be shown that the hypotheses of static and dynamic generation of dislocations yield approximately the same results. In section 3 theoretical relations are derived between the concentrations of the various kinds of lattice defects formed during the deformation, and the amount of plastic * Received February 21, 1955. t Philips Research Laboratories, fabrieken, Eindhoven, Holland. ACTA

MET_1LLURGICA,

N. V. Philips’

VOL. 3, NOVEMBER

2. FRANK-READ

UNDER

VARYING

STRESS

We consider a dislocation source of length 1. Several kinds of sources have been proposed in literature: the classical Frank-Read source that emits dislocation loops in one atomic plane only, and sources of a more general type, that produce loops on successive atomic planes.‘*2 The following treatment applies to all these sources. The critical shear stress for activation of a source is q=-

aGb 1 ’

(1)

where G is the shear modulus, b the Burgers vector of the source, and a: a numerical constant of order unity. Let the number of emitted dislocation loops be n and the area covered by them be A. This area is determined by the local stress 7 near the source in the following way.

Gloeilampen19.55

SOURCE

519

520

ACTA

METALLURGICA,

Let the density of randomly distributed dislocations in the crystal be Do. Presumably these dislocations are arranged in a spatial network, the elements of which can each in their turn act as sources. A loop of area A has crossed ADO dislocations, and has formed ADo jogs. The jogs have run a distance of the order At, and, assuming that the dislocation loop retains its shape on expansion, have left in their wake a number of defects (vacancies or interstitials) which is given by P-4 ‘D,

f=7

(2)

where /3is a numerical constant which depends on the shape of the loop and on the efficiency with which the jogs form defects. For a circular loop moving in a field of pure screw dislocations perpendicular to its plane p= 2/31r5/2=0.04; in general p will be smaller, say, 0.02. It may be argued2 that in actual cases the efficiency of defect formation is much less, as jogs in successive loops can eliminate each other. However, if the loops follow each other in distances of the order of a thousand atomic distances, this process will not take place. Also, jogs may move along the loop instead of with them, or may perform a combined motion. In order not to produce defects, this motion must confirm to rather severe restrictions, which makes it improbable that this effect reduces the efficiency of defect formation appreciably. Let the mean energy of formation of a defect be U. It can easily be shown that the energy needed to form these defects in copper for instance, is larger than that involved in the formation of jogs and increases much faster than the latter and than that involved in the elongation of the dislocation line. The energy of defect formation is then the leading factor in the energy-balance governing the expansion of a dislocation loop. Thus, in sufficient approximation, this expansion is associated with the storage of an amount of energy of about fU,. The energy suppliedybyzthe local shear stress is simply E=rAb. It is known from various experiments3 that only part of the supplied energy is stored in a metal in the form of lattice defects which (at low temperature) remain after the deformation, the other part being lost as heat. The fraction retained is XE, where X is a numerical constant, its value depending on the method of deformation. The ultimate area reached by a dislocation loop under a shear stress 7 is thus limited by the equation P-4 *Do XrAb= ---r, b

VOL.

3, 195.5

from which it follows that : A=

Xrb2 2 . 1@DoU, I

(3)

The emitted loops produce a back-stress at the site of the source. The component of this stress that opposes the local activating shear stress is given by T,,=-

(4)

A+ ’

where y is a constant of order 0.1, depending on the shape of the loops and on the value of Poisson’s ratio. It was silently assumed that all loops reach the same area A, this is certainly not so. However, on the average the area covered by the loops will indeed be of the order of A as given by (3). The time necessary for the loop to attain its final area after the emission has taken place will be very short. Even if the loops run with a velocity of the order of only 1 per cent of that of sound, this time is of the order of 1W sec. In this time the local shear stress has not changed appreciably, and one can insert in (4) the value of A which follows from (3). Thus, one finds rb=

BrGDo Uf n Xb

i

(5)

The source will be activated again when the shear stress at the site of the source, T-U,, attains anew the value TV,i.e., when /3yGDoUf n cllGb -=----* 7(6) Xb r 1 It is frequently assumed that, according to an hypothesis of Fisher, Hart and Pry,4 a source once activated continues emitting dislocation loops until the resultant stress at the site of the source has dropped to a value as low as about *T,. This should happen when the dislocations move with approximately the velocity of sound in the material, so that their kinetic energy attains an appreciable value. This dynamic generation of dislocations can be opposed to the static generation, in which at each activation process only one loop is emitted. Equation (6) holds for both cases. However, in the case of static generation, n takes successively the values 1,2,3,. . . , whereas in the case of dynamic generation the loops are emitted in sudden bursts, containing ek loops at the Rth activation process. The numbers nk are limited by the condition that the back stress exerted by these loops alone shall be equal to about $71, as the stress at the site of the source at the moment of activa-

VAN

tion is precisely

BUEREN:

TV.According

FORMATION

OF

D = L?Tn6A 4, 2

-=TTl,

Xb

Tk

or TlTk.

nkz=-

3 PrGDoUf The number of loops per burst is thus proportional to the activating stress rk. We shall show below that after a few per cent of strain, the total number of loops emitted is large-of the order lOoand it is easily demonstrated that even in the case of dynamic generation, the error involved by replacing the discontinuous relation (6) by a continuous relation, is small. There is, then, no difference between the two forms of behavior of dislocation sources and we arrive at

(7) where 10stands for the expression

and ~1 is again equal to aGb/l. The physical significance of the quantity ZO is observed as follows : In order to activate a source of length I, the critical semicircular shape of the dislocation segment must be produced by the acting shear stress. This segment has an area ?rP/S; it contains, according to (2), ,fcr=bD,Jb. (&rP)+defects. The energy needed to form these defects must be smaller than the supplied energy hGb2/1. (+I%); thus one has : (9) When

this condition

is not fulfilled-that

is, when

(10) the source cannot be activated. maximum length of a dislocation Frank-Read source. 3. DENSITY

One may thus read for la the segment that can ever act as a

OF DISLOCATIONS, AND INTERSTITIALS

VACANCIES

Suppose the number of activated sources of length 1 is N per cm3, and that only one glide system is present (single glide). These sources contribute to the total strain of the material the amount : t = NbnA . The total number

01)

of defects formed by these sources is

(12)

521

DEFECTS

and the total length of dislocation

to (S), one has

PrGDoU, ‘Bk

LATTICE

formed is (13)

where 6 is a numerical constant depending om the shape of the loops. For circular loops, 6= 247~; in general 6 will be somewhat larger-say 6 = 5. In these formulae the dependence on the stress is governed by the quantities X, n and A. Now these quantities are functions of the local stresses r near the various sources. These local stresses are unknown and their dependence on the applied stress involves the workhardening characteristics of the solid. We therefore want to eliminate 7 from the expressions (ll), (12) and (13). Assuming that T/T1 is at least so much larger than unity that in (7), the term 7,‘~~ can be neglected in comparison with (T/Q (this assumption holds after the first few per cent of strain), one finds with the help of (3) and (7) : a p3yXGDo” c5/4 F= (14) u’ SI;;b6 ’ and ’ &VU,D, 3/4 D=S\ i --t (15) TXGb” ’ It is seen that these formulae contain 1 only implicitly, viz., in the quantity ~1’. As LY occurs only as 11’2,the rise of N on increasing deformation, when increasingly shorter sources are being activated, has only a minor influence on the magnitudes of F and D, and can therefore be neglected. That means that, effectively, the relation between defect concentration and strain is independent of the length of the sources, and one may take for h: the total number of activated sources (of arbitrary length) per cm3. The occurrence of the fourthdegree root has also the important effect that no precise knowledge of the numerical values of the parameters /3, y, X, ~1’and U, is needed. Formulae (14) and (15) have been derived under the assumption that single glide occurs. Then the quantity Do, the density of crossing dislocations, can be regarded as a constant. However, in multiple glide the “forest” of dislocations crossed by the expanding loops becomes thicker as the deformation proceeds, owing to the dislocations emitted by sources on intersecting glide systems. Suppose there are R active glide systems; the total number of activated sources being Jr, the density of active sources on a given glide plane is then of the order of magnitude A = %d/g

(16)

sources per cm2, if d is the mean mutual distance between two active glide planes of one system. It is a simple problem to compute the average number of dislocation zones cut through by a given zone on a certain glide plane. Assuming that all zones have the same area A, and that the deformation can be considered as homogeneous, this average number is found to be

ACTA

522

METALLURGICA,

about: lvi=lvki+,

(17)

a numerical factor of order unity being neglected. Every zone cut through contains n dislocations. Thus the total number of dislocations crossed by a given dislocation loop becomes on the average equal to D&+NnAs. That is to say, one should, in the case of multiple glide, replace the constant DO by Do’= Do+ NnA*.

(18)

Taking N = 10” cm+, n = 50, A = 10F6 cm2, values that probably occur after a few per cent of strain (compare next section), it is seen that the second term at the right-hand side of (18) has a value of the order of .5X10s cme2. As DO only amounts to about lo8 cmP2, this can be neglected. The energy balance describing the expansion of a loop now reads: /3NnA2 XrAb= -U,, b

from which is follows that Xrh2

&=L..I-

(20)

PNUr for multiple

glide. Thus, hb3

e=Nb.nA=---r

(21)

PUf

and from (12) and (18) it follows that Fcf.

3,

1955

for the various constants occurring in them. We shah confine ourselves to the case of copper. As already mentioned, for the parameters (Y,0, y and 6 one can put: (~=l, @=0.02, r=O.l, 6=5, with a fair degree of accuracy. The value of X is very uncertain. From the measurements quoted in3 it follows that for deformation at room temperature X is observed to be about 0.05. At very low temperatures this value would probably be found somewhat greater, as then no defects diffuse out of the material. We shall tentatively put X=0.1, which means that 10 per cent of the strain energy remains stored in the metal. G is known to be 4X10n dynes/cm2, b is known from the lattice dimensions of Cu to be 2.5X 10es cm. The energy of formation of a vacancy is somewhat smaller than 1 eV; that of an interstitial is about 5 eV5. As probably many more vacancies than interstitials are formed in view of the large difference in the energy involved, we shall take lJ,= 2eV=3X lo-l2 ergs. In single glide, the dislocation density can, as shown above, be put equal to Do, tEe density of originally present dislocations in the network ; various authors687 agree that Do-lo8 cmm2. The number N of active Frank-Read sources in a metal can be estimated to be of the order of a few tenths of D&-that is, of the order 10n cme3. As said above, the rather large uncertainty in N does not appreciably influence the results. We thus find, taking the uncertainties into account : F = (0.5 - 1.5) X lOr,~~~*cmw3,

(24)

D= (OS-

(25)

1.5)X1011~384cmm2,

From (8) we find for the length lo

X2b3 (,4)2=-_2

(22) PUf2

VOL.

*

Hence in the case of multiple glide the relation replaced by the extremely simple relation

(14) is

lo= (4- 10)X 10d4 cm.

From (3), (7) and (8) one can easily deduce that after the first few per cent of strain: A= (0.8-2)XlO-‘%~

Fr_‘BE2.

cm2

n= (200- 500) X et.

‘b3

An expression corresponding to (15) is not so easily derived. We shall not go into this here. It may be mentioned that neither (21) nor a corresponding expression that can be derived from (3), (7) and (11) can be considered as describing the workhardening characteristics of the material. The quantity r has not a direct physical significance ; it is only the mean local shear stress near an activated source and may differ appreciably from the applied shear stress. Further, all results apply only when the influence of thermally activated processes is neglected, that is at very low temperatures. OF THEORY

WITH

EXPERIMENT

The various expressions occurring in section 3 can be evaluated numerically by inserting appropriate values

(27) (28)

The value of la corresponds to a critical shear stress for the beginning of plastic deformation which is given by: aGb To=-. 10

One finds from the theory TO= (1- 3) X lo7 dynes/cm2. For pure copper single crystals this stress is observed to be about 0.1 kg/mm2= lo7 dynes/cm2.8 In the case of multiple glide the relation between the concentration of point defects and the strain followc from (23) to be for copper: F= 1.3X1021g2 cme3.

4. COMPARISON

(26)

(29:

With respect to the order of magnitude, during the first few per cent of strain there is not much differenct between (24) and (29). On further strain, however, I

VAN

BUEREN:

FORMATION

increases much faster in multiple glide than in single glide. The dimensions of the dislocation zones around activated Frank-Read sources are in good agreement with the characteristics of the elementary structure found by W’ilsdorf and Kuhlmann-Wilsdorft if one interprets this structure as the surface markings produced where activated sources, of one glide system only, dissect the surface. The length of these lines as observed on copper and aluminium is a few times 1O-3 cm, the step height lies between 20 and 100 atomic distances ; according to the theory given, after 10 per cent of strain, the length of the elementary lines should be of the order of As= 10M3cm, and the step height should be on the average equal to YZ=100 atomic distances. According to the theory, the mutual spacing of the lines should be of the order of (NA)-’ cm, which amounts to a few times 1tY5 cm, whereas (2-5) X 10v6 cm was observed for this spacing. However, near the surface the density of activated sources is probably much greater than in the interior. By far the most important information on the concentrations of lattice defects is obtained by the measurement of the electrical resistivity as a function of the strain. From the work of JongenburgerrO and Hunter and Nabarro,‘l it follows that one vacancy per cm3 in copper contributes 1.5X 1O-21 &cm to the resistivity, one interstitial atom about three times as much, viz., 4.5 X lo-?’ pfi cm, and a density of D dislocations per cm2 the amount of 0.4X lo-l4 0~52 cm. Inserting in (24) and (25) yields for the theoretically expected resistivity-strain dependence in single glide : Ap = (0.02 - 0.05) X E~‘~+(2 - 8) X 104e3’4.

(30)

It follows that the contribution of the dislocations to the resistivity would be very slight and could be neglected. For multiple glide it can easily be estimated that the contribution of dislocations to the resistivity is also negligible compared to that of the point defects. Thus we have from (29) : Ap-2~~. (31) The observed line materials the form

resistivity-strain curves for polycrystalcan all be described by a power law of Ap=ae’,

(32)

where a is about 0.05, and r lies between 1.2 and 1..5.12 Blewitt13 observed the resistivity-strain relation in copper single crystals and found a relation of the form Ap=O.Ole?.

(33)

He applied large strains of order unity. The value of the coefficient a in both cases, as well as that of the exponent r in the case of polycrystals, are in much better agreement with the theory as applied to the case of single glide than with the theory of multiple

OF

LATTICE

523

DEFECTS

glide. This seems at first sight rather unexpected. However, the theory as presented here deals only with the phenomenon of the so-called fine slip in slightly deformed metals; the strong sliplines, as observed on most heavier deformed metals are probably formed in another way. They should be considered as more or less accidental effects that are not considered in the above treatment. The number of point defects formed in this manner is presumably much less than the number formed in fine slip (probably the circumstances favouring coarse slip are simply the absence or reduced frequency of formation of point defects). From the occurrence of multiple coarse slip it cannot be concluded that the fine slip, considered locally, also takes place on different glide planes. In the experiments of Kuhlmann and Wilsdorfg elementary lines belonging to one glide system only are usually locally observed. Therefore, formula (30) is probably better applied to the actual case than formula (31), especially when the density of the original dislocation network is high. In Blewitts’ experiments on single crystals, this density may have been exceptionally low, favoring the occurrence of some kind of multiple fine slip. The low absolute value of the extra resistivity may also be connected withan exceptionally low value of Do. Van Bueren and Jongenburger lot. cit. alternately extended and twisted polycrystalline copper and silver wires. They observed that intermediate twisting influenced the resistivity-strain relation on subsequent extension in a very special way : the exponent Y remains constant (except for the first few per cent of strain after the twist), the coefficient a, however, becomes multiplied by a factor m which depends on the amount of twist and is 1.3 at a torsional surface strain tt of 0.1. According to Paxton and Cottrell, such a strain causes an increase of the number of homogeneous distributed dislocations by an amount

D,2 rb’

where r is the radius of the wire (0.25 mm). From this it follows that Dt after et=O.l was 1.6X108 cmv2. According to (14), the introduction of Dt homogeneously distributed dislocations has the effect of multiplying the resistivity-strain relation by a factor m.=

when it is assumed that only point defects contribute to the resistivity. From the known values of D, and m, Do can be calculated; it was found to be 3.8X10* cmw3, in reasonable agreement with the expectations. From this and the foregoing it seems that experiment and theory can be made to agree best when it is accepted that dislocations do not contribute more than say 10%

524

ACTA

METALLURGICA,

to the low temperature resistivity. This is in contradiction to the result of recovery experiments12 that indicate that dislocations are responsible for about half of it. This contradiction is at the moment unsolved.* REFERENCES 1. B. A. Bilby, Paper on Bristol Conference on Defects in Crystalline Solids (1954). 2. H. Suzuki, J. Phys. Sot. (Japan) 9, 531 (1954). 3. G. J. Taylor and H. Quinney, Proc. roy. Sot. A143,307 (1934) ; L. M. Clarebrough, M. E. Har leaves, D. Michell, and G. W. West, Proc. roy. Sot. A215, 50 s (1952). 4. J. C. Fisher, E. W. Hart, and R. H. Pry, Phys. Rev. 87,958 (1952). * Note added in proof.-Recent observations by the author on the resistivity change of deformed copper in a magnetic field seem to confirm the conclusion that dislocations are indeed far more effective than concluded above. The only way to explain this is to assume that the theoretical scattering cross-section of dislocations is in error by a factor of 10, as the defect densities computed in (24) and (2.5) are rather to be regarded as upper limits.

VOL.

3,

1955

5. H. B. Huntington

and F. Se&, Phys. Rev. 61, 311 (1941); H. B. Huntington, Phys. Rev. 61, 325 (1941) ; C. J. Meechan and R. R. Eggleston, Acta Met. 2, 680 (1954). ! F. Seitz, Advances in Physics 1, 43 (1952). A. H. Cottrell, Didocations and Plastic Flow in Crystds (Oxford University Press, Oxford, 1953). 8. G. Masing, Lehrbuch der allgemeinen Metallkunde (Springer, Berlin, 1950), and many other authors. Z. angew. Phys. 4, 9. H. Wilsdorf and D. Kuhhnann-Wilsdorf, 361, 409 (1952); Acta Met. 1, 394 (1953). Paper on Bristol Conference on Defects in Crystalline Solids (1954). Appl. sci. Res. B3, 237 (1953); P. Jongenlo* P. Jongenburger, 11 burger, Nature 175, 54.5 (195.5). S. C. Hunter and F. R. N. Nabarro, Proc. roy. Sot. A220,.542 l2

f.9z!‘Pry and R. W Hennig Acta Met 2 318 (1954) ; C. W. Berghout, unpublished; J! Molenaar and’ W. H. Aarts, Nature 166,609 (1950); M. J. Druyvestyn and J. A. Manintveld, Nature 168, 868 (1951); H. G. van Bueren and P. Jongenburger, Nature 175,544 (195.5); W. H. Aarts and R. K. Jarvis, Acta Met. 2, 87 (1954). A review of all data is given by the author in the Zeitschrift ftir Metallkunde 46, 272 (19.55). 13. T. H. Blewitt. Private communication. 14. S. Paxton and A. H. Cottrell, Acta Met. 2, 6 (1954).