MICROSTRAIN IN POLYCRYSTALLINE METALS*

N. BROWN? and K. F. LUKENS Jr.?

A theory was developed for describing the dependence of microplastic strain on stress and gram size of polycrystalline metals. The microplastic strain is given by

y = CpD3(a - Q,~)=/&T,~

where C w l/2. p is the density of sources, u the applied stress, u,, 0 stress to move first dislocation, a

shear modulus and D grain size. The theory was checked by measuring the stress-strain curve in the micro region using a capacitance

type extensometer. The metal investigated was ingot iron with grain sizes from 44 to 140 p. The agreement between the theoretical equation and the experimental data was excellent not only in the case of the iron but for data on copper and zinc from the investigation of others. The density of sources

in ingot iron and copper was 2.5 and 6.9 x lo6 sources/ems, respectively. The range of the microstrain

region has been calculated in terms of the amount of microstrain that occurs at the macroscopic yield point.

As a useful sidelight it turns out that annealing ingot iron at 650°C after a microstrain completely removes the work hardening so that the specimen returns to its initial state. Consequently, the same specimen may be used repeatedly for tests in the microstrain region.

MICRO-DEFORMATION DANS DES METAUX POLYCRISTALLINS

Les auteurs ont developpe une theorie decrivant la dependance de la micro-deformation plastique, de la tension et de la dimension des grains dans un metal polycristallin.

La micro-deformation plastique est don&e par:

y = CpD3(cr - u~“)~/Qu,~

oti C w l/2 p est la densite des sources, u la tension appliquee, u 0° la tension necessaire pour mouvoir la premiere dislocation, (7 le module de cisaillement et D la dimension du grain.

Cette theorie est confirmee par la determination des courbes tension/deformation dans des petite8 regions en utilisant un extensometre type capacitance. Le metal utilise Btait un lingotin de fer dont les dimensions des grains variaient de 44 a 140 ,u. La concordance entre l’equation theorique et les resultats

experimentaux s est revelee excellente non seulement dans le cas du fer mais aussi grace It des resultats experimentaux obtenus par d’autres chercheurs dans le cas du cuivre et du zinc.

La densite des sources dans le fer et dans le cuivre Btait respectivement 2, 5 et 6,9 x IO” sources/cm3. La zone de micro-deformation a et& calculee en partant de la quantite de micro-deformations apparais-

sant aux points de decrochement macroscopique. 11 ressort de ces essais que le recuit a 650°C d’un fer ayant subi une micro-deformation detruit le durcis-

sement. L’echantillon revient a son &at initial. En consequence, le meme echantillon peut Btre utilise a

plusieurs reprises pour des essais dans la region de micro-deformation.

MIKRODEHNUNG IN VIELKRISTALLINEN METALLEN

Die Abhangigkeit der plastischen Mikrodehnung van Spannung und KorngrijDe wird an Hand einer neu entwickelten Theorie beschrieben. Die Plastisohe Mikrodehnung ist gegeben durch

y = CpDa(u - u,“)~/Gu,~.

Dabei ist C w l/2, p die Quelldichte, u die angelegte Spannung, c 0° die Spannung, urn die erste Verset- zung zu bewegen, # der Schubmodul und D die KorngriiOe.

Die Theorie wurde durch Messungen der Spannungs-Dehnungs-Kurven im Mikrodehnungsgebiet gepriift, wobei ein kapazitives Extensometer beniitzt wurde. Das untersuchte Metal1 war GuDeisen mit KorngroBen van 44 bis 140 ,a. Die Ubereinstimmung zwischen der theoretischen Gleichung und den experimentellen Ergebnissen war hervorragend, nicht nur im Fall des Eisens, sondern such fur Daten von Kupfer und Zink aus anderen Untersuchungen. betragt 23 bzw. 6,9 x lo6 Quellen/cm

Die Quelldichte in GuDeisen und Kupfer 3. Die Ausdehnung des Mikrodehnungsbereichs wurde berechnet

mit Hilfe des Mikrodehnungsbetrags, der an der makroskopisohen FlieSgrenze auftritt. Als niitzliches Nebenergebnis stellt sich heraus, daB eine Gliihung des GuBeisens bei 65O’C nach

einer Mikrodehnung die Verfestigung vollstiindig abbaut, so da6 die Probe wieder ihren Ausgangszu- stand annimmt. Folglich konnen die gleichen Proben wiederholt zu Versuchen im Mikrodehnungsbereich verwendet werden.

The very early stage of plastic deformation is can only be observed with a high sensitivity extens- called the microstrain region. Small plastic strains on ometer. There are several investigations of this the order of 1O-4 usually occur before the onset of phenomenon in single crystals of aluminum(r3) and macroyielding. The details of the microstrain region zincc3) and in polycrystalline ironc4) and copper.c5)

The only published investigation on the effect of * Received October 29, 1959; revised December 28, 1959. t Metallurgy Department, University of Pennsylvania,

grain size is the one by Thomas and Averbach(5) on

Philadelphia, Pa. copper. Thomas and Averbach. found that for a

ACTA METALLURGICA, VOL. 9, FEBRUARY 1961 106

BROWN AND LUKENS JR.: MICROSTRAIN IN POLYCRYSTALLINE METALS 107

given applied stress the amount of microstrain varied as P, where D is the grain diameter, Unfortunately, the theory by Thomas and Averbach does not properly explain the dependence of the microstrain on the grain size or the stress. In this paper a proper quantitative theory is presented which explains the shape of the microstress-strain curve and its depen- dence on grain size. The theory is consistent with additional data on iron which was obtained in the present investigation as well as with data on copper and zinc from other investigations.

THEORY

First, we would like to show that the Thomas and Averbach theory is wrong. On page 73 of their paper(s) it is assumed “that an average of one Frank- Read source has been active in each grain.” The authors then give the following equation for the total plastic strain :

A E. &x&z nb 4 L-3 (1)

where n, the number grain, is given by

nN

7,. is applied stress and

of piled-up dislocations per

WV - 7,) (2)

7, is the stress to activate a source. A, is the area swept out by a dislocation as it goes from the source to the grain boundary and A, is the cross-section area of the specimen. Since,

A,ND2 (3)

and b and A, are constants, and substituting equations (2) and (3) in (I), it is seen that according to Thomas and Averbach the amount of microstrain varies linearly with the stress and is independent of grain size. This result is contrary to the experimental observations. Since the microstrain varies non- linearly with stress and as 03.

The present theory starts with the assumption that the number of sources per unit volume is uniform throughout the specimen and independent of grain size. Thus, the strain per ith grain is

yi = n,,D2hpD3/Al (4)

where n( is the number of dislocations emitted by each source in the ith grain, D2 is the cross-section areas of the grain, b is Burgers vector, p is the density of sources, A is the cross-section area of the specimen and 1 is the gage length. As in the Thomas and Averbach theory a linear response is assumed between the number of piled-up dislocations per source and the back stress on the source where($)

n, = (0. - tT,“)KD/Gb (5)

cr is the applied stress troi is the stress to activate a source in the ith grain, K is a constant, -2 and G is the shear modulus. The total strain, 7, in the specimen is equation (4) summed over the total number of grains that contribute

y = ~Y~~~AZ~~Aff~i (6) i

fiAa,” is the fraction of the grains whose sources are activated by a stress between aai and cgi + Aa,‘. Since a random orientation of grains is assumed, the most reasonable form for fi is shown in the foIlowing figure. aao is the stress to activate a source in the most favorably oriented grain and aoM in the least favor- ably oriented grain. For the b.c.c. and f.c.c. metals aoM/~Oo is about 2 .(‘) Joining equations (4), (5) and (6) using the above distrjbution function, and converting the summation to an integral, the final equation for the plastic microstrain as a function of applied stress and the grain size is obtained.

y = CpP(a - aoo)21Gaoo

where C is a constant, *l/2.

(7)

Thus, the above equation predicts (1) the stress to first activate a source in the specimen is independent of grain size, (2) the strain for a given stress varies as Ds and (3) the microstrain varies parabolically with stress. The above model assumes that the only barriers to dislocations in the microstrain region are the grain boundaries. Consequently, in the experi- mental work every effort was made to produce specimens with variable grain size and negligible structure within the grains.

EXPERIMENTAL

Commercial Armeo ingot iron cold drawn to l/2 in. diameter rod was used as material. The rod was cut to approximately 6 in. lengths and heat treated for grain size in an argon-2% hydrogen atmosphere. A typical heat treatment for grain size consisted of holding a specimen at 1200°C for a given time, furnace cooling to 900°C and placing the second specimen in the furnace along with the first one. These were held a specified time at temperature and then furnace cooled to SOO”C, and the third specimen was placed in the furnace and held at temperature. All specimens were furnace cooled together. This method was used so that all specimens would have a similar matrix.

Grain size was determined by taking micrograp~ of a cross-section at x 100 and counting grains. Large grained specimens were not as uniform as the small grained specimens. Substructure or veining was apparent in some specimens under the microscope if

108 aCTA alETALLURGICA, VOL. 9, 1961

they were properly etched. The sub-grains seemed to

be uniform in size from specimen to specimen.

Microbeam X-ray analysis by Warrington at the

Cavendish Laboratory showed that all specimens

had the same structure interior to the grain within

the resolution of the microbeam method. Any

subgrain boundaries that existed had an angular tilt

of less than about l/3”.

Specimens and grips were designed to minimize

bending. Specimens were carefully machined and

ground to insure their straightness. The gage section

FIG. 1. Capacitance gage, grips end specimen.

was 1.0 in. by 0.25 in. with a shoulder radius of

l/4 in. The shoulders of the specimen were 7116 in.

and the ends machined to accommodate split rings for

gripping (Fig. 1). Specimens having threaded ends

were found to be inadequate. The split ring was held

in a ball bearing that was seated in a carefully

machined and polished socket. Specimen shoulders

were machined to give a snug fit in the ball. The ball

and socket were greased to aid in alignment under

load.

Strain measurements were made using a parallel

plate capacitance strain gage (Fig. 1) with a sen-

sitivity of lo-” in/in. The linear portion of the stress-

strain curve was used to calibrate the gage.

Testing was done on an Instron tensile testing

FIG. 2. Typical step-load vs. strain curve for iron as taken from the zy-recorder.

machine at a strain rate of O.Ol/min. Tests to investi-

gate microyield point and microstrain were of the

continuous load and step load type. Microcreep was

used as a measure of microyield point in the step

loading test. Specimens annealed at 650°C after a

microstrain test showed good reproducibility in

microyield point. Complete tests through the

macroyield point were made after the reproducibility

and microyield point tests were completed. All of the

stress-strain data was recorded in an xy-recorder.

EXPERIMENTAL RESULTS

The sensitivity of the method with respect to

determining non-linearity in the load-deformation

curve is shown in Fig. 2. The method can detect the

proportional limit to a strain sensitivity of about

1OW. Once non-linear behavior was observed during

loading, the unloading curve wasalways linearso that a

permanent strain resulted. Hysteresis loops were not

observed after a microstrain. The proportional limit

determined by the method of Fig. 2 was about the

900

FIG. 3. Continuous load-strain curve for iron, taken from zy-recorder.

BROWN AND LUKENS JR.: MICROSTRAIN IN POLYCRYSTALLINE METALS 109

same as that determined by a step loading process

and waiting for measurable creep to occur. The

waiting time was always l& min. In the micro-

strain region the creep was logarithmic so that the

waiting time was not very critical.

The reproducibility of the proportional limit in a

specimen after a 600°C anneal is interesting and

useful because it permits successive determination on

the same specimen and also indicates that the disloca-

tions which were introduced were simply arrayed.

Continuous stress-strain curves from zero stress

through the macroscopic yield point are shown in

ASTM

22: 20

16 a 8 p ‘4

FIQ. 4. Upper yield point vs. inverse square root of grain size for iron.

Fig. 3. The outstanding feature is that for a given

stress the larger grain sizes show more microstrain.

As expected the macroyield point increases with

decreasing grain size in accordance with Fig. 4:

urn,, - D-112.

(8)

The above relationship has been observed often

in ironc8) and other metals.(g*lO) The relationship is

based on the theory that the larger grain size permits

more dislocations to be piled up so that the resulting

stress concentration is essentially the applied stress

times the number of piled-up dislocations.

From Fig. 3 it appears that the proportional limit

increases with decreasing grain size. It must be

remembered that the experimentally determined

proportional limit is governed by the sensitivity with

which strain is measured. If one dislocation moves in

only one grain from the center of the grain to the

grain boundary, an undetectable strain of lo-l2

would occur. In the following section the theory will

be applied to the above data on iron as well as to

other data on copper and zinc which were obtained by

other investigators.

iron

13

I

0 IO 12 14 16 18 20

stress. 1000ps.i.

FIG. 5. Square root of plastic strain vs. stress for various grain sizes in iron.

ANALYSIS

It turns out that all the available stress-strain

curves in the microstrain region can be described by

equation (7). A plot of y* against applied stress

should be linear. This linearity is exhibited by our

data on iron in Fig. 5, by the data of Thomas and

Averbachc5) on copper in Fig. 6 and by the data of

Roberts who worked on zinc at the University of

Pennsylvania (Fig. 7). The curves in Figs. 5 and 6 tend

to converge toward a common value of uoo, the stress

to produce the first plastic strain. There is however a

small, but definite, grain size effect on ooo, which

might be expected because a smaller grain is not

likely to contain as long a dislocation source as a larger

one. This last point is not accounted for by the

9 N “, 8 b - 7 x .c 6

$ 5 .v 4 z 5 3

2

0 I 2 3 4 stress. 1OOOp.s.i.

FIG. 6. Square root of plastic strain vs. stress for various grain sizes in copper. Data taken from

THOMAS and BVERBACH’~).

110 ACTA METALLURGICA, VOL. 9, 1961

zinc

/ 1 , I I I I I I II

0 2 4 6 8 IO 12 14

Stress, 1OOp.s.i.

FIG. 7. Square root of plastic strain vs. stress for & zinc specimen with grain size 0.41 mm.

theory. For zinc, data on only one grain size was

available so that only the parabolic relationship

between stress and strain can be verified.

A plot of the log of the slope of curves in Figs. 5

and 6, vs. log D should have a slope of 3/Z as

required by the theory. Fig. 8 shows that this

point is very well substantiated.

From the above data the density of dislocation

sources and the number of sources per grain have

been calculated (Table 1). For the fine grain speci-

mens most of the grains do not contain a source so

that up to the macroyield point these grains are

deformed only elastically. The theory could be

checked more completely if a direct experimental

count could be made of the density sources. The

electron microscope offers the possibility of making a

direct count of the density of sources.

The theory and experiments support the following

conclusions : (1) For a given stress the strain varies as D3. (2) The square root of the strain varies linearly

with the stress.

5’

I ,/ , I I I I I 0 I 2 3 4 5 6

In (D, cm. x 10s)

FIG. 8. Plot of log slope of Figs. 5, 6 and 7 vs. log grain diameter for iron and copper.

(3) The stress to move the first dislocation is sub-

stantially independent of grain size.

The above conclusions lead to the following

characterization of the microstrain region:

(1) The number of sources per grain vary as 03.

(2) There is a linear relationship between the back

stress on a source and the number of piled-up

dislocations.

(3) The grain boundary is the primary barrier to

dislocations in a polycrystalline metal which

has no substructure.

f

so cf %

FIG. 9.

DISCUSSION

The microstrain region begins with the stress, ue”,

which first activates a source; the corresponding

strain is on the order of 10-12. The microstrain region

ends at the macroyield point. The condition for

macroyielding may be given by

no = fscf (9)

where n is the number of piled-up dislocations (T is the

applied stress, and oo is the stress to activate or possibly create sources which cannot be activated or

created directly by the applied stress. Relationship (9)

leads to the observed dependence of the macroscopic

yield point on grain size (equation 8).

The microstrain at the macroscopic yield point may

TABLE 1

Copper I Iron

kg/cm2 I 90.8 1 83.5-88.5

sources/cm3 / 6.9 x lo8 / 2.5 x 106

Grain size

(p)

Sources Grain size Sources per grain (/A) I per grain

BROWN AND LUKENS JR.: MICROSTRAIN AND POLYCRYSTALLINE METALS 111

be obtained by connecting equations (7), (9) and (5).

In using equation (5) the maximum number of

dislocations piled up will be used. The grain with

(IO8 = coo corresponds to a maximum in n. Thus the

microstrain at the macroyield point is given by

yM = (!?!I?) [ po2 -g=yz - uoq2. (10)

In general it is expected that macroyielding

coincides with the generation or activation of new

sources of dislocations with the piled-up dislocations

providing the required stress concentration to

activate these new sources. It is also expected that

slip on more than one slip system might begin at the

onset of macroyielding because the maximum

resolved shear stress produced by the pile-up would

not in general coincide with the maximum resolved

shear stress produced by the applied stress. From a

phenomenological viewpoint the microstrain region

sometimes is separated from the macrostrain region by

readily observed discontinuity in the stress-strain

curve (a drop in load). Other times when the transition

from the microstrain region to the macrostrain region

appears to be smooth upon direct observation of the

stress-strain curve, an analytic representation of each

region will show a discontinuity. Such a discontinuity

between the microstrain and macrostrain region has

been demonstrated by Roberts and Brownc3).

In conclusion, the microstrain region is separated

from the macrostrain region in the sense that at the

macroyield point there is a discontinuity in the

generation of dislocations. Consequently the macro-

yield point is looked upon as a unique event in the

plastic history of a crystalline material.

ACKNOWLEDGMENTS

J. M. Roberts not only gave valuable experimental

assistance and advice, but the data on zinc was also

provided through his courtesy. Professor N. F. Mott

and Dr. P. B. Hirsch were most encouraging during

the theoretical part of the investigation which was

carried out while one of the authors (N. B.) was a

Guggenheim Fellow at the Cavendish Laboratory.

D. Warrington kindly made and analysed the micro-

beam examination of the iron. One of us (K. F. L.)

was supported by a fellowship from the Wilbur B.

Driver Company. We thank Dr. J. Harwood for his

continued encouragement and support,. The Office of

Naval Research has been the sponsor for this research.

REFERENCES

1.

2.

3. 4.

5.

6.

8. 9.

0.

N. THOMPSON, C. K. COOQAN and J. G. RIDER, J. Inst. Met. 34, 73 (1955-56). P. CHASSLEY and N. THOMPSON, Phil. Mag. 3, 1398 (1958). J. M. ROBERTS and N. BROWN, J. Metal, N.Y., in press. W. S. OWEN, M. COHEN and B. L. AVERBACH, Trans. Amer. Sot. Met. 50, 517 (1958). D. A. THOMAS and B. L. AVERBACH. Acta Met. 7. 69

-r

(1959). A. H. COTTRELL, Dislocations and Plastic Plow in Crystala p. 111. Clarendon Press, Oxford (1953). C. S. BARRET, Stmctwe qf Metals p. 346. McGraw-Hill, New York (1952). N. J. PETCE, J. ITOTZ S. Inst. 173, 25 (1953). F. E. HAUSER. P. R. LONDON and J. E. DORN. Trans. Amer. Inst. Min. (Metall.) Engrs. 206, 589 (1956): ~- R. P. CARREKER and W. R. HIBBARD, JR., Acta Met. 1, 659 (1954).