INFLUENCE OF INTRINSIC STACKING FAULT ENERGY ON CYCLIC HARDENING

M. R. JAiMES and A. W. SLEESWYK

Laboratorium voor Fysische Metaalkunde. Materials Science Centre. Nijenborgh 18. Groningen. The Netherlands

(Received 13 March 1978)

Abstract--It was shown in a preceding paper [7] that the effect of the experimentally determined non-equilibrium plastic strain 8. after reversal in low cycle Fatigue increases with the number of reversals n: the strain to be subtracted from the absolute strain in each half-cycle is equal to w.*& The factor w n varies from 1 to 2 and may be represented bq the empirical formula (I). = 2 - exp(C.(I - n)/(n, - I);; n, is the number of reversals to saturation, C an adjustable factor which is a constant of the material. The method is applied to experimental results obtained on AISI 310. z-brass, Al, Cu and Ni. A single stress vs (corrected) strain curve is obtained for each material from a monotonic tensile test and cyclic tests at three strain amplitudes. The values of the constant C are linearly correlated to stacking fault energy values. These findings are discussed in terms of the Schoeck-Seeger cross slip model and cell formation.

R&m&-On a montrb dans un article p&&dent-Sleeswyk et al. 1978-que I’effet de la dtformation plastique experimentale hors d’iquilibre /?., aprts renversement de la sollicitation dans des essais de fatigue 6 has nombre de cycles. augmente avec le nombre de renversements n: la deformation que l’on doit soustraire de la deformation absolue dans chaque demi-cycle est tgale a w:j3.. Le facteur rz~” varie de 1 g 2 et peut etre represent6 par la formule empirique w, = 2 - exp {C*(l - n)/(n, - 1); : nr est le nombre de renversements a saturation, C est une constante qui dCpend du mattriau et que I’on peut ajuster. On applique cette methode B des rCsultats experimentaux obtenus dans I’acier AISI 310. le laiton z, Al, Cu et Ni. On obtient une seule courbe de la contrainte en fonction de la deformation (corrigee), pour chaque matkriau, B partir d’un essai de traction monotone et d’essais de deformation cyclique pour trois amplitudes de dCformation. Les valuers de la constante C sent likes lineairement aux valeurs de I’bnergie de defaut d’empilement. On discute ces r&.ultats a I’aide du modile de glissement d&i& de Schoeck et Seeger et de la formation des cellules.

Zusammenfassung-In einer vorausgehenden Arbeit (Sleeswyk et al. 1978) wurde gezeigt. da13 der Effekt der experimentell bestimmten plastischen Nichtgleichgewichts-Dehnung 8. nach Lastumkehr in zyk- lischer Verformung mit der Anzahl n der Lastwechsel ansteigt: die von der absoluten Dehnung in jedem Halbzyklus abzuziehende Dehnung ist gleich wn*Bn. Der Faktor (3, variiert von 1 bis 2 und kann mit der empirischen Beziehung o, = 2 - exp(C.(l - n)/(n, - 1)) dargestellt werden; II, ist die Lastwechselzahl bis zur Slttigung, C ein anpassbarer Parameter, der fiir ein bestimmtes Material kon- stant ist. Die Methode wird auf experimentelle Ergebnisse an AISI 310, x-Messing, Al, Cu and Ni angewendet. Fiir jedes Material wird eine einfache Spannungs- (korrigierte) Dehnungskurve aus einem Zugversuch und aus Ermiidungsversuchen bei drei Dehnungsamplituden erhalten. Die C-Werte hLngen linear mit den Werten fir die Stapelfehlerenergie zusammen. Diese Ergebnisse werden anhand des Schoeck-Seegerschen Quergleitmodells und der Zeilbildung diskutiert.

1. INTRODUCTION

The phenomenon of saturation in cyclic deformation at constant strain amplitude may be plausibly attri- buted to a balance between work hardening. occurring predominantly in the later part of each half-cycle and the softening which mainly takes place immediately after stress reversal. Work hardening results when part of the mobile dislocations that produce the strain are halted or react in the crystal. thus forming new obstacles. The lack of cyclic hardening during satu- ration indicates that the mobile dislocations accom- modating the strain do not produce a net increase in the density of obstacles.

Most explanations of cyclic saturation involve pri- marily the reversible movement of dislocations. One of these explanations is the cell shuttling model [l-4],

in which the same free mesh lengths of dislocations are thought to shuttle to and fro between cell walls. which would account for the saturation strain. The mobile dislocation segments sweeping through the cell volume have to surmount the obstacles in the cells thereby accounting for the flow stress. AU cell shuttling models are stationary in the sense that they do not account for the transition between cyclic har- dening-or softening-and saturation.

Mughrabi-et al. [j] pointed out that in reality dis- location motionis only quasi-reversible: dislocations may annihilate each other or leave the crystal at the free surface, and new dislocations may be created while the same dislocation configuration is main- tained. Even so, a cell shuttling model may.provide a useful, although simplified description of dislocarion movement confined to the cell volume from u-hich

1721

1777 __ J.AME.5 ASD SLEESWYK: STACKING FAULT ENERGY ASD CY-CLIC HARDENING

it is at least evident that part of the dislocation motion even in pre-saturation cycling possesses a

reversible component which does not contribute to the cumulative plastic strain.

The crucial problem of how to account quantitat-

ively for the reversible strain was first attacked by

Daniel and Horne [6]. They suggested that the rever- sible component was equal to the full Bauschinger

strain p’. defined as the plastic strain occurring at

complete reversal of the prior maximum stress. In their model the Bauschinger effect results from the motion of unpaired dislocations when the stress is

reversed. at least in stage II hardening of single crys-

tals. A dislocation rearrangement of this sort will have

little effect on the stress-state. Although in principle

their model provides a feasible method to correct the cumulative strain by subtracting from it the Bausch- inger strain occurring in each half-cycle. it was never applied to experimental data.

More recently. Sleeswyk et al. [7] determined the

reversible plastic strain component pi using elements of the theory of obstacle hardening proposed by

Orowan [S]. The strain /_$ can be determined from the experimental data: it is determined by the strain

value at which the magnitudes of the flow stresses in the forward and reverse sense in one cycle are

equal to one another. Application of this correction leads to the following expression for the irreversible cumulative eir. which of course very much resembles

the one given previously by Daniel and Horne [6]:

Eic = 2M, - 1 a&, (1) i=l

where<, is the strain amplitude. taken as half the strain

difference between reversal points, and n is the

number of half-cycles. Comparison with experiment

showed that for the first reversal p, gave an excellent correction on the cumulative strain with wr = 1; on

the other hand, it was easily shown that in saturation

a”, = 2, i.e. there is a transition from j to 2p between

the first cycle and saturation. An acceptable fit to experimental data obtained in

low cycle fatigue on AISI 310 specimens was obtained using an empirical transition function of the form:

,.=2-exp(-C&+).

where n, is the number of cycles to saturation. It was found that 0 as a function of l ic correlates well with the monotonic stress-strain curve, using a single value of C independent of E, and n,. The aim of this paper is to extend the method to a number of other f.c.c. metals and alloys.

2. EXPERIiMENTAL

Polycrystalline specimens of aluminium, copper. nickel, stainless steel and z-brass (see Table 1) were tested in tension only and in alternating uni-axial tensionjcomparison in the low cycle-high amplitude

Table I.

Materinl

99.999”” Al 99.95”,, CU 99.950” Xi AISI 310 Cu-35”” Zn

Heat treatment Grain sizr ( Cl (hi bn)

350 24 140 400 I 70 800 3 80 600 2-J 10 100 I 32

fatigue region. After machining the specimens to a fine surface finish they were vacuum-annealed at

various temperatures resulting in the grain sizes

recorded in Table I. The gauge section of each speci- men, diameter 1 mm and length 10 mm, was carefully-

examined under an optical comparator at a magnifi- cation of 50 x The average diameter was determined to within + 0.005 mm and the gauge length was

found by comparing the dimensions before and after testing: the samples were not cycled to fracture.

Testing was done on a laboratory-built 5 ton screw driven tensile tester adapted for cyclic deformation. The specimens were mounted in the lower end of a

tube-and-rod extension of the crossheads in which centering diaphragms are utilized to prevent plastic buckling. as described elsewhere [9].

The signal from the semi-conductor load cell (Baldwin-Lima-Hamilton type USG- I, max. capacity

10,000 lb) was recorded on both a two-pen strip-chart recorder (Goerz ER 250) and an X-Y recorder

(Hewlett-Packard 7004 B) on the Y-channel. Parallel to the specimen and close to it were mounted two

LVDT’s (Hottinger-Baldwin type WlE/O-5) between the tube and the rod of the extension. The signal from these LVDT’s was also recorded on both recorders

after amplification by a Hottinger-Baldwin

KWS/3S-5 amplifier operating at a carrier frequency of 5000 Hz. The X-Y recorder was equipped with an

automatic step-wise chart advanoe (HP type 17005 AM) which shifted the chart 7 cm in the X-direction

after each cycle. Thus, even in the saturation region each stress-strain was recorded separately.

The nominal crosshead speed employed in these plastic deformation tests was + 2.10-’ cmjmin. corre- sponding to a nominal strain rate of _ 3 x IO-‘s-t. Cyclic deformation was carried out in the push-pull

mode about zero mean strain at a constant plastic strain amplitude E,. The strain control was carried out in a manner similar to that employed by Wilson and Robinson [lo] in that the direction of the loading ramp was reversed when the plastic extension signal

became equal to one of two reference signals corre- sponding to the required plastic extension limits. The tests were carried out at 178 K in a cryostat fed with liquid N2 from a pressure reservoir through a magne- tically controlled valve (Cryoson system). An NTC resistor acted as a sensing element for the valve con- trol system, the temperature was separately moni- tored by means of a Pt-Pt/Ir thermocouple of which

J.A.MES ASD SLEESWYK: STACKING FAULT ENERGY AND CYCLIC HARDENING 1713

the output was read from a Kipp Microva AL1 gal-

vanometer. In the cryostat the temperature difference between

the two heads of the specimen was limited to c. +‘C. and the temperature variation with time, once equilib-

rium has been reached. was less than C*C. Before

reaching temperature equilibrium the load on the specimen was limited automatically to less than

20 kgf. A continuous load record of the cooling-down period was recorded on the strip chart recorder : typi- cally, this period is 6 h. In view of the temperature

sensitivity of the load cell and the effect of tempera- ture variations on the testing frame. the testing was carried out in an air-conditioned room with supple- mentary air circulation.

The analog data recorded on the chart recorders was converted to digital form using an antiplotter. The load-elongation curve was converted to true stress-true strain on a Control Data Cyber 71/6 com- puter and plotted on an incremental drum plotter. The plastic strain amplitude was constant to i: 0.003

and the parameter /3. determined from the replotted data as described in Ref. 7. was estimated accurate to + 0.001.

STRESS I kgf/mrn’)

3. RESCLTS

As has repeatedly been shown the uncorrected

cumulative plastic strain is entirely unsuitable as a

parameter characterizing the mechanical state in any type of stress-strain law. This failure of the cumula- tive strain is most evident in the saturation region. where it steadily increases while no flow stress changes or significant variations in dislocation density occur. In addition, it is rapidly found by any trial that the strain hardening coefficient dg:de is not an adequate substitute.

A most promising approach is due to Feltner and Laird [l l] who observed a marked parallelism between the deformation processes in stage II unidir-

ectional deformation and high amplitude fatigue

when they compared various physical parameters at equivalent flow stresses. However. only under well- specified circumstances is the flow stress a sufficient

parameter: in general at least one other parameter is needed to fully describe the mechanical state.

The method reported earlier [7] to account for the reversible part of the plastic strain (equation 1) places Feltner and Laird’s findings in a framework where

)I’, ” ” ( ” 8 ’ c 1 ’ I I ‘I c 002 lxx 006 a08 a10 aI2 ai4 016 al.18 a20

Eic

Fig. 1. End points of cyclic tests at three different amplitudes fitted to the unidirectional tensile curve (drawn) for Al, Cu and Ni, using equations (I) and (2). The values of /?. u-ere determined from the experimental data as described by Sleeswyk et al. [7]. the fit of the cyclic data to the tensile curve

was obtained for one value of the adjustable parameter C in equation (2) for each metal.

1724 JAMES AND SLEESWYK: STACKING FAULT ENERGY AND CYCLIC HARDENING

STRESS lkgflmm .‘I

80 1

OL 0.01 002 0.03 O.OL 005 0.06 a07 0.08 0.09 010

Eic

Fig. 2. Results similar to those presented in Fig. 1 obtained by the same methods, for the alloys stainless steel AISI 310 and x-brass.

they are valid and furnishes the irreversible cumula-

tive plastic strain eic as a parameter which allows di- rect comparison between unidirectional and cyclic hardening. A curve of flow stress versus Eic (equations 1 and 2) in cyclic deformation was found to be quali- tatively similar to a unidirectional work hardening curve in AISI 310. Similar sets of curves are given

C

10 -

9-

e.-

7-

I

0 20 LO 60 80 100 120 IL0 160 180 200

Fig. 3. The C-values used to fit the results of cyclic tests to the tensile curve as shown in the preceding diagrams plotted as a function of the stacking fault energy 7. The values of 7 were taken from the work of Segall [I?] for Cu. Al and Ni. from that of Thorton er al. [I33 for the x-brass. while the recent work of Bampton et ai. [I43 fur-

nished an estimate for 7 in AISI 310.

in Figs. 1 and 2 for the f.c.c. metals and alloys studied here. The transition parameter C was adjusted for each material to fit the cyclic data to the unidirec- tional hardening curve. The same value of C was used for all strain amplitudes in a given material. Within experimental scatter. a unique curve independent of strain amplitude describes the cyclic behaviour for each material and this curve follows the unidirec- tional strain hardening curve. The variation is largest in the Ni data because a definite saturation limit was never reached; an extremely small but tinite increase in flow stress was taking place even after 350 reversals for E, = 0.0052. In general it proved not difficult to assign +values to the recorded cyclic deformation curves, with an accuracy which we estimate was within + 10%. Of course, n, varies with the strain amplitude E, which makes it remarkable that a single value of the parameter C for each material was suffi- cient to bring the cyclic deformation curves and the unidirectional tensile curve to asymptotic coincidence.

4. DISCUSSION

Equation 2 is of course no more than an empiri- cally chosen transition function which is simple and produces a reasonable fit between the cyclic and monotonic data. Its major merit is that it contains only one adjustable parameter, viz. the constant C. It may well be that a different function will give a better fit: in the diagrams there are some signs of systematic deviations, but these are in the present stage a secondary effect only.

JAMES AND SLEESWYK: STACKING FAULT ENERGY AND CYCLIC HARDENING 1725

Whatever its precise form, however, it is mandatory

that the single parameter in the transition function

bears a functional relationship to the parameters

determining dislocation motion if the function is to have a physical significance. That is what was actually

observed : there exists a functional relationship between the C-values which were necessary to bring about coincidence of the monotonic and cyclic curves

for the various materials. and the stacking fault energy 7 as is obvious from the diagram presented in Fig. 3. It must immediately be added that if C

is not highly accurate-only within _ 0.5-the y-data are even less accurately known: in spite of that there can be little doubt that there is a correlation.

The influence of the stacking fault energy on cross- slip of perfect dislocations is well-established since the

work of Schoeck and Seeger [ 151: the larger the stacking fault energy, the more the dislocations are contracted and the less energy it costs to contract them further in order to enable them to cross-slip. So the tendency evident from Fig. 3 is to be inter-

preted as follows: if cross-slip is relatively easy o, approaches the value of _ 2 in fewer cycles relative

to the number of cycles to saturation than if cross slip is difficult.

In any rationalization of the experimental observa- tions-and more than that cannot reasonably be attempted at present-this finding must be given a

crucial place. In addition. the shift of the value of w, from 1 in the first cycle to 2 in saturation must

be explained. The following is a qualitative descrip-

tion of the train of events which is supposed to lead to saturation.

In the first half-cycle a number of dislocations glid- ing in opposite directions on intersecting glide planes halt each other. They may interact and form more or less stable planar arrays, braids etc. The energy gained during these interactions and the stability of

the resulting configurations depends on the ease with which the dislocations may cross slip and interact. Initially. these configurations are created more or less

randomly where groups of dislocations meet each other. but they do not rigorously confine dislocation movement. The larger the amplitude of the 1st half- cycle. the greater the opportunity for dislocations to entrap each other.

In the next half-cycle part of these configurations will disintegrate and part will remain to act as semi- permeable barriers to dislocation movement. As deformation progresses after reversal, a number of dislocations will be halted at the barriers: that will cause these to grow both in extent and in constitutive dislocation content. As a result, the barriers gradually become less penetrable to dislocations, thus raising the flow stress.

In subsequent cycles these processes continue at de- creasing rate and the barriers are slowly transformed into cell walls that become more and more impenetr- able to dislocations. After each reversal part of the dislocations accreted in the previous half-cycle in the

cell walls are released again. When they have crossed

the ceil volume they are trapped again: in other

words a large fraction of the mobile dislocations are

temporarily trapped in the cell walls in the later

stages of cycling. When dislocation release from the cell walls in a

half-cycle balances the dislocation accretion in the previous cycle the situation becomes stationary: satu-

ration is reached. The model of accretion and release presupposes an attractive interaction between the cell walls and the mobile dislocations: this is entirely in

accordance with the supposed role of cross slip in the early stage of barrier formation.

A consequence of a positive interaction between mobile dislocations and cell walls-which presumably

would be a short-range interaction-would be that the dislocations would travel at high speed across the cell. At any moment only a small fraction of the mobile dislocations is actually crossing the cell

volume. This qualitative model explains the following fea-

tures: the decrease of cell size with increasing strain

amplitude. due to the greater density of barriers formed in the first cycles;

the influence of high stacking fault energy on the facility with which barriers form and retain their dis- location content after stress reversal;

the change of o,, due to an increasing impermeabi-

lity of the cell walls and increasing confinement of dislocation motion;

the phenomenon of saturation, which results from

the balance between accretion and release of disloca-

tions from cell walls;

the electron microscopical evidence that the density of dislocations traversing well-developed cells is very low.

It seems that this qualitative model unites many of the known features of the plasticity aspects of low cycle fatigue. Our future task will be to put the model on a more quantitative footing by further experimen-

tation. In this connection we may mention a recent

paper by Baudry et al. [16] who employed an elegant special technique to determine reversibility by in situ magnetic measurements on reversible stress-induced martensite formation in low cycle fatigue. Their method of analysis makes use of a single ‘reversibility factor’ after the first reversal-roughly comparable to

the product wi/$-which does not lead to consistency of the results for amplitudes E, > 0.005. This approach is not very divergent from ours, and is easier to apply, but does not appear to lead to results as coherent as the ones presented in Figs. 1 and 2.

Acknowledgemenrs-The authors wish to thank Mr. H. J. Bron for his capable technical assistance in preparing specimens and Mr. W. S. T. Maathuis for help in perform- ing the first series of experiments. The investigation was carried out as a part of a research program on Low Cycle Fatigue and Dislocation Dynamics supported by the National Research Foundation F.O.M.

1726 JAMES .kb’D SLEESWYK: STACKING FAULT

REFERENCES 9

1. 0. F. Watt and R. K. Ham, iVuture. &?nd. 211, 734 10

(1966). II. 2. C. E. Feftner and C. Laird. rlcra meralf. 15. 1633 (1967). 12 3. J. C. Grosskreutz. Ph,rs. Srarlrs Sofidi (Bf 47, I1 (1971). -1. J. M. Finney and C. Laird. Phil. ‘wag. 31. 339 (1975). l3 5. H. Mughrabi. K. Herz and X. Stark. .&a merczff. 24.

659 (1976). 14. 6. R. C. Daniet and G. T. Home, Me&f. Trans. 2. I161

(1971). 15. 7. A. W. Sleeswyk. M. R. James, D. H. Plantinga and

W. S. T. Maathuis. .-lcta mcm/l. 26. 1265 (1978). 16. 8. E. Qrowan, Internal Stresses and Fatigue in MrtaIs.

(edited by G. M. Rassweiler and W. L. Grube), p. S9. Elsevier, New York (1959).

ENERGY AND CYCLIC HARDENING

A. W. Sleeswyk, Rec. scient. insrrrcm. St, 793 (1960). C, J. Wilson and A. Robinson. J. Phys. E: Scimt. In- strum. 1% I?9 [ 1977). C. E. Feltner and C. Laird. /Icra nrerafl. 15, 1621 (1967). R. L. Segall. Adcances in Marerials Research (edited bv H. Herman), Vol. 3. p. 109. Wiley. New York (1965). d. R. Thorton. T. E. iMitchell and P. B. Hirsch. Phil. ,%lug. 7. I349 (1962). C. C. Bampton. 1. P. Jones and M. H. Loretto. Acta metaff. 26, 39 (1978). G. Schoeck and A. Seeger. Rep. Cc@ Defects in S&fs. p. 340 (3ristol). London Phys. Sot. (1955). G. Baudry, D. McLean and A. Pineau, Scripta metall. 11, 987 (1977).