The Phy..ics of Metals and Metallograp")~ Vol. 8/. No.4. /996. pp. 374-380.OriKinal RU.I'..ian Text Copyright to /996 by Fizika Metallov i Metallovedenie. Greenberg. Ivanov. Barabash. Blokhin.EnKIi.." Tran..lation Copyright to 1996 by MAJ1K HaYKallnrerperiodica Publishing IRussia).

THEORYOF METALS

Comparative Analysis of Stress Jumps in Metals andIntermetallic Compounds: I. Description of Two-Stage Straining

B. A. Greenberg*, M. A. Ivanov**, T. o. Barabash**, and A. G. Blokhin** Institute of Metal Physics, Ural Division, Russian Academy of Sciences,

ul. S. Kovalevskoi 18, Ekaterinburg, 620219 Russia

** Institute of Metal Physics, National Academy of Sciences of Ukraine, pro Vernadskogo 36, Kiev, 252680 Ukraine

Received November 2, 1995

Abstract-The Cottrell-Stokes two-stage straining of fcc metals and intermetallic compounds whose yieldstress exhibits an anomalous temperature dependence is analyzed theoretically. The magnitude of stress jumpsbrought about by changes in temperature is expressed via characteristics of dislocation transformations to short-and long-lived barriers. The manner in which the thermally activated blocking of dislocation sources affects themagnitude of stress jumps in intermetallic compounds is investigated. Several alternative combinations of con-ditions are discussed, under which stressjumps similar to those observed in metals occur in intermetallic com-pounds with changing temperature.

INTRODUCTION

The well known experiments of Cottrell and Stokes[I], in which stress jumps that arise as a result ofchanges in temperature or strain rate are measured,make it possible to distinguish between the contribu-tions made to the flow stress by various obstacles inhib-iting the movement of dislocations. Above all, these arecontributions due to long-range and short-range obsta-cles. Usually such contributions are supposed to beadditive; they are responsible for irreversible andreversible changes in flow stress, respectively, withchanges in temperature or strain rate [2].

Recently, the field of research on stress jumps hasexpanded considerably owing to prestraining experi-ments on intermetallic compounds. A rather motleypicture has emerged. For example, in their experimentswith TiAI, Stucke et at. [3] found that the change instress due to a decrease in temperature seemed to besimilar, to a certain extent, to what is observed in Cot-trell-Stokes experiments on fcc metals. On the otherhand, in their study of Ni3AI, Dimiduk and Parthasar-athy [4] noted a decrease in stress, and an unusuallylarge one for that matter.

The anomalous temperature dependence of yieldstress observed in the intermetallic compounds consid-ered brings us to several questions: How does this tem-perature anomaly affect stress jumps upon changes intemperature (strain rate)? What are the conditionsunder which the changes in stress can be almost thesame for intermetallic compounds and metals, despitethe above anomaly (part I of this study)? Why is it thatthe stress jumps occurring in intermetallic compoundsare so large (part II)?

We attempt to describe stress jumps in metals andintermetallic compounds within a unified scheme. Indoing so, we find it efficient to use the concept we haveset forth in [5, 6]. It offers a way to consider varioustypes of dislocation transformations both aided andunaided by thermal fluctuations. Our analysis suggestsseveral experiments that offer an insight into the phe-nomenon we are concerned with (see part II).

1. EQUATIONS OF PLASTIC DEFORMATION

The dislocation transformation diagram shown inFig. 1 is the simplest of those that schematically depictthe reactions prerequisite for stress jumps to occur. Inwhat follows, it will be referred to as the C diagram, asis done in [6]. It includes an elementary diagram of thelobe type, which describes g ;= s transformations,and an elementary diagram of the ray type, whichdescribes g - S' transformations. They demonstratethat glissile (g) dislocations can go over into sessileconfigurations of the s and S' types. These configura-tions differ in the ratio between their lifetime and

observation time t, equal to E/E. Morespecifically,

t.. ~ t,

t.. ~ t,(1.1)

that is, s-type barriers are short-lived, and s'-type barri-ers are long-lived. In other words, g dislocations con-tinuously go over into s dislocations and back, whereass'-type dislocations do not transform back to glissiledislocations over the observation time. That is, they areindestructible barriers.

374

COMPARATIVE ANALYSIS OF STRESS JUMPS: I.

For the C diagram, the equations of detailed balancemay be written as

Nfl = -NflvfI.,+ N"v"g-Ngvgs,+M,

N.. = Ngvgs-Nsvsg, (1.2)

N,,' = Ngvgs"

Here N. is the density of glissile dislocations; Ns and Ns'are the fIdensities of sessile dislocations of the s and s'types; M is the strength of dislocation sources; vgs' vgs"

and V"fIare the frequencies of g - s,g - s',ands- g transitions, respectively, vsg being inverselyproportional to the lifetime ts introduced previously.

As will be recalled, Ngis connected to the strain rate

E by a relation of the form,

E = bvfNg, (1.3)

where b is the modulus of the Burgers vector, v is thevelocity of a glissile dislocation, and f is the Schmidfactor. Solutions to simultaneous equations (1.2) areanalyzed in [5, 6], where the applicable approximationsare also formulated. Here, we will limit ourselves to theexpressions essential for the subsequent analysis ofstress jumps.

Because Ng ~ N" [6], the total dislocation density pfor the C diagram may be written as

p =N" + Ns" (1.4)

Whatever the blocking mechanisms involved, a glis-sile dislocation must overcomethe elastic opposition ofits environment that consists of s and s' dislocations.Therefore, a ne<;essarycondition for plastic flowto takeplace is

0' = KJP, K = aJlblf. (1.5)

Here, Jl is'the shear modulus and a is a nondimensionalparameter, which depends, among other things. on thedislocation orientation. Later, we will go back tothe form in which the relation between 0' and p can bewritten.

For convenience, we introduce partial stresses O'sand 0'",

2 20'" = K N", (1.6)

which describe the transformation of the glissile dislo-cations into the short-lived (lobe-type diagram) andlong-lived (ray-type diagram) barriers. Because NsandN".are additive, it follows that

2 2 20' =0'" + 0'"" (1.7)

Using the solutions to (1.2) for Ns and Ns'and rela-tions (1.6), we immediately obtain

2 1.0', = --::;-£t"

, y11." '

(1.8)

THE PHYSICS OF METALS AND METALLOGRAPHY

375

whereE

Y = b{, (1.)= !fJ.dE.K t..,,' £ t..,f'o

Here, th~.free path lengths t..sand t..s'are inversely pro-portionalto the previouslydefinedfrequenciesvgs andvgs',respectively.

Thus, for the C diagram we have

(1.9)

2 - £[

1.,

(I

)E

]0' ---+--Y t.." t...,.£ .

As (1.10) implies, the contributions to 0'2 fromshort- and long-lived barriers, determined by lIt..s and

I/t..s'. respectively. have different weight factors. They

are ts and £1£. respectively. It is important that ts ~

E/e, according to (1.1). Therefore, if I/t..s and I/t..s'

are comparable in value or even I/t..s' < l/t..s. then thedominant contribution to 0' should come from theg- s' transformation to long-lived barriers. On thecontrary,the g s transformation to short-livedbarri-ers can be dominant only if the corresponding free pathis considerably shorter than it is for the g - s'trans-formation. In other words, the ray diagram easily winsover the lobe diagram.

In view of the above statement, we assume that

(1.10)

O's~ 0'.,"

Then, equation (1.7) takes the form

(1.11)

(1.12)

Using (1.8), we write ~O'as

1 ft.,~O' = 2 t..~' (1.13)Y " .,

In form, (1.13) is similar to the expression knownfrom [7]

0' =O'G+ O's. (1.14)

Here. O'Gand O'sdetermine the contributions from long-range and short-range obstacles, respectively (seeIntroduction). The main difference between (1.12) and(1.14) is that in (1.12) the elastic opposition of disloca-tions is described by the entire expression, whereas in(1.14) this is done solely by the term O'G'According to(1.7), the contributions 0'.,and O's'form a quadratic sum,and it is only in the approximation defined in (1.11) that

~s g s

Fig. 1. C diagram of dislocation transformations.

Vol.81 No.4 1996

376 GREENBERG et al.

T

Fig. 2. Schematic representation of two-stage straining.

they are additive, whereas (1.14) does not use such anapproximation at all. Moreover, in (1.14) only Os isassociated with thermally activated processes. In(1.12), both terms are involved because g - s'trans-formations can be thermally~activated,for example, inintermetallic compounds. Lastly, the contributionsfrom long-range and short-range obstacles in (1.14) areseparated, whereas dO in (1.12) contains both Osand ox"

2. STRESS JUMPS DUE TO CHANGESIN TEMPERATURE OR STRAIN RATE

IN METALS

The Cottrell-Stokes experiment is usually carriedout as follows. At first, the specimen is deformed attemperature TI to a certain predetermined amount ofstrain E (a few percent). Then, the temperature ischanged and the specimen is deformed at T2likewise toE. We will limit ourselves to the case (Fig. 2) whereTJ> T2 so as to avoid considering the recovery of thedislocation structure. Using the expressions derivedabove, let us calculate the stress jumps occurring upona decrease in temperature at stage II of work hardeningin fcc metals.

2.1. Description of Stage II of Work Hardeninginfcc Metals

Suppose that two types of dislocation transforma-tion take place at stage II. One happens becauseLomer-Cottrell barriers arrest dislocations. The otheroccurs when dislocations intersect the "forest". Assum-ing that the Lomer-Cottrell barriers are indestructible,we will describe the arrest of dislocations as g - s'transformations, for which the corresponding stress Os'is given by (1.8). In describing the intersections of theforest by dislocations as g ==s transfonnations, we willassume that at stage II the characteristic time ts satisfiesrelation (1.1) and that the applicable stress Osis givenby (1.8).Of the two types of transformations we are con-cerned with only the s - g transformation, which rep-resents the intersection of a forest by dislocations and isa thermally activated process. Its t.fmay be written as

-(U'fll

)t.f= t.,exp .kT '(2.1)

where Usgis the activation energy for an s - g trans-formation.

Because a great number of Lomer-Cottrell barriers

are formed at stage II, the term I/As' is apparentlysmaller than l/As' albeit only slightly. Then, as followsfrom (1.10), the g - s' transformation is a dominantone, and the stress ° is defined by (1.12). Note that,because (1.11) is satisfied, dO is such that

dO ~ O.f" (2.2)

Similar to [7], we take it that A.f' is inversely propor-tional to the strain E as reckoned from the onset ofstage II, that is, A.f'= A.,./E.Then, using (1.9), we imme-diately obtain

aJ1~os' = 9E, 9 = f ~m;.'

where 9 is the work-hardening rate. I

(2.3)

2.2. Analysis of Stress Jumps

In two-stage straining, the stresses oland 02 at thefirst and second stage are defined by expression (1.12)taken at the appropriate temperatures, with os' definedby (2.3) and dO, by (1.13). In keeping with (2.2), theincrements dO I and d02 have almost no effect on theflow stress at a fixed temperature. Otherwise, the strain-hardening factor at stage II would vary with tempera-ture. Upon a change in temperature, however, the stressjump

da<7)=02 - 01 (2.4)

is determined solely by the difference of small incre-ments .

da<7)=d02 - ~o I' (2.5)

This is because the contribution os' from the dominantg - s' transformation, equal to 9E, is independent oftemperature, unless one takes into consideration thetemperature dependence of the shear modulus. How-ever, the stress jump "memorizes" the dominant trans-formation. Indeed, as can be seen from (2.5) and (1.12),the dc:f.7)is not the difference between the stresses Osatdifferent steps of deformation. It determines the corre-sponding increment dO only if it is multiplied by theratio os/20s" Thus, g ==s transformations, which donot reveal themselves at stage II, that is, remain "hid-den", show up when a change in temperature takesplace. The difference between the associated incre-ments, ~02 - dOl' is not infinitesimal when a strong

1 To allow for a nonuniform distribution of dislocations, that is, forthe fact that a number of dislocations, say n of them, are piled up

at a barrier, it will suffice to multiply 1Cin (1.5) by j;,. Accord-

ingly, we multiply by j;, the right-hand side of expression (2.3)for the work hardening rate, which thus becomes identical (towithin a certain numerical coefficient) to the expression knownfrom [7].

Vol. 81 1~6THE PHYSICS OF METALS AND METALLOGRAPHY No.4

COMPARATIVE ANALYSIS OF STRESS JUMPS: I.

change in temperature takes place and governs themagnitude of 6a<T).

Using (1.13), we may re-write (2.5) as

6cr(T) = 6crl(t/T2' cr2)-I ).t.,(T(, crl)

Hence, it immediately follows that 6a<T)> 0 at Tz < T(because, according to (2.1), the lifetime t.. increaseswith decreasing temperature.

In a similar way, we find the stress increment dcr(£)

caused by an increase in the strain rate £

6cr(£) = dcrl(~2t,(crZ) I).EIt,( cr)

Because crz= crl + 6a<T)-,~expression(2.6) is an equa-tion from which one can find 6a<T)if one knows thebehaviorof A." and V,"IIwith changing cr.For example,ifwe assume that A...is inversely proportional to EZand theactivation energy V'II only slightly varies with cr, weobtain 6a<7)/cr==const, which is the well-known Cot-trell-Stokes relation [I, 2]. To demonstrate, it immedi-ately follows from (1.13) and (2.3) that, under theassumptions made, dcr/cr,f'is independent of E.There-fore, noting that in the case at hand t" defined by (2.3)is independent of cr and crt ==crs.(T), we immediatelyobtain the Cottrell-Stokes relation from (2.6).

It is likewise possible that the Cottrell-Stokes rela-tion will be satisfied under a different set of conditions.We will not analyze solutions to equation (2.6). Instead,we will only attempt to define the role that dislocationtransformations to long-lived and short-lived barriersplay in plastic deformation under both fixed and vary-ing external parameters.

(2.6)

(2.7)

3. TWO-STAGE STRAINING OFINTERMETALLIC COMPOUNDS

3.1. Experimental FindingsExperiments similar to Cottrell and Stokes's were

made on intermetallic compounds displaying a temper-ature anomaly of flow stress. In their studies of thestraining behavior of Ni3Al, Davies and Stoloff [8] andCopley and Kear [9] found that at T) > Tz the stressdecreased rather than increased as in the case above;that is, 6a<7) < O.

The increments 6cr(£) caused by changes in thestrain rate look more usual. Above all, as Thorntonet al. [10] have found in Ni3AIand Takeuchi and Kura-moto [II] in Ni3Ga, an increase in the strain rate isaccompanied, similar to metals, by a positive change in

stress 6cr(£), but the stress change is small compared to

the stress itself. Moreover, dcr(£) in intermetallic com-pounds was found to be significantly smaller than inmetals.

THE PHYSICS OF METALS AND METALLOGRAPHY

377

500

400

296K

onen

~ 200en

100

0.5 1.0 1.5Total strain, %

2.0 2.5

Fig. 3. Stress-strain curves for TiAI single crystals. Lowercurve, compression at room temperature; upper curves, two-stage straining [3].

In the analysis that follows we will also use theresults reported by Thornton et al. [10] for Ni3(AI,Cr),where the specimen was prestrained (to about 20% atroom temperature), and the yield stress was then mea-sured within the anomalous behavior region. Theyobserved that up to about 600°C the specimen retainedthe high yield stress imparted by prestrain.

Stucke et al. [3], and Dimiduk and Parthasarathy [4]experimented with TIAI and Ni3AI, respectively, usinga scheme (see Fig. 2) similar to the one used in Cot-trell-Stokes experiments. These intermetallic com-pounds were found to respond differently to two-stagestraining, although both exhibited an anomalous behav-ior of their yield stress in a broad temperature range.Stress-strain curves for TiAI and Ni3Al are given inFigs. 3 and 4, respectively. In both cases, T), which cor-responds to the first straining stage, falls within theinterval where cry(1)shows an anomalous behavior, andthe second stage proceeds at room temperature. Also,Figs. 3 and 4 show stress-strain curves taken at roomtemperature without prestrain.

Referring to Fig. 3, it can be seen that in the case ofT1A.Ichanges in temperature leave the stress almostunchanged. According to Stucke et al. [3], the changein stress, da<7),is about 10 MPa. Hence, the observedbehavior of TlAllooks more like that of metals in theconventional Cottrell-Stokes experiments, as notedpreviously.

In the case of Ni3AI,however, as can be seen fromFig. 4, a sharp decrease in stress is observed on passingto the second stage. If we take as crzthe stress at whichthe hardening factor settles at a constant value, thencr)/crz=::3.5. The strain-hardening factor at the second-stage temperature Tzlikewise responds to prestrain dif-ferently in the intermetallic compounds involved. Com-parison of the curves in Figs. 3 and 4 shows that pre-strain leaves this factor almost unchanged in TiAI, butincreases it significantly in Ni3Al.

Vol.81 1996No.4

378 GREENBERG et ai.

3.2. Blocking of Dislocation Sources

Intermetallic compounds owe their anomalousstress-strain behavior with temperature first of all to thefact that thermal activation is responsible not only forthe transformations of blocked dislocations into glissiledislocations, which is usual, but also for the dislocationblocking processes itself. In other words, thermal acti-vation causes dislocations to transform to barriers,whichmay be regarded as indestructible in a certain tempera-ture interval. In our opinion, dislocations are blocked assoon as they multiply, and this occurrence determinesthe stress required to activate dislocation sources [12].

Among all the possible types, let us choose disloca-tion sources that have a critical configuration. Such aconfiguration must be achieved before the subsequentloss of stability can lead to the multiplication of dislo-cations. Let there be, as an example, an ordinary Frank-Read dislocation source, in which the spacing betweenthe points of pinning is 21and which emits a superdis-location whose Burgers vector is 2b. Unless it isblocked, the superdislocation will acquire a criticalshape (in the simplest case, this will be a semicircle) ata stress crf equal to Jlb 11.

Suppose that some thermally activated g - s~transformation, with a characteristic free path /...F'results in dislocation barriers that may be taken as inde-structible at a given temperature. Then, if 1> /...Fta dis-location segment will be too long (compared to /...F)toachieve a critical shape; instead, it will acquire ablocked configuration. If, on the other hand, 1< /...Ftthesegment will have enough time to pass through a criti-cal configuration before it is blocked.

The values of 1at which superdislocations can mul-tiply and at which dislocations are blocked as they bowout are separated by a smeared boundary. However, theboundary becomes smeared near /...Ftso it is safe toadopt /...Fas the critical size of a dislocation segment.

200

1 a---. . .......................-

o 123Plastic shear sttain, %

4

Fig. 4. Resolved shear stress plotted against shear strain inNi3Al: (1) at 300 K; (2) at 875 K (at the first stage) and300 K (at the second stage) [4].

Then, the stress required to activate Frank-Readsources may be written as

(3.1)

In writing (3.1), no other dislocation blockingmechanisms are considered except the thermal activa-tion. As (3.1) implies, the dislocation blocking signifi-cantly changes the situation. Indeed, in the absence ofblocking, an increase in their length makes it increas-ingly more convenient for dislocation segments toserve as sources. In the presence of blocking, thelengths of these same segments fall in the forbiddeninterval 1> /...Ftwhere sources just cannot be switchedon. Moreover, if the barrier remains indestructible, noincrease in stress can make the source for which 1> /...Foperative. Thus, whether or not it is possible to activatea source depends on the ratio between the values of thesource length and the free path related to the transfor-mation of a dislocation to an indestructible barrier.

For a thermally activated g - s~ transformation,')..~7) may be written as

(3.2)

where U:S. is the activation energy for blocking a dis-location that belongs to the source. Using (3.1) and(3.2), we immediately obtain an expression that makesallowance for the anomalous temperature behavior ofthe stress required to activate dislocation sources

(3.3)

However, the switching-on of dislocation sourcesalone is not sufficient to initiate plastic flow. Its onsetmay be visualized as follows. Dislocations do not beginto multiply uniformly throughout the crystal; rather,they do so separately, most favorable regions initiallyisolated from one another. Later, the regions jointogether and take up a sizable proportion of the space,similar to what happens in the case of what is known aspercolation. By extending the analogy, this instant maybe thought of as marking the start of plastic flow. Forthis to happen, however,it is necessary that the increas-ing dislocation density and the external stress should beconnected by a relation of the form (1.5). In that case,glissile dislocations will be able to overcome the elasticopposition of their environment. To sum up, the condi-tion for plastic flow to take place may be written as

(3.4)

For brevity, this condition may be termed a "doublekey"; it permits the plastic flow to start as a macropro-cess.

THE PHYSICS OF METALS AND METALLOGRAPHY Vol.81 No.4 1996

COMPARATIVE ANALYSIS OF S1RESS JUMPS: 1.

It is not at once that the relation defined in (1.5)takes effect after dislocation sources become operative.A time interval is needed for the growing dislocationdensity and the operative stress to strike a balance. Thisinterval determines the transition region between elas-tic and plastic deformation in the stress-strain curve.

In a way, there is an alternative to the above view.Frequently, the operative stress is written as

<1= <10+ KJP, (3.5)

where <10is a certain starting stress, say, <1F'In what fol-lows, we will compare the possibility of using either the"double key" (3.4) or relation (3.5) in interpreting theexperimental findings.

3.3. Two-Stage Straining without a Changeof the Slip System

Consider two-stage straining when the temperatureTl lies within the region where the yield stress exhibitsan anomalous behavior. We will use the dislocationtransformation diagram shown in Fig. 1. Here, theg = s transformations are intersections of a forest, asin Section 2, but the g - s' transformationsarethethermally activated transformations, typical of interme-tallic compounds, that convert dislocations to barriersindestructible at the temperatures under study. Thestress cr.fassociated with a forest intersection is signifi-cantly lower than the stress <1s'associated with the for-mation of barriers. That is, relation (1.11) is satisfied,and the resultant stress can be found from (1.12)and (1.13).

In finding cr.f"we assume that the initial condition is<10= crF' If we limit ourselves to a small amount ofstrain, then, in view of (1.8), we will get

crAT) = crF(T) + 9(T)e,

1e(T) = - ~ --- .--'

(3.6)

Let us write A.f'as

-(u ,

)AAT) = A...expk~ '(3.7)

where UIIS'is the activation energy for a g -- s'trans-formation. In contrast to the activation energy U:S.defined above, we are concerned here with the blockingof dislocations no longer associated with a source.Therefore, the temperature dependence of the workhardening rate takes the form

F

e(T) = aexp( (Ug'f'k-TUgs'»). (3.8)

As follows from (3.8), for the function e(1) to show ananomalous behavior with temperature, it is necessary

Fthat U11.<'< UII.f"

THE PHYSICS OF METALS AND METALLOGRAPHY

379

Thus, the stress <11(T1)achieved toward the end ofthe first straining stage may be written as

(1)(T1) = <1..,(T\)+ ~<1I(T1),

~<1I(T1) = 1 Et.f(T1)2YA..<1AT1) .

Here, criT1) and tiTI) are defined by (3.3), (3.6), (3.8),and (2.1), respectively.

As it cools to T2,which is the temperature of the sec-ond straining stage, the specimen retains the micro-structure produced at the first stage and having a totaldislocation density PI' For this reason, if the slip systemoperative at the second stage is the same as before, themicrostructure will present to the dislocations of this

system an opposition KJP; , equal to crI' Hence, if plas-tic flow at the second stage begins at a stress <12(T2),equation (3.4) will take the form

<12(T2)::::max{cr~T2), crl(T1)}. (3.10)

In compliance with (3.3), we have <1~T2)< <1~T\).Therefore, noting (1.12) and (3.6), we have cr~T2) <<11(T\).This immediately implies that

<12(T2)::::<11(T1). (3.11)

On the contrary, if equation (3.5) applies, writing itconsecutively for the end of the first stage and for thestart of the second immediately reveals that a decreasein temperature is accompanied by a decrease in stressby an amount approximately equal to the differencebetween the stresses required to make a slip system gooperative, that is,

<11(T1)- <12(T2) ==cr~T1) - cr~T2). (3.12)

As the curves of Fig. 3 show, equation (3.11), ratherthan equation (3.12), holds for TlAl. In a way, this val-idates the concept of the "double key".

The difference between the stresses cr\ and cr2mark-ing the end of the first and the start of the second stageof straining in an intermetallic compound without achange of the slip system is governed solely by g = stransformations. Therefore, we may write ~cr<1)in aform similar to (2.6). In contrast to metals, however, thestress crs"which, according to (3.9), has a direct bearingon ~crl(T1),shows an anomalous behavior with temper-ature in intermetallic compounds. As (2.6) implies,~cr<1)becomes positive as the temperature decreases. It,thus, turns out that the anomalous behavior of cr(1)does not reverse the sign of the stress jump.

If the strain rate varies within the region where cr(1)

shows an anomalous behavior with temperature, ~cr(t)will, as in the case of metals, be defined by equation(2.7) [in which ~<11is given by equation (1.13), whichcontains the stress crs.].Owing to this, in changing E,the memory of the previous g - s'transformationsisretained, although ~cr(t) is governed, similar to Llcr<1),

(3.9)

Vol.81 No.4 1996

380 GREENBERG et 'al.

by g = s transformations, which remain "hidden" aslong as the strain rate is held constant. As in metals,

Llcr(E)is positive upon an increase in the strain rate.Equation ( 1.13)implies that the anomalous behavior ofcr,\,.(7)will produce a definite effect on the temperature

(')dependence of Acr E .

In intermetallic compounds whose yield stressexhibits an anomalous behavior with temperature, largestress jumps (macrojumps) AdD comparable with thestress itself may occur. Stress macrojumps in interme-tallic compounds are discussed in part II of this study.

REFERENCES

I. Cottrell, A.H. and Stokes, RJ., Effects of Temperatureon Plastic Properties of Aluminum Crystals, Proc. R.Soc., 1955,vol. 233, no. 1192, pp. 17-34.

2. Basinski, S.J. and Basinski, Z.S., Plastic Defonnationand Work Hardening, in Dislocations in Solids,Nabarro, ERN., Ed., Amsterdam: North Holland, 1979,vol. 4, pp. 261-362.

3. Stucke. M.A., Dimiduk, D.M., and Hazzledine, P.M.,The Influence of Prestrain on the Flow Stress Anomalyin TiAI Single Crystals, in High Temperature Orderedlntermetallic Alloys V. Mater. Res. Soc. Symp. Proc.,vol. 228, Baker, I. et ai., Eds., Pittsburg,PA:MRS, 1993,pp.471-476.

4. Dimiduk, D.M. and Parthasarathy, T.A., Implicationsfrom Pre-Straining Experiments on Emerging Kink-

Based Models for Anomalous Flow in Ll2 Alloys, Phi-los. Mag. Lett., 1995, vol. 71, no. 1, pp. 21-31.

5. Greenberg, B.A. and Ivanov, M.A., New Concepts ofAnalyzing Plastic Deformation of TiAI and Ni3AI Inter-metallic Compounds, Mat. Sci. Eng.. A, 1992, vol. A153,pp. 356-363. '

6. Greenberg, B.A. and Ivanov, M.A., Dominant Disloca-tion Transformations and Temperature Dependence ofFlow Stress in Intermetallic Compounds, Phys. Met.Metallogr., 1994, vol. 78, no. 3, pp. 247-265.

7. Berner, Rand Kronmiiller, H., Plastische Verformungvan Einkristallen, Berlin: Springer, 1965.

8. Davies, RG. and Stoloff, N.S., Influence of Long-RangeOrder upon Strain Hardening, Phi/os. Mag., 1965,vol. 12, no. 116,pp. 297-304.

9. Copley, S.M. and Kear, B.H., Temperature and Orienta-tion Dependence of the Flow Stress in Off-Stoichiomet-ric Ni3AI (y' Phase), Trans. Metall. Soc. MME, 1967,vol. 239, no. 7,pp.977-984.

10. Thornton, P.H., Davies, RG., and Johnston, TL, TheTemperature Dependence of the Flow Stress of the y'Phase Based upon Ni3AI, Metall. Trans., 1970, vol. I,no.l,pp.207-218.

II. Takeuchi, S. and Kuramoto, E., Temperature and Orien-tation Dependence of the Yield Stress in Ni3Ga SingleCrystals, Acta Metall., 1973, vol. 21, no. 4, pp. 415-425.

12. Greenberg, B.A. and Ivanov,M.A., Blocking of Disloca-tions and Dislocation Sources in TiAl, in Gamma Tzta-niumAluminides, Kim,Y.-w.,Wagner,R, Yamaguchi,M.,Eds., Warrendale, Pa.: TMS, 1995, pp. 299-306.

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