the critical resolved shear stress of peak-aged aluminium–lithium single crystals

THE CRITICAL RESOLVED SHEAR STRESS OF

PEAK-AGED ALUMINIUM±LITHIUM SINGLE CRYSTALS

A. KALOGERIDIS, J. PESICKA{ and E. NEMBACH{Institut fuÈ r Metallforschung der UniversitaÈ t MuÈ nster, Wilhelm-Klemm-Strasse 10, 48149 MuÈ nster,

Germany

(Received 20 July 1998; accepted 12 January 1999)

AbstractÐThe critical resolved shear stress (CRSS) of single crystals of aluminium-rich aluminium±lithiumalloys has been measured at 283 K. These alloys are strengthened by coherent precipitates of the metastableL12 long-range-ordered d' phase. The specimens had been subjected to long aging treatments such thatthey were in the peak-aged state. This contrasts with the earlier investigation (Schlesier and Nembach,Acta metall. mater., 1995, 43, 3983) in which underaged Al±Li single crystals have been studied. The contri-bution tp of the d' precipitates to the present peak-aged specimens' total CRSS has been found to be afunction only of the d' volume fraction f, i.e. tp was independent of the radius of the d' precipitates. Theexperimental data tp(f) are well represented by a relation which had been developed for the description ofpeak-aged g'-strengthened nickel-base superalloys. These latter materials are isomorphous to d'-strength-ened Al-rich Al±Li alloys: coherent L12 long-range-ordered particles are embedded in a disordered f.c.c.matrix. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Aluminum alloys; Transmission electron microscopy (TEM); Phase transformation, growth;Mechanical properties, plastic; Dislocations, mobility

1. INTRODUCTION

Aluminium-rich aluminium±lithium alloys are

strengthened by coherent spherical precipitates of

the metastable L12 long-range-ordered d' phase [1±

14]. Hence Al±Li alloys are isomorphous to g'-strengthened nickel-base superalloys: L12-ordered

precipitates are embedded in a disordered or short-

range-ordered f.c.c. matrix [15±24]. In both classes

of materials, matrix dislocations with (a0/2)h110iBurgers vectors glide in pairs, where a0 is the lattice

constant. The leading dislocation D1 of such a pair

creates antiphase boundaries (APB) in the L12-

ordered particles and the trailing dislocation D2 re-

stores their order. Due to the isomorphism between

Al±Li alloys and superalloys, models which relate

the critical resolved shear stress (CRSS) of superal-

loys to the dispersion of their g' particles are also

applicable to Al±Li alloys [2±6, 10±13].

The actually measured CRSS of both classes of

alloys comprises two contributions: tp and ts. tp is

needed to overcome the L12-ordered particles (sub-

script p for ``particles'') and ts stands for the CRSS

of the solid solution matrix (subscript s for ``sol-

ution''). The superposition of tp and ts will be dis-

cussed in Section 3.2.1.

Three states of aging have to be distinguished [21±24]. During an isothermal heat treatment, theunderaged, peak-aged and overaged states are

reached in succession; they are sketched in Fig. 1for a constant volume fraction f of spherical par-ticles. In order to put the evaluations and discus-sions given in Sections 3 and 4 on a ®rm basis, the

de®nitions of these three states and the relevantequations are summarized below.

1.1. Underaged state

The L12-ordered particles are sheared by dislo-cation pairs. In the critical moment when D1 cuts

through the particles, their spacing Lc1 (subscript cfor ``critical'') along D1 is larger than the squarelattice spacing Lmin of the particles in the glide

plane:

Lmin � 1��nsp Ar �1�

where ns is the number of particles intersecting theunit area of the glide plane and r their average

radius.

1.2. Overaged state

Particles are bypassed by dislocations; hence theyare not paired. The Orowan process operates [25].

1.3. Peak-aged state

This is the optimum state of the material: atgiven f, tp has its maximum value. There are two

Acta mater. Vol. 47, No. 6, pp. 1953±1964, 1999# 1999 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/99 $20.00+0.00PII: S1359-6454(99)00018-X

{Permanent address: Department of Metal Physics,Charles University, Ke Karlovu 5, 12116 Prague 2, CzechRepublic.

{To whom all correspondence should be addressed.

1953

quite di�erent types of peak-aged

states [5, 19, 21, 24, 26±28]. They are explained with

reference to Fig. 1, in which three functions

tp (r,f = const.) are shown. The ascending line

describes the underaged state (the particles are

sheared and Lc1 exceeds Lmin). The horizontal

straight line refers to the case where the particles

are sheared and Lc1 equals its minimum possible (at

given r and f) value Lmin. tp[Lmin (r,f= const.)]

is independent of r [see equation (3a) introduced

below]. The descending curve is based on the

Orowan process. The two types of peak-aged statesare the following ones:

Scenario (i). This case is sketched in Fig. 1(a).

The ascending curve referring to the underaged

state and the descending Orowan curve intersect

each other above the straight horizontal line

representing tp(Lmin). Hence the maximum of the

actual CRSS is given by tp[Lmin(r,f = const.)].

The double arrow indicates the range of peak-

aged states, in which tp equals tp(Lmin). This

range may be wide [28].

Scenario (ii). The peak-aged state is deter-

mined by the transition from particle shearing to

bypassing [Fig. 1(b)]. The ascending and descend-

ing curves intersect each other below the straight

horizontal tp(Lmin) line. In Fig. 1(b) the arrow

marks the maximum of tp. Actually this maxi-

mum will become rounded o� and there will be a

range of peak-aged states. In such specimens

Orowan loops and dislocation pairs are found

side by side.

Unfortunately, the conceptual di�erences between

the two above-described peak-aged states are not

always clearly expressed in the literature. It depends

on the system under investigation which of the two

scenarios leads to a consistent description of the

peak-aged state. For the commercial nickel-base

superalloys NIMONIC PE16 [21] and NIMONIC

105 [28] it was Scenario (i). But even if Scenario (i)

is applicable and equations (3a) and (b)Ðsee

belowÐhold, local ¯uctuations of the interparticle

spacings may render the operation of the Orowan

process possible in some parts of the glide plane: at

each particle con®guration encountered by D1, D1

will choose that mechanism which requires thelower stress, either particle shearing or circumvent-

ing. The probability of the operation of the latter

mechanism increases with the average particle

radius r (Fig. 1). Hence the larger r is, the more

likely it is to ®nd Orowan loops left behind by dis-

locations.

If during an isothermal Ostwald ripening treat-

ment, r increases monotonically and f stays con-

stant, the three above-named states of aging are

easily discerned in the function tp(t), where t is the

duration of the aging treatment: @tp/@t and @tp/@rare positive, zero and negative for underaged, peak-

aged, and overaged materials, respectively. This is,

for instance the case for g'-strengthened nickel-basesuperalloys (see e.g. Refs [21, 24, 28]), in which f is

virtually constant during Ostwald ripening.The CRSS contribution tp of g'-strengthened

nickel-base alloys has been related to their dis-

persion of spherical g' particles. In the present in-vestigation the same relations are applied to d'-strengthened Al±Li alloys.

1.4. Underaged specimens [18±24, 29]

Dislocation pairs cut through the long-range-

ordered particles:

2btpf� A*

1g3=2

�r

fS

�1=2� g

ÿA*

2xÿ au� �2a�

with

A*1 � 2C1

o 3r

poq

� �1=2

�2b�

and

A*2 �

2C1C2o 2r

poq�2c�

and g= speci®c energy of APBs, b =a0/Z2 = magnitude of the Burgers vector of the

dislocations D1/D2, a0=lattice constant, S= linetension of D1, x= w/(orr), and w = average rangeof the interaction force between one g' particle and

D1 [21, 24, 30]. or and oq are statistical coe�cients;they are governed by the distribution function of in-dividual particle radii. For the Wagner [31]±Lifshitz±Slyozov [32] distribution or and oq equal

0.82 and 0.75, respectively [21, 24]. C1 and C2 arenumerical factors involved in the theories ofstrengthening, where C1=0.94 and

C2=0.82 [21, 24, 30]. These values lead to A*1=0.91

and A*2=0.44. Lmin introduced in equation (1)

equals [r(poq/f)1/2]. au (subscript u for ``underaged'')

depends on the con®guration of the trailing(a0/2)h110i dislocation D2 of a pair. There are twolimiting cases: if D2 is straight, au equals unity; ifD2 lies outside of all g' particles, au vanishes. In the

derivation of equation (2a) it has been assumedthat f does not exceed10.2.In a recent paper on d' strengthening of an Al±Li

alloy, which contained a minor addition of Ag, Leeand Park [33] based their evaluations on anequation which is quite similar to equation (2a), but

the authors presupposed that au as well as x vanish.The authors justi®ed this as follows.

1. a= 0. Pretorius and RoÈ nnpagel [34] simulated

the glide of dislocations in the g'-strengthendnickel-base superalloy NIMONIC PE16 in acomputer and concluded that au vanishes.

2. x= 0. In their computer simulations of the glideof a dislocation through a random array ofextended obstacles, Schwarz and Labusch [30]allowed for the range w. Haasen and

KALOGERIDIS et al.: CRITICAL RESOLVED SHEAR STRESS1954

Labusch [35] applied the results of these compu-

ter simulations to g' hardening: this led to the

term [A*2x] in equation (2a). Lee and Park

claimed that Labusch statistics are more appro-

priate for di�use obstacles. Labusch [36] stressed,

however, that w should be always allowed forÐ

unless it has been proved for the system under

investigation that w can actually be disregarded.

In order to keep all subsequent discussions as

general as possible, the terms [A*2x] and au are kept.

The decision whether they are necessary or not is

left to the experiment. For the g'-strengthenednickel-base superalloy NIMONIC PE16, it turned

out that the term [A*2xÿ au] is negative [21],

whereas for underaged d'-strengthened Al±Li alloys

it has been found to be positive [11, 13]. Evidently,

neither x nor au should be disregarded a priori. This

discussion will be taken up in Section 4.1.

Lee and Park rightly remarked that Schwarz and

Labusch [30] treated only weak obstacles. The for-

mer authors based their evaluations on Friedel's

formula [24, 37, 38], which too is correct only for

weak obstacles: Lee and Park aimed at improving

on S appearing in this formula. Their evaluations

and the present ones di�er in three aspects, namely

au, x, and the dislocation line tension S. The latter

point will be dealt with in Sections 3.2.2 and 4.1.

1.5. Peak-aged specimens [19±24, 28, 39]

Equations (3a) and (b) refer to the Scenario (i)

described above: the long-range-ordered particles

are sheared by dislocation pairs and Lc1 equals Lmin

2btpf� g

�B

f 1=2ÿ ap

��3a�

with

B � 2o rÿpoq

�1=2 � 1:07 �3b�

On the basis of the results of Foreman and Makin's

computer simulations [19, 40], some authors reduced

B by a factor close to 0.8 [22, 41], which is supposed

to allow for the random arrangement of particles in

the glide plane. Actually, however, the interparticle

spacings (Fig. 2) do not vary as widely as they do

in a random array [24]. Though ap (subscript p for

``peak-aged'') has an analogous meaning as auappearing in equation (2a), it cannot be taken for

granted that ap equals au. Equation (3a) does not

involve the particle radius. This is evident in Fig. 1

where the function tp[Lmin(f= const.,r)] is constant.

In the present paper measurements of the CRSS

of peak-aged d'-strengthened binary Al-rich Al±Li

single crystals are reported. This contrasts with ear-

lier work the focus of which was on underaged Al±

Li crystals [11]. The problem with Al±Li alloys is

Fig. 1. Schematic sketches of the three states of aging: underaged, peak-aged, and overaged. The CRSStp is plotted vs r1/2, f is supposed to be constant. Particle shearing by paired dislocations, underaged,Lc1>Lmin: ÐÐÐ; particle shearing by paired dislocations, peak-aged, Lc1=Lmin: - - -; single, unpaireddislocations bypass the particles: Ð Ð Ð. The peak-aged state (arrows) is governed (a) by the glide ofdislocation pairs and by the condition Lc1=Lmin [Scenario (i)], and (b) by the transition from particle

shearing to bypassing [Scenario (ii)].

KALOGERIDIS et al.: CRITICAL RESOLVED SHEAR STRESS 1955

that during isothermal aging r as well as fincrease [3, 4, 10±13, 42, 43] with time t. Therefore, iffor a given Al±Li alloy tp is plotted vs t or r, it isvery di�cult to recognize peak-aged specimens.

This will be detailed in Section 3.2.2.

2. EXPERIMENTAL METHODS AND RESULTS

2.1. Specimen preparation and measurements of theCRSS

The preparation of the single-crystal specimenswas the same one as in Ref. [11], only their orien-tation di�ered slightly: h1 5 11i instead of h1 2 10i.The overall atomic fractions ct of Li in the presentalloys were 0.0114, 0.0454, 0.0753, and 0.108. Thesevalues have been calculated from the constituents:

0.99999 Al and an Al±0.469 Li master alloy. Thecompositions have been checked by atomic absorp-tion spectroscopy. At the present aging temperatureof 433 K, the two ®rst alloys are homogeneous

solid solutions and the latter two alloys are d'strengthened. The specimens were compressiontested at 283 K; the resolved strain rate was

1.1�10ÿ4/s.The CRSSs ts of the solid-solution single crystals

were measured in two states:

1. (a)-specimens: directly after quenching from thehomogenization treatment at 843 K;

2. (b)-specimens: after compression testing the (a)-

specimens by 2±4%, they were re-homogenizedand subjected to an aging treatment at 433 K.

The present d'-strengthened specimens will be

referred to by Px/y where P stands for ``present'',and x and y indicate the d' particle dispersion:x1100f, y1r in nanometres. All results are listedin Table 1. Each of the CRSS entries t (equal to ts

or tt) is the average over 3±4 non-predeformed spe-

cimens; the quoted error limits of t are the standard

deviations of the averages.

The average radius r and the volume fraction f of

the d' precipitates have been determined by trans-

mission electron microscopy (TEM) [42, 43].

Examples are presented in Fig. 2. The d' precipitatesare spherical and hence they comply with the

assupmtion made in the derivations of

equations (2a)±(c) and (3a) and (b). The error limits

of r and f are estimated to be below, in most cases

even well below, 5 and 20%, respectively. f

increases with r. The f values of the present speci-

mens have not been smoothed; hence the function

f(r, ct=const.) does not always increase monotoni-

cally. In contrast, the f values in Ref. [11] had been

smoothed.

Besides the present experimental results, Schlesier

and Nembach's [11, 13] and Jeon and Park's [12, 13]

data will be analysed. Since all present CRSS data

and those published by Jeon and Park have been

Fig. 2. Dark ®eld images of d' particles: (a) f = 0.058, r= 27 nm, undeformed; (b) f = 0.169,r= 19.7 nm, deformed. Along the glide planes marked by arrows the d' particles have been sheared by

dislocations.

Table 1. Experimental results: ct=overall atomic fraction of Li,f = d' volume fraction, r= average radius of the d' precipitates,

t = CRSS (ts or tt) at 283 K

100ct 100f r[nm] t[MPa]

ssa a 1.14 5.620.2ss b 1.14 3.520.6ss a 4.54 8.820.3ss b 4.54 4.320.2P4/5 7.53 4.48 5.24 10.321.3P5/9 7.53 5.49 8.9/19.0 23.821.9P6/16 7.53 6.07 16.4 52.625.4P6/27 7.53 5.77 27.0 60.723.7P7/7 10.8 7.28 6.55 65.822.5P13/11 10.8 12.8 10.8 75.825.0P17/20 10.8 16.9 19.7 104.328.0P25/30 10.8 24.6 30.4 121.524.9

ass denotes homogeneous Al-rich solid solutions.


taken close to ambient temperature, only those ofSchlesier and Nembach's data will be discussed

which have been measured at 283 K. In Ref. [13]these data have been compiled from the literature.They will be referred to by Sx/y (S for

Schlesier [11, 13]) and Jx/y (J for Jeon [12, 13]); xand y have the same meaning as in Px/y.The following data will be excluded from all

evaluations; these data are obviously de®cient.

P4/5: tt is far too small.

P5/9: the d' particle dispersion is inhomo-geneous; in most parts r is 8.9 nm, but there areregions of about 3 mm diameter in which r isclose to 19 nm; f is 0.055 everywhere. On the

basis of the growth law of the d' particles, i.e. r3

varies linearly with aging time [31, 32], it hasbeen proved that 8.9 nm is the right value.

P7/7: tt is too high. This will be discussed inSection 4.1.S3/5: f= 0.0322, r = 4.67 nm [11, 13]; same as

specimens P7/7.J6/11: f = 0.057, r= 10.9 nm [12, 13]; r of

these specimens is too large. It should be around9 nm instead of 10.9 nm; this follows from a plot

of the function f(1/r) [42]. The di�erence betweenthe two r values amounts to about 20%, which isfar beyond the estimated limits of error.

It can be ruled out that the de®ciencies of speci-mens P4/5 and P7/7 are caused by some Li segre-

gation during single-crystal growth. As stated aboveeach entry for tt is the average over 3±4 specimens.Each has been cut from di�erent parts (top, middle,

bottom) of di�erent crystals.

2.2. TEM of dislocation con®gurations

After compression testing the present d'-strength-ened specimens to technical strains of about 2±4%,their dislocation con®gurations have been studiedby TEM. Thin slices were cut by spark erosion ap-

proximately parallel to the primary {111} glideplanes. Typical examples of the observed dislo-cation con®gurations are shown in Fig. 3. Three

di�erent types of con®gurations have been found:(1) dislocation pairs, (2) single, i.e. unpaired dislo-cations, and (3) dislocation loops. The abundanceof these three types of con®gurations is character-

ized semi-quantitatively by the parameter F listedin Table 2. F= 0 indicates that the respective typewas completely absent and F = 5 means that the

respective type was highly dominant. There is a ten-dency of F of dislocation pairs to decrease as thesquare lattice spacing Lmin of the d' precipitates

[equation (1)] increases. F of single dislocations andloops shows the opposite variation with Lmin.Sheared particles like those shown in Fig. 2(b)

have been found in all present specimens. Singledislocations are probably relics of paired dislo-cations which became uncoupled due to cross-slipof D1 [14].

3. DATA EVALUATIONS

3.1. CRSS of homogeneous solid solutions

In Fig. 4, ts is plotted vs c2=3s ; cs is the atomicfraction of Li in these solid solutions. Equation

(4a) [11, 44] has been least squares ®tted to the sixdata:

tsÿcs� � ts

ÿcs � 0

�� t00c2=3s �4a�The results of the ®t are: ts (cs = 0) = 4.5 MPaand t00 = 21.3 MPa. The scatter of the data is

rather wide. In Section 3.2.1, however, it will beshown that this has no serious e�ects on the evalu-ations of the data taken for d'-strengthened speci-

mens. ts of the ``recycled'' (b)-specimens, whichhave been heat treated at 433 K (Section 2.1), is sig-ni®cantly lower than ts of the other present solid-

solution single crystals, which had only been hom-ogenized once. This lowering of ts may be due torecovery of some damage incurred by the specimensduring quenching from the homogenization tem-

perature or during spark machining. If the ``re-cycled'' specimens are disregarded ts(cs=0) and t00equal 5.0 MPa and 32.4 MPa, respectively.

Jeon and Park [12] represented their ts data by alinear relation:

tsÿcs� � ts

ÿcs � 0

�� t00cs �4b�with ts(cs=0) = 1.7 MPa and t00=81 MPa.

Equation (4b) yields slightly lower CRSS ts valuesthan equation (4a) does.

3.2. CRSS of d'-strengthened specimens

The most important numerical results presentedbelow have been compiled in Table 3.3.2.1. Derivation of tp. The total, actually

measured CRSS tt (subscript t for ``total'') of a d'-strengthened specimen involves two contributions:that of the d' precipitates (tp) and that of the matrix(ts). ts is calculated with the aid of equations (4a)

and (b). Equation (4b) is only used for Jeon andPark's [12] data Jx/y. The atomic fraction cs of Liin the matrix is derived from the specimen's overall

atomic fraction ct (subscript t for ``total''), from theatomic fraction cp (subscript p for ``particles'') inthe d' particles and from f:

cs � ct ÿ fcp1ÿ f

�5�

This is the lever rule. cp is close to 0.23 [45±47] andcs of all analysed specimens is between 0.058(S6/11) and 0.098 (P7/7).

The stress tp follows from the empiricalequation (6) [21, 24]:

tp �ÿtkt ÿ tks

�1=k �6�For underaged as well as for peak-aged g'-strength-ened single crystals of the commercial nickel-basesuperalloy NIMONIC PE16 [21], the exponent k


Fig. 3. TEM of dislocation con®gurations in deformed peak-aged d'-strengthened specimens. Both longdislocation pairs in (a) and (b) have identical Burgers vectors ~b: (a) f = 0.128, r= 10.8 nm; (b)

f = 0.169, r= 19.7 nm; (c) f = 0.246, r= 30.4 nm; (d) f = 0.061, r= 16.4 nm.

Table 2. TEM results of the present peak-aged specimens Px/y: ct=overall atomic fraction of Li, Lmin=square lattice spacing of the d'precipitates, r= their average radius, F= abundance of the dislocation con®gurations (pairs, single dislocations, Orowan loops); F = 0,

absent; F= 5, highly dominant; g= speci®c APB energy

No. 100ct Lmin [nm] r [nm] Abundance F g (J/m2) FigurePairs Single Loops

P4/5 7.53 38 5.2 4 1 1P5/9 7.53 58a 8.9/19.0 4 2 1P6/16 7.53 102 16.4 2 3 3 0.105 Fig. 3(d)P6/27 7.53 172 27.0 1 2 4 0.126P7/7 10.8 37 6.6 5 0 0P13/11 10.8 46 10.8 5 0 0 0.107 Fig. 3(a)P17/20 10.8 73 19.7 4 0 1 0.131 Fig. 3(b)P25/30 10.8 94 30.4 1 4 4 0.127 Fig. 3(c)

aLmin calculated for r= 8.9 nm.


has been found experimentally to equal 1.23. For

the reasons given in Ref. [11], it was not possible todetermine k of underaged d'-strengthened Al±Li

alloys experimentally. Therefore k = 1.23 has also

been used for these [11, 13]. Equation (6) with

k = 1.23 is also applied to the present d'-strength-ened Al±Li single crystals. The choice of k has only

minor e�ects on tp because ts is rather low. For thesame reason, the uncertainties of ts evident in Fig. 4,

do not a�ect tp seriously.

3.2.2. Identi®cation of peak-aged specimens. Since

during isothermal aging, f of d'-strengthened Al±Li

alloys increases with time and hence with

r [11, 42, 43], it is very di�cult to recognize peak-

aged specimens (Section 1). In the present investi-gation the following approach has been chosen for

identifying them. According to equation (2a), which

has been meant for underaged specimens, the term

[2btp/f] of all d'-strengthened specimens is plotted vs

[r/(fS)]1/2 in Fig. 5; here and in the following the

specimens listed at the end of Section 2.1 have been

disregarded.

The dislocation's line tension S is written as:

S � KSgb2 ln

�R0

b

��7a�

KSg is the geometric mean of the elastic prefactors

of edge and screw dislocations. This choice has

been justi®ed in Ref. [21]. The present TEM study

of deformed peak-aged specimens revealed that the

characters of the paired dislocations covered the

entire range from screw to edge [Figs 3(a) and (b)].

The magnitude b of the Burgers vector is

0.286 nm [48]. R0 is the outer cut-o� radius; Lmin

[equation (1)] will be inserted for it. KSg depends on

the atomic fraction cs of Li in the matrix; KSg has

been published by Schlesier and Nembach [11]:

KSg � KSg0

ÿ1� rcs

� �7b�

Fig. 4. CRSS ts of homogeneous Al±Li solid solutions vsthe atomic fraction c2=3s of Li. Present specimens: onlyhomogenized (a)-specimens, D; after deformation, re-hom-ogenization, and aging ``recycled'' (b)-specimens, H; fromRef. [11], q. The straight line represents equation (4a)

®tted to the data.

Table 3. Results of the evaluations of the CRSSs: k = exponent in equation (6), R0=outer cut-o� radius in equation (7a), ts(cs=0) andt00=parameters in equation (4a), D = average deviation of the data from the ®tted straight lines and [A*

2xÿ au] = term in equation (2a)

State of aging No. Evaluated specimens k R0 ts(cs=0)/t00 [MPa] g (J/m2) D (%) [A*2xÿ au] Figure

Underaged 1 Sx/y, Jx/y 1.23 Lmin 4.5/21.3 0.0842 4.2 2.0 Fig. 52 Px/y, Sx/y, Jx/y 1.23 Lmin 4.5/21.3 0.0811 9.3 2.23 Sx/y, Jx/y 1.00 Lmin 4.5/21.3 0.0786 5.4 2.14 Sx/y, Jx/y 1.23 LF 4.5/21.3 0.0827 4.1 2.25 Sx/y, Jx/y 1.23 Lmin 5.0/32.4 0.0817 4.4 2.26a Sx/y, Jx/y 1.23 Lmin 4.5/21.3 0.107 4.3 1.67b Sx/y, Jx/y 1.23 Lmin 4.5/21.3 0.0665 4.2 2.6

Peak-aged 8 Px/y 1.23 4.5/21.3 0.119 8.9 Fig. 6(a)9 Px/y 1.00 4.5/21.3 0.112 9.810 Px/y 1.23 5.0/32.4 0.117 9.411 Huang/Ardell [5] 1.23 4.5/21.3 0.135 9.2 Fig. 6(b)12c Jeon/Park [12] 1.23 1.7/81.0c 0.112 5.3 Fig. 6(a)

aS [equation (7a)] of screw dislocations.bS of edge dislocations.cEquation (4b).

Fig. 5. [2btp/f] vs [r/(fS)]1/2 of d'-strengthened Al±Li singlecrystals: present data Px/y, .. The straight line representsequation (2a) ®tted to the data Sx/y, q [11, 13] and Jx/y,

D [12, 13], taken from the literature.


where KSg0 and r equal 2.01 GPa and 2.7, respect-

ively. These authors also listed the parameters KS0

and r of pure edge and pure screw dislocations.

The straight line in Fig. 5 represents equation (2a)

least squares ®tted to Schlesier and Nembach's

(Sx/y) and Jeon and Park's (Jx/y) data only. The

results for the speci®c APB energy g and the term

[A*2xÿ au] are 0.0842 J/m2 and 2.04, respectively.

On an average, the quoted authors' data deviated

by 4.2% from the straight line. In Fig. 5, all ®ve

present data Px/y lie more than 11% below the

straight line. On an average they are too small by

21%. The latter two statements hold independently

of the value inserted for the elastic prefactor of S

[equation (7a)]. On the basis of Fig. 1(a), it is con-

cluded that the present specimens Px/y are in the

peak-aged state: their CRSS falls in the range

marked by the double arrow in Fig. 1(a).

3.2.3. The CRSS of peak-aged specimens. Ifequation (3a), which refers to peak-aged specimens,

is least squares ®tted to the ®ve data of the peak-aged specimens Px/y, ap turns out to be slightlynegative. That is unreasonable and caused by the

scatter of the data. Therefore ap is set equal to itsminimum value of zero and with the aid ofequation (3a) the speci®c APB energy g is calculated

for each tp datum. The resulting individual g valuesare listed in Table 2; the average over these ®vedata is 0.119 J/m2. The average deviation from the

latter value amounts to 28.9%. This is well withinthe error limits estimated on the basis of those of tpand f.In Fig. 6(a), [2btp/f] of the ®ve present peak-aged

specimens has been plotted vs 1/f1/2. The straightline shows equation (3a) with g = 0.119 J/m2,B = 1.07, and ap=0.0. Evidently equation (3a) rep-

resents the data very well; this proves that (a) thesespecimens are in the peak-aged state and Scenario(i) (Section 1) applies (Section 4.2.1) and that (b)

ap=0.0 is correct.In the disregarded specimens P5/9, whose d' par-

ticle dispersions are inhomogeneousÐr= 8.9 nm

and r = 19 nm (Table 1)Ðdislocation pairs predo-minate (Table 2). If r= 8.9 nm is inserted for thesespecimens and their CRSS tp is compared with tpof other underaged specimens, tp of specimens P5/9

turns out to be too low by more than a factor oftwo. The same result is obtained if these specimensare assumed to be peak-aged. Evidently these speci-

mens ®t neither scheme.

4. DISCUSSION

4.1. Underaged specimens

According to Fig. 5, equation (2a) represents thefunction tp(f,r) of underaged specimens well. The

present result g = 0.0842 J/mm2 for the speci®cAPB energy of the underaged specimens Sx/y andJx/y agrees well with the former resultg = 0.0780 J/m2 [13]. The small di�erence arises

from the present disregard of specimens S3/5 andJ6/11 and from the slightly modi®ed evaluation ofts. Sluiter et al. [49] performed ®rst-principles calcu-

lations and obtained g= 0.085 J/m2. This valuealmost coincides with the present one.The reliability of the result presently obtained for

underaged specimens has been checked by tryingseveral alternatives; they are listed here and the re-spective results have been compiled in Table 3:

1. as described above;2. the present peak-aged specimens Px/y are treated

as underaged ones;

3. unity is inserted for k appearing in equation (6);4. the Friedel length [22±24, 37, 38] is inserted as

outer cut-o� radius R0 in equation (7a); and5. the coe�cients in equation (4a) are varied.

Fig. 6. [2btp/f] vs 1/f1/2 of peak-aged specimens. Thestraight lines represent equation (3a). Note the di�erentscales: (a) w, present data Px/y; D Jeon and Park's [12]data; g= 0.119 J/m2, B= 1.07, ap=0.0. (b) Data com-piled by Huang and Ardell [5] (Cassada et al.'s [54] datum

is not shown); g = 0.135 J/m2, B = 1.07, ap=0.0.


The scatter of the results obtained for g is small;this demonstrates their reliability. In all evaluations,

the term�A*

2xÿ au�turns out to be positive: it is

close to 2.0. Hence, since au is non-negative, x mustbe positive, i.e. the ®nite range w = (xorr) (Section

1) of the interaction force between d' particles anddislocations must be allowed for. If au as well as xare supposed to vanish, the straight line in Fig. 5

must pass through the origin of the coordinate sys-tem. Evidently this would lead to systematic devi-ationsÐ at low/high values of the abscissa, the data

points represented by open symbols would lieabove/below, respectively, the straight line.Other authors [5, 10, 12] who analysed underaged

d'-strengthened Al±Li alloys found somewhat higher

g values; at least part of the di�erence is caused bythe higher line tension S inserted by theseauthors [11, 13, 33].

The underaged specimens P7/7 and S3/5 have toohigh CRSS tp values. Both specimens have thesmallest radii r at given Li concentration, ct. In the

case of specimens S3/5, it has been suggested [11]that there may be some contribution from chemicalstrengthening. This may also be the case for the

present specimens P7/7. Moreover, the following in-terpretation appears to be possible. Specimens P7/7have the highest Li concentration cs in the matrix:cs equals 0.098. This is more than twice the Li con-

tent of the most concentrated homogeneous solidsolutions presently investigated (cs=0.0454;Table 1). Sato and Kamio [50] found in a quenched

binary Al±0.079 Li solid solution some precursoryorder. The view that there may be some order insome of the present matrices is supported by Poduri

and Chen's [51] recent theoretical calculations. Ifthe matrix of specimens P7/7 is somewhat ordered,ts calculated for them with the aid of equation (4a)will be too low and consequently tp will have been

overestimated.For unknown reasons the CRSS tp of specimens

P4/5 is far too small. The TEM work revealed no

anomaliesÐthere were predominantly dislocationpairs and a few single dislocations and loops(Table 2).

4.2. Peak-aged specimens

4.2.1. Applicability of Scenario (i). As shown in

Section 3.2.3, equation (3a) represents the tp dataof the present peak-aged Al±Li specimens well. Thisproves the applicability of Scenario (i), for which

this equation had been derived. The observation ofsheared d' particles [Fig. 2(b)] and of paired dislo-cations (Fig. 3, Table 2) in all present specimens

lends further support to Scenario (i). At constant ct,there is a tendency of g to increase with Lmin

[equation (1)].

4.2.2. Discussion of possible e�ects of the Orowanprocess and of single dislocations on tp. If the CRSSof the peak-aged specimens with large Lmin valueswere signi®cantly lowered by the operation of the

Orowan process, the respective g values calculated

with the aid of equation (3a) would turn out to be

too low. Actually specimens P6/27, which have the

highest Lmin value and in which some single dislo-

cations and very many loops have been found in

the TEM study (Table 2), have a g value above

average. The applicability of Scenario (ii) (Section

1) will be ruled out in Section 4.3.

Similarly the occurrence of single dislocations has

hardly any e�ect on the CRSS. The g values of spe-

cimens P6/27 and P25/30, which primarily di�er in

their abundance of single dislocations (Table 2), are

the same within 20.4%. Since single dislocations

need approximately twice as high shear stresses for

their glide as dislocation pairs need, one may con-

clude that unpaired dislocations are immobile and

plastic ¯ow is carried on by the remaining pairs.

4.2.3. Alternative evaluations. The results of some

slightly di�erent evaluations of the data obtained

for the present peak-aged specimens Px/y are given

in lines 8±10 of Table 3. These results have been

found to be very stable.4.2.4. Summary. Scenario (i) and equations (3a)

and (b), which are based on it, lead to a consistent

description of the function tp(f) of peak-aged d'-strengthened specimens. The three important suppo-

sitions are:

(a) the d' particles are sheared by dislocations;

(b) dislocations glide in pairs; and

(c) the critical length Lc equals Lmin

[equation (1)].

This view is in agreement with the TEM obser-

vations (Section 2.2): in all present specimens Px/y

sheared d' particles have been found and there were

dislocation pairs (Table 2).

4.2.5. Possible variation of g with r. As can be

seen in Table 2, there isÐat constant overall Li

concentration ctÐa tendency of g of the peak-aged

specimens to increase with r. It is open to discus-

sion whether this increase is actually signi®cant.

Below this issue will be analysed further. As men-

tioned in Section 4.2.1, there is an analogous vari-

ation of the function g(Lmin, ct=const.). Since Lmin

is proportional to r [equation (1)], the increase of

the function g(Lmin, ct=const.) probably re¯ects

that of the function g(r, ct=const.).

4.2.6. Discussion of data of other authors. Jeon

and Park [12] have published data for two peak-

aged binary Al±Li specimens with ct=0.0735:

f = 0.072; ttÿts=49.6 GPa;

f = 0.080; ttÿts=58.8 GPa.

Evaluating these data as described in Section 3.2.3

yields g = 0.11220.006 J/m2. This result is in excel-

lent agreement with that obtained for the present

peak-aged specimens namely 0.11920.011 J/m2.

This agreement is also evident in Fig. 6(a), where

Jeon and Park's data are shown together with the

present ones.


Huang and Ardell [5] have compiled several ttand ts data of supposedly peak-aged Al±Li alloys

from the literature [2±4, 52±54]. There are problemswith some of these data: f had to be amended andyield stresses had to be converted to CRSSs. Since

some of the data [3, 4, 54] compiled by the quotedauthors for underaged specimens have been foundto be unreliable [11], one must be sceptical about

the supposedly peak-aged ones. Moreover, theCRSS of single crystals of a particle-hardened ma-terial and the yield strength of polycrystals of the

same material do not always peak for the same par-ticle dispersion [24, 55].Nevertheless, with the only exception of the

datum originally published by Cassada et al. [54],

all data compiled by Huang and Ardell for suppo-sedly peak-aged Al±Li alloys have been evaluatedwith reference to equation (3a) with ap=0.0.

Equation (6) with k = 1.23 has been used to deducetp from tt and ts. The result for the average g is0.135 J/m2. This value is only 13% higher than that

obtained for the present peak-aged specimensnamely 0.119 J/m2. In Fig. 6(b), all evaluated tpdata of Huang and Ardell's compilation are shown.

The straight line represents equation (3a) withg = 0.135 J/m2, B = 1.07 and ap=0.0. The averagedeviation of individual g values from the average gamounts to 29.2%; this is very close to the scatter

of the g values of the present peak-aged specimens(8.9%, Section 3.2.3). In their own evaluation,Huang and Ardell obtained 0.144 J/m2 (excluding

Cassada et al.'s datum). This is only 7% more thanthe result of the present evaluation (0.135 J/m2).The evaluation of these authors di�ered from the

present one in two respects: B was reduced inequations (3a) and (b) by the statistical factor 0.81and unity was inserted for k appearing inequation (6). The e�ects of these two di�erences

mostly compensate.

4.3. Comparison of g values derived from underaged

and peak-aged specimens

The above evaluations for underaged and forpeak-aged d'-strengthened Al±Li alloys yield for g0.084 J/m2 and 0.119 J/m2, respectively. As inRefs [11, 13], the error limits are estimated to be20.02 J/m2. Evidently the two present results for gdo not agree within these margins. This di�erence isnot caused by the fact that actually Scenario (ii)instead of (i) (Section 1) applies. If the CRSS of the

peak-aged specimens were governed by Scenario(ii), deriving g on the basis of equations (3a) and(b), which had been meant for Scenario (i), would

yield too low g values, i.e. the observed di�erencebetween g(underaged) and g(peak-aged) would have theopposite sign: g(underaged)>g(peak-aged). Three possibleinterpretations of the discrepancy are discussed.

4.3.1. Minor de®ciencies of equations (2a)±(c)

and (3a) and (b). Though these equations have

been proved to correctly represent the functional

variation of tp with f and r, there may be cali-

bration errors of, e.g. C1 and B in equation (2b)

and equations (3a) and (b), respectively. Reducing

B appearing in equations (3a) and (b) by a statisti-

cal factor close to 0.8 raises g(peak-aged), i.e. this

makes the discrepancy between g(peak-aged) and

g(underaged) even worse. Huang and Ardell reduced B

by the factor 0.81.4.3.2. Dislocation line tension S. According to

equation (2a), the result for g(underaged) of underagedspecimens depends on S as g(underaged) is approxi-

mately proportional to S1/3. So far the geometric

mean of S of edge and screw dislocations has been

inserted (Section 3.2.2); this has been justi®ed in

Ref. [21]. In general, the actual CRSS contribution

tp is governed by that type of dislocation which

requires the higher stress to glide [24]. In the pre-

sent case, these would beÐat given gÐedge dislo-

cations: they have the lower value of S and hence

lead to the higher value of tp. Inserting the respect-

ive values yields for the underaged Sx/y and Jx/y

specimens, g(underaged, edge)=0.067 J/m2 (line 7 in

Table 3). Evidently this value is even smaller than

the one obtained above for the geometric mean S,

namely 0.084 J/m2. In their evaluations Huang and

Ardell [5] and Park and coworkers [12, 33] inserted

S of screw dislocations and obtained high values

for g(underaged). With S of screw dislocations, g(under-aged, screw)=0.107 J/m2 (line 6 in Table 3) is obtained

for the Sx/y and Jx/y specimens. This result is

rather close to the present one for g(peak-aged),namely 0.119 J/m2. But it is hard to justify the

insertion of S of screw dislocations. In the present

TEM study of peak-aged specimens, dislocation

pairs of edge, of screw and of mixed character have

been observed; they were strongly bent [Section 2.2,

Figs 3(a) and (b)]. The same holds for the micro-

graph of a deformed underaged bulk specimen

shown in Fig. 7, i.e. because of the strong bending,

the dislocations' character varies over the whole

range.

4.3.3. g varies with r. Another tentative interpret-

ation is that the overall degree of L12 long-range

order of the d' particles increases with the duration

of the Ostwald ripening treatment and hence with r.

In this case, g of peak-aged specimens exceeds g of

underaged ones, because the latter ones have d'particles with smaller radii. This interpretation

is in line with the above-mentioned tendency of

g(peak-aged) to increase with r (Table 2). In this view

g(under-aged) and g(peak-aged) refer to speci®c APB

energies averaged over di�erent ranges of r.


5. CONCLUSIONS

Equations (3a) and (b) describe the e�ects of L12long-range-ordered d' precipitates on the CRSS ofpeak-aged Al-rich Al±Li single crystals well.

In all present peak-aged d'-strengthened speci-mens, sheared d' particles and dislocation pairshave been observed. In the majority of peak-aged

specimens, also unpaired, single dislocations anddislocation loops have been found. The applicabilityof equations (3a) and (b) together with the TEMobservations prove that the critical resolved shear

stress of peak-aged specimens is governed by themobility of dislocation pairs, which cut through thed' particles, and that Scenario (i) described in

Section 1 applies.

AcknowledgementsÐThanks are due to Dr H. RoÈ sner,now with the Forschungszentrum Karlsruhe, for providingFig. 7. Financial support by the DeutscheForschungsgemeinschaft is gratefully acknowledged.

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the critical resolved shear stress of peak-aged aluminium–lithium single crystals

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