-
Dislocation Multiplication from the FrankRead Source in Atomic
Models
Tomotsugu Shimokawa1,+ and Soya Kitada2
1School of Mechanical Engineering, College of Science and
Engineering, Kanazawa University, Kanazawa 920-1192, Japan2Division
of Mechanical Science and Engineering, Graduate School of Natural
Science & Technology, Kanazawa University,Kanazawa 920-1192,
Japan
Dislocation multiplication from the FrankRead source is
investigated in aluminum by applying atomic models. To express the
dislocationbow-out motion and dislocation loop formation, we
introduce cylindrical holes as a strong pinning point to the
dislocation-bowing segment.The critical configuration for
dislocation bow-out in atomic models exhibits an oval shape, which
agrees well with the results obtained by theline tension model. The
critical shear stress for the dislocation bow-out in atomic models
continuously increases with decreasing length L of theFrankRead
source (even at the nanometer scale). This is expressed by the
function L¹1 lnL, which is obtained by a continuum model based
onelasticity theory. The critical shear stresses for the FrankRead
source are compared with those for grain boundary dislocation
sources, as well asthe ideal shear strength.
[doi:10.2320/matertrans.MA201319]
(Received September 3, 2013; Accepted October 15, 2013;
Published November 29, 2013)
Keywords: dislocation multiplication, FrankRead source,
dislocation bow-out, dislocation source, atomic simulation,
mechanical property
1. Introduction
Plastic deformation of crystalline materials at low
temper-atures occurs by dislocation movements; hence, the
resistanceto dislocation motion controls material strength. For
coarse-grained polycrystalline metals with a grain size d larger
than1 µm, yield strength increases as grain size decreases.
Thisphenomenon is known as the HallPetch relation,1,2) wherethe
grain boundaries act as a strong barrier to the
dislocationmovement. One of the theoretical models available
todescribe this behavior is the dislocation pile-up model,which
assumes that a dislocation pile-up will induce a slip inthe
adjoining grain if the stress at the spearhead of the
pile-upreaches a critical value.3) On the other hand, if the grain
sizedecreases below 1µm, the strength cannot be expressed bysimple
extrapolation of the conventional HallPetch slope.4)
This result implies that we should consider other size effectsof
plastic-deformation phenomena, such as the competitionbetween
critical shear stresses for dislocation nucleationfrom
intragranular and intergranular dislocation sources,57)
which is believed to be a dominant plastic-deformationmode in
nanocrystalline metals (d < 100 nm).8) Atomic-levelresolution is
required to investigate grain-boundary-mediatedplastic
deformations; therefore, atomic simulations thatsimultaneously
treat intra- and intergranular dislocationsources are expected to
be a powerful tool to elucidate theunique mechanical properties of
ultrafine grained metals.911)
The FrankRead source is the well-known intragranulardislocation
source,12) in which a dislocation segment lying onan active slip
plane ® and whose ends are strongly pinned bythe other parts of the
dislocation lying outside the plane®bows out under an applied
stress. At critical stress, thisdislocation segment can generate
dislocation loops, as aresult of which dislocation multiplication
can occur.Although many studies on the FrankRead source have
beenreported to date, including the use of analytical
expressionswith polygonal shapes,13) numerical calculations
investigat-ing its equilibrium shape under an external stress14)
with self-
stress,15) or dislocation dynamics simulations,1618) only afew
atomic simulations on this system are currently availablein the
literature.19,20) This is due to difficulties in obtainingstrong
pinning points in the atomic models for the dislocationbow-out and
the subsequent dislocation multiplication.21)
In this study, we attempt to realize the FrankRead source(which
can induce stable dislocation multiplication) in atomicmodels and
compare the critical shear stress for this sourcewith previously
reported values13,14,19,22) to evaluate theirvalidity at the
nanometer scale. Furthermore, the criticalshear stresses for
dislocation nucleation from the FrankRead source are compared with
those from grain boundariesreported in Ref. 7).
2. Methodology
Figure 1 shows our developed FrankRead source model.The embedded
atom method for aluminum, proposed byMishin et al.,23) is adopted
for the atomic interactions. Thecrystal orientation is set to be
the same as that used in themodel reported by de Koning et al.19)
However, it shouldbe noted that the Burgers vectors of the two
dislocationscreated on the {111} planes and the two dislocations
createdon the {112} planes have directions opposite to those
givenin the de Koning model because of the different
proceduresapplied to make the FrankRead sources. The
FrankReadsource in our model is created by introducing an atomic
plateinto the {110} atomic layers. The dimensions of the
atomicplate are L and H in the y and z directions,
respectively.
We performed a preliminary simulation by applying theFrankRead
source model developed by de Koning usingH = 4 nm and L = 10 nm. We
observed cross slips whichhave their stating point at the pinning
arms of {112}dislocations, a migration of the pinning points, and
thedislocation bow-out from the dislocation segment on the{112}
planes before reaching the critical conditions. Theseresults
suggest that no perfect dislocation bow-out motionoccurred when
applying the model by de Koning. Thisunexpected phenomenon has also
been reported by Liet al.20) and Tsuru et al.11) To obtain strong
pinning points,+Corresponding author, E-mail:
[email protected]
Materials Transactions, Vol. 55, No. 1 (2014) pp. 58 to
63Special Issue on Strength of Fine Grained Materials ® 60 Years of
HallPetch®©2013 The Japan Institute of Metals and Materials
http://dx.doi.org/10.2320/matertrans.MA201319
-
cylindrical holes with a diameter of 1 nm are introduced
byremoving the dislocation core structure on the {112} planes.The
dislocation segments on the {111} planes are thusterminated by the
free surfaces of the cylindrical holes.Weinberger et al. reported
that cylindrical holes have enoughdislocation-pinning potential.21)
Furthermore, to prevent across slip, whose starting point is the
pinning segment alongthe [111] direction, the x-directional
displacements of theatomic clusters located next to the slip plane
of thedislocation segments on the {111} planes are controlled.The
displacement-controlled regions around the cylindricalholes are
shown in Fig. 1(a) as hatching regions and inFig. 1(b) as solid
marked atoms. Each FrankRead sourcecorner has four atomic clusters,
as shown in Fig. 1(b), and thex-directional displacement of each
cluster is taken from theaverage x-directional displacement of each
atom in thatcluster. The other directional displacements of the
atomicclusters are not controlled.
The dimensions of the analytical models in the x, y and
zdirections (lx, ly and lz) are set to 40, 40 and 20
nm,respectively, and periodic boundary conditions are adopted inall
directions. H is set to 4, 6 and 8 nm, and L is set to begreater
than or equal to H. Each analytical model is relaxedfor 20 ps under
no external stress. The analysis temperature iskept at 10K by the
velocity scaling method,24) and eachdimension lx, ly and lz is
controlled to keep the nominal stressat zero.25) The equilibrium
atomic configuration of the FrankRead source is shown in Fig. 1(c),
where the dark- and light-gray spheres correspond to atoms in the
local hexagonalclose-packed (hcp) configuration and defect
structures,
respectively; the atoms in the face-centered cubic structureare
not shown. The classification of atomic structures isperformed by
common neighbor analysis.26) For each relaxedmodel, shear
deformation tests are performed at _£zx ¼ �5�108 1=s at 10K to
investigate the dislocation bow-out motionfrom the FrankRead
source.
3. Results and Discussions
3.1 Dislocation multiplication from the FrankReadsource
Figure 2 shows the changes in shape of the upperdislocation
segment with H = L = 4 nm under externalloading. The lower
dislocation segment behaves similar tothe upper one. Figure 3 shows
the variations in the number ofdefect atoms (Ndef ), including hcp
atoms. The letters in Fig. 3correspond to the shapes of the
dislocation bowing-outsegment in Fig. 2. Initially, the value of
Ndef increasesgradually, until the dislocation bow-out state in
Fig. 2(b) isreached, and after that it continues to grow
dramatically.Finally, a perfect dislocation loop is formed, as
shown inFig. 2(f ). Because periodic boundary conditions are
adoptedin all directions, the dislocation loop is eventually
destroyedby neighboring dislocation loops in image cells
(dislocationannihilation), and therefore, the defect structures in
theanalytical model return to their initial state with
plasticdeformation of one of the Burgers vectors. As shown inFig.
3, the dislocation segment can regenerate a dislocationloop under
an external loading, which confirms that ourdeveloped FrankRead
source has strong pinning points for
Fig. 1 (a) Analytical model for the FrankRead source showing a
dislocation segment L terminated by cylindrical holes with height
H.(b) Displacement control regions to prevent cross slip of the
bowing dislocation segment. (c) Atomic configurations of the
FrankReadsource under no external load. The light and dark colored
spheres represent defect and hcp atoms, respectively.
Dislocation Multiplication from the FrankRead Source in Atomic
Models 59
-
the dislocation segments and can suppress the cross slips.Thus,
our results demonstrate that dislocation multiplicationcan be
realized by atomic simulations.
3.2 Critical dislocation bow-out configurationIn situation (b)
in Fig. 3, Ndef increases dramatically once
the critical conditions for the dislocation bow-out motion
arereached. The stress in situation (b) can therefore be
attributedto the critical shear stress, ¸MD.27) Figure 4 shows the
shapesof dislocation segments with different L values in the
criticalstate. The value of H is fixed to 4 nm in all models.
Thewhite broken lines in Fig. 4 represent the critical
dislocationconfiguration in an isotropic elastic body, with a
Poisson ratioof 0.33, estimated by de Wit et al.28) Because of the
differentline tensions between the edge and screw
dislocationcomponents, the critical configuration does not show
aperfect circular form, as estimated by the constant linetension
model, but rather an oval shape with a ratio of theminor axis
length to the major axis length of 1 ¹ ¯, where ¯is the Poisson
ratio. The critical configuration obtained byatomic simulations
agrees well with the results reported by deWit et al. because the
anisotropic factor, calculated by c44/(c11 ¹ c12), is 1.25 for
aluminum (expressed by the atomicpotential); hence, we can assume
that this metal showsisotropic elastic properties. Hatano et al.29)
also reported the
same agreement: critical dislocation bow-out shapes, thatends
are pinned by nanovoids, in copper calculated by atomicsimulations
agree well with the results reported by de Witet al. The obtained
results suggest that the influence of thedisplacement control
regions on the dislocation bow-outmotion is neglected in our
models.
3.3 Critical shear stress for dislocation bow-outFigure 5 shows
the relationship between the critical shear
stress ¸MD and the length of the dislocation segment L.The value
of ¸MD reported by de Koning et al. (withL = 5 nm and H = 2.8 nm)
is also plotted. We also showthree critical shear stresses, namely,
¸OR, ¸HL, and ¸SB, where¸OR is estimated by a constant line tension
approximationcommonly known as the Orowan stress30) [eq. (1)], ¸HL
isderived from analytical expressions for the energies of
thedislocation segments with a semi-regular hexagon config-uration
[eq. (2) by Hirth and Lothe13)], and ¸SB represents thevoid-row
bypassing stress, void spacing L, and diameter D,and is numerically
evaluated by considering a dislocationself-interaction and a
dislocation-void interaction in eq. (3)by Scattergood and
Bacon.22)
¸OR ¼®b
Lð1Þ
¸HL ¼®b
2³Lð1� ¯Þ 1�¯
2ð3� 4 cos2 ¢Þ
h iln
L
r0� 1þ ¯
2
� �
ð2Þ¸SB ¼
®b
2³Lln
r0D
þ r0L
� ��1þ 1:52
� �ð3Þ
Here, ®, b, r0 and ¢ are the shear modulus, the magnitude ofthe
Burgers vector, the dislocation core radius, and the anglebetween
the Burgers vector and the original dislocationstraight line,
respectively. In this study, r0 and ¢ are set to band 0,
respectively, and because eqs. (1) to (3) are valid forisotropic
elastic bodies, the shear modulus ® and Poisson’sratio ¯ are
estimated by the Voigt approximation31) foranisotropic elastic
bodies, expressed by the atomic potential.We thereby obtain ® =
29.2GPa and ¯ = 0.33. In eq. (3),although D originally expresses
the void diameter, we assumethat D is ly ¹ L because the FrankRead
sources areperiodically arranged in the y direction in Fig. 1.
Generally, considering the dislocation self-interactiondecreases
the critical bow-out stress, that is, ¸HL and ¸SB issmaller than
¸OR, as shown in Fig. 5. Moreover, ¸HL shows a
Fig. 2 Dislocation multiplication from the FrankRead source
under external loading in the atomic simulations. L = H = 4 nm.
Fig. 3 Relationship between the number of defect atoms Ndef
(includinghcp atoms) and the simulation time for the dislocation
multiplicationconditions shown in Fig. 2 (the letters represent the
different dislocationbow-out states indicated in Fig. 2).
T. Shimokawa and S. Kitada60
-
non-monotonic dependence on L, as shown in Fig. 5.
Thedislocation bow-out from the FrankRead source with
thisnon-monotonic dependence is expected to be a dominantmechanism
for explaining the inverse HallPetch relationfound for a number of
nanocrystalline metals.32) However,the ¸MD value obtained by atomic
simulations shows amonotonic dependence on L, which means that
¸MDcontinuously increases with decreasing L. It is interestingthat
the ¸MD values obtained by our atomic models fit the ¸SBvalues
obtained by continuum model simulations, whichinclude several
approximations regarding the dislocation coreradius used (r0 = b)
and the dislocation-void interactionconsidered using reasonable
surface energy.22) Osetsky et al.also found the same agreement for
the motion of an infinite,straight but flexible edge dislocation
through a row of voidswith diameters ranging between 0.9 and 4 nm
in body-centered cubic iron.33) These results suggest that
thecontinuum model based on elasticity theory has enoughpotential
to describe dislocation bow-out motions, even in thenanometer
region. Our atomic simulations studies also showthat it is
difficult to explain the inverse HallPetch relation byusing the
dislocation bow-out motion from the FrankReadsource.
3.4 Influence of dislocationdislocation interactions onthe
dislocation bow-out
As shown in Fig. 5, the ¸MD values, including the results ofthe
pinning arms of the {112} dislocations reported by deKoning et
al.,19) decreases with increasing H, which is thedistance between
the upper and lower dislocation segments
in the absence of external loading. If we assume that
theinfluences of the pinning points of both the cylindrical
holesand the {112} dislocations on the dislocation bow-out
motionare similar, the interactions between the upper and
lowerdislocations will influence the value of ¸MD. Foremansuggests
that the critical bow-out stress for an edgedislocation segment is
approximately described by:
¸c ¼ A®b
2³Lln
L
r0
� þ B
� �; ð4Þ
where A is almost 1 for the edge and 1.5 for the screwsegments,
and the constant B depends on the orientation ofthe side-arms and
the presence of local stress fields.14) In thissection, we assume
that ¸MD is expressed by eq. (4) andevaluate the influence of
dislocationdislocation elasticinteractions by comparing the
critical stress ¸FO obtainedby Foreman14) for a dislocation segment
L with verticaldislocation arms to the glide plane of the segment
using theBrown self-stress method.15) Figure 6 shows the
relationshipbetween the critical shear stress ¸MD (in units of
®b/L: ¸OR)and L/r0 (r0 = b). The broken and solid lines represent
¸FOand eq. (4), respectively. The values of the constant B,fitted
to each H value, are also shown in Fig. 6. Although theupper and
lower dislocation segments in the atomic modelsinteract with other
dislocation segments in the image cells(owing to periodic boundary
conditions), the relationshipbetween ¸MD/¸OR and L/r0 is found to
fit the form eq. (4),with A = 1, for all the studied H values. A
similar influenceof the periodic boundary conditions is observed in
discretedislocation simulations.18)
Fig. 5 Critical shear stress as a function of the FrankRead
source length. Fig. 6 Critical shear stress (in units of ®b/L) as a
function of the FrankRead source length (in units of r0).
Fig. 4 Critical configurations of the dislocation bow-out for
different source lengths. The white broken lines were obtained by
the linetension model.
Dislocation Multiplication from the FrankRead Source in Atomic
Models 61
-
Because the Foreman model contains one dislocationsegment for
the bow-out motion, ¸FO is regarded as a result ofH¼ ¨. The
constants A and B for ¸FO are approximately 1and 0.66,
respectively; hence, we can simply considerthe relationship between
B and H. Figure 7 shows a plotof B vs. H¹1 for ¸MD and ¸FO. It is
interesting to notethat the value of B increases linearly with H¹1,
as shownby the broken line. This relationship is expressed by
theequation B = 2.6/H + 0.66; however, it is not easy
toanalytically understand. The critical shear stress ¸MD
fordislocation multiplication from the FrankRead source inatomic
models is described by a form of eq. (4), and thevalue of ¸MD when
H¼ ¨ is possibly expressed by ¸FOassuming that the influences of
the pinning points of thecylindrical holes and dislocations on the
dislocation bow-outmotion are similar.
3.5 Intra- and intergranular dislocation sourcesThe critical
shear stresses for dislocation nucleation from
the FrankRead source, ¸FR (= ¸MD), are compared withthose from
the h112i symmetrical tilt grain boundaries, aswell as the ideal
shear strength. The critical shear stresses ofthe intergranular
dislocation sources, ¸GB, are evaluated usingreported values of the
critical tensile or compressive stressesfor dislocation nucleation
from structural units in the tiltgrain boundaries.7) The ideal
shear strength, ¸IS, is estimatedusing the critical shear stress
for homogeneous nucleation ofdislocation loops in a perfect crystal
structure whose crystalorientation is the same as that used in the
FrankRead sourcesimulation. All the critical shear stresses are
evaluated usingthe (111)[110] shear stress component. The analysis
temper-ature is maintained at 10K for all the simulations, and
thestrain rates are _£ ¼ 5� 108 1=s and _¾ ¼ 1� 109 1=s for
thehomogeneous dislocation nucleation model and the grainboundary
model, respectively. The obtained ideal shearstrength, ¸IS,
evaluated by the homogeneous nucleation ofdislocation loops, is
3.65GPa.
We introduce a characteristic length of the dislocationsources S
to compare the critical shear stresses for variousdislocation
sources. S is determined by the source length L,regular intervals
of the structural units h,7) and Burgersvectors b for the FrankRead
source, the grain boundarydislocation source, and the homogeneous
dislocation nucle-ation source, respectively. It should be noted
that S has noconnection with grain size, except for the FrankRead
source.Figure 8 shows the relationship between the critical
shearstress and S for each dislocation source. ¸FR
continuouslyincreases with decreasing length S. The solid line
obtained bya continuum model in eq. (4) (with A = 1 and B =
1.64)indicates that ¸FR becomes larger than ¸IS if S is smallerthan
1 nm. However, we performed a FrankRead sourcesimulation using S =
3 nm and H = 4 nm in the atomic
Fig. 7 Constant B in eq. (4) as a function of H¹1. The symbols ,
,and © have the same meaning as in Fig. 6, and represents the
resultsobtained by Foreman14) when H¼¨.
Fig. 8 Relationship between critical shear stress and
characteristic length of the dislocation source. The symbols , ( )
andrepresent the critical shear stresses ¸FR (= ¸MD), ¸GB under
tensile loading (compressive loading), and ¸IS, respectively. Grain
boundarystructures, in descending order by regular intervals of
structural units S (= h), are 73, 77, 15 and 59 for tensile
loading, and 105,125, 31, 21 and 35 for compressive loading.
T. Shimokawa and S. Kitada62
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model; we observed that dislocation multiplication occurred(¸FR
µ 1.81.9GPa), but the critical configuration of thedislocation
bow-out was different from that shown in Fig. 4.This is due to the
interaction between the bowing-outdislocation and another
dislocation nucleation from thecylindrical hole, which is observed
when ¸ reaches 1.7GPa.Furthermore, in the case of S = 2 nm and H =
4 nm, an edgedislocation dipole terminated by cylindrical holes, as
shownin Fig. 1(c), cannot be constructed after relaxation because
ofthe short distance between the two surfaces of the
cylindricalholes (i.e., approximately 1 nm). Therefore, the
minimumlength of S is 4 nm for our developed FrankRead source,
andthe maximum ¸FR value is 1.36GPa.
In the case of the grain boundary dislocation source, ¸GB
increases upon decreasing the regular interval of
structuralunits S. Because edge dislocations are nucleated from
thestructural units, the elastic interactions between grainboundary
dislocations existing at the structural unit7) stronglyinfluence
¸GB. The minimum ¸GB value is approximately1GPa in the case of 105
under compressive loading; thisvalue is smaller than that of ¸FR
for S = 5 nm. However,almost all ¸GB values are larger than the
maximum ¸FR value,which suggests that the dominant dislocation
source is theFrankRead source if a crystal contains FrankRead
sourceswith a source length S larger than approximately 5
nm.However, it should be noted that there is a possibility
ofdecreasing the value of ¸GB, obtained by atomic simulationsusing
bicrystal models under nearly athermal condition,7)
if we consider the thermally activated process under areasonable
strain rate used in conventional experiments andstress
concentration at grain boundaries and triple junctions.
4. Conclusion
Dislocation multiplication from the FrankRead sourceis realized
by atomic models. The obtained critical shapeand stress for the
dislocation bow-out motion are comparedwith previously reported
results to evaluate them. Our mainfindings are summarized as
follows:
(1) Dislocation multiplication from the FrankRead sourceis
realized by atomic models by adopting cylindrical holes asa strong
pinning point to the dislocation-bowing segment andalso suppressing
cross slips, which have their starting pointat the surface of the
pinning holes.
(2) The critical configuration for the dislocation
bow-outobtained by atomic simulations exhibits an oval shape with
aratio 1 ¹ ¯ between the minor and major axis lengths. Theobtained
shape agrees well with the results obtained by theline tension
model.
(3) The critical shear stress for the dislocation
bow-out,determined by atomic models, is found to
continuouslyincrease with decreasing values of the FrankRead
sourcelength L. This can be expressed by the function L¹1 lnL,which
is obtained by a continuum model based on elasticitytheory, even at
the nanometer scale.
(4) The minimum FrankRead source length L realized inthe
developed atomic model is 4 nm and the maximumcritical shear
stress, which is one third of the ideal shearstrength, is larger
than that for dislocation nucleation fromh112i tilt grain
boundaries with long structural periodicities.
Acknowledgements
This research was supported by the Ministry of
Education,Culture, Sports, Science and Technology (MEXT)KAKENHI
Grant Number 22102007, the Japan Scienceand Technology Agency (JST)
under Collaborative ResearchBased on Industrial Demand
“Heterogeneous StructureControl: Towards Innovative Development of
MetallicStructural Materials”, and the Japan Society for
thePromotion of Science (JSPS) KAKENHI Grant Number24560091.
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Dislocation Multiplication from the FrankRead Source in Atomic
Models 63
http://dx.doi.org/10.1088/0370-1301/64/9/303http://dx.doi.org/10.1016/j.actamat.2009.05.017http://dx.doi.org/10.1016/j.actamat.2009.05.017http://dx.doi.org/10.1016/j.actamat.2006.08.060http://dx.doi.org/10.1016/j.actamat.2006.08.060http://dx.doi.org/10.1016/j.msea.2009.03.035http://dx.doi.org/10.1103/PhysRevB.82.174122http://dx.doi.org/10.1016/j.actamat.2007.07.052http://dx.doi.org/10.1016/j.actamat.2007.07.052http://dx.doi.org/10.1103/PhysRevB.75.144108http://dx.doi.org/10.1103/PhysRevB.75.144108http://dx.doi.org/10.1103/PhysRevB.83.214113http://dx.doi.org/10.1103/PhysRevB.83.214113http://dx.doi.org/10.2320/matertrans.MH201313http://dx.doi.org/10.2320/matertrans.MH201313http://dx.doi.org/10.1103/PhysRev.79.722http://dx.doi.org/10.1080/14786436708221645http://dx.doi.org/10.1080/14786436408224223http://dx.doi.org/10.1023/A:1008730711221http://dx.doi.org/10.1023/A:1008730711221http://dx.doi.org/10.1016/j.ijplas.2006.10.002http://dx.doi.org/10.1016/j.ijplas.2006.10.002http://dx.doi.org/10.1103/PhysRevLett.91.025503http://dx.doi.org/10.1103/PhysRevLett.91.025503http://dx.doi.org/10.1016/0001-6160(82)90188-2http://dx.doi.org/10.1103/PhysRevB.59.3393http://dx.doi.org/10.1103/PhysRevB.59.3393http://dx.doi.org/10.1103/PhysRevLett.45.1196http://dx.doi.org/10.1021/j100303a014http://dx.doi.org/10.1021/j100303a014http://dx.doi.org/10.1103/PhysRev.116.1113http://dx.doi.org/10.1103/PhysRevB.72.094105http://dx.doi.org/10.1016/0921-5093(93)90422-Bhttp://dx.doi.org/10.1016/0921-5093(93)90422-Bhttp://dx.doi.org/10.1080/14786430310001603364http://dx.doi.org/10.1080/14786430310001603364