,dcra metall.
Vol.
Printed in Great
33. No.
9. pp.
16014612,
1985
0001-6160/85
S3.00 +O.OO
Press Ltd
Copyright 0 1985 Pergamon
Britain. All righIs reserved
THE CONCEPTS OF LATENT HARDENING
STRAIN HARDENING
IN METALLIC
SINGLE CRYSTALS
AND
P. FRANCIOSI
Laboratoire
P.M.T.M.,
C.N.R.S.,
Universitt
Paris-Nerd,
France
Avenue
(Received 24 March
1985)
I. B. Clkment,
93430 Villetaneuse.
Abstract-Starting
from the rate independent theory of the single crystal plasticity (Schmid law), one here
tries to both give the most realistic description of the concept of intracrystalline latent hardening-or
hardening anisotropy-through
the analysis, at the microstructural scale of the dislocation densities, of
various and complementary experimental data, and to underline the excessive restriction introduced by
the classical assumption of pure strain hardening in the determination of the most general single crystal
hardening variation law for the wider range of loading processes. These developments about the
microstructural features of single crystal hardening, involving mobile and nonmobile dislocation density
evolutions (multiplications) and interactions, are correlated at the macroscopic scale of the slip systems
with a strain and load process and orientation hardening law which is shown on few examples to give
a better fit with observed behaviour than pure strain hardening. A third part shows that, even starting
from a rate dependent assumption for the single crystal plastic flow, the present analysis of latent
hardening remains basically unchanged. It shows too that the load process and orientation effect on
hard&&
which is implicitly present in such approaches although introduced according to specific
assumptions on the correlations between slips and dislocation densities. is one of the most significant
features in the single crystals hardening evolution during plastic deformation, whatever the theoretical
frame is chosen.
R&mnLPartant
de la th&rie de la plasticitt d’un monocristal indcpcndante de la vitesse (loi de S&mid),
nous essayons ici de dormer la description la plus &aliste possible du concept de durcissement latent
intracristaUin-+u anisotropic de durcissement-par
I’analyse, g l’&helle microstructurale des dens& de
dislocations, de r&It&s exp&imcntaux divers et complCmentaires, et de souligner la tr& grande
___.________~_C____r~~
~
de la loi de variation du ~&~&&~~&m
pour une t&s large gamme de
m&anismes de charge. Nous relions ces dcveloppemcnts concernant ks traits microstructuraux du
durcisscment monocristallin, m&ant en jcu des tvolutio~ (multiplication) de la densit6 de dislocations
mobiles et non mobiles, et kurs interactions, g l%chelle macroscopique dcs syst&mes de glissement, avec
une la d’&rovissage par k deformation. le chargement et l’orientation dont on montre sur quelques
exemplea qu’elle donne un meilleur accord avec le comportement observ6 qu’tm durcissement de
dtiormation pur. Dans une troisi&me partie, nous montrons qua, m&me en partant de l’hypoth&se d’un
Ccoukment plastique monocristallin d&pendant de la vitesse, n?tre analyse du durcissement latent reste
fondamentalement in&a&e. Nous montrons &akment que I’effet du m6canisme de mise en charge et
de l’orientation sur le du&semez& qui est implicitement pr&scnt dam de telles approches bien qu’il soit
introduit selon des hypoth6ses sp&ilIques sur les corr&lations entre 1e.sglissements et les den&a de
dislocations, est un des traits les plus cara&ristiques de l%volution du durcissemcnt des mono cristaux
au tours de la d&formation plastique, quelque soit le cadre thtorique choisi.
Baasmog-Ausgehend
von der Raten-unabhiingigen The&e der Einkristallplastixitt (Schmidtsches Gesetx) wird versucht, eine m6glichst realistische Reschmibung des Konzeptes der intrakristallmen
latenten Verfestigung-odcr der Vufcstigungsanisotropie-xu
geben, Hicrzu werden die Vcrsctzungsdichten
und die verschiedenen sich ergiinzenden Ergebnisse analysiert. AuBerdem wirddie auBergew6hnliche
Einsch&kung deutlich gcmacht, die mit der klassischen Annahme einer reinen Delmungsverfestigung
eingefuM wird, wenn das allgemeinste Gesetz fir die Vtinderung
da Einkristallverfestigung fir einen
weiten tkreich von Relastungsproxessen bestimmt werden soll. Die Entwickhmg der mikrostrukturellen
Eigenschaften der Einkristallverfestigung, welche die Entwicklung der beweglichen und nicht beweglichen
Versetzungsdichte (Multiplikation) und au& Wechsclwirkungen einschlieDt, werden auf dcr makroskopischen Ebene der Gleitsysteme mit einem Dehnungs-und Lastprozell und einem Verfestigungsgesetz
korreliert. Dieses Gesetz crlaubt-wit mit cinigen wenigen Reispielen gezeigt wirdcine bessere Anpassung
an das beobachtete Verhaiten wie bei der reinen Dehnungsmfcstigung.
Im dritten Teil wird gezeigt. dat3
diese Analyse im wesentlichen gkich bleibt, such wenn van einer Geschwindigkeitsabhgngigkeit des
EinkristallRieBens ausgegangen wird. Aulkrdem
wird gezeigt, dal3 der Lastproxell und der
OrientierungseinlIufa auf die Verfestigung, v&he impliit bei solche-n Niiherungen such bei der Einfihrung durch spezidle Annahmen iiber den Zusammenhang xwischen Gkitung und Versetxungsdichten
vorhanden ist, die herausragendsten Eigenschaften in der Entwickhmg der Einkristallverfestigung sind,
unabhingig vom theoretischen Ansatz.
1601
t6a2
FRANCIOSI: CONCEPTS OF LATENT AND STRAIN HARDENING
YNTRODUCTION
oaly involves the slip amplitudes together with a
The usual meaning
of single crystal latent
physical hardening moduli matrix fh 1 whi& is ashardening-in
the rate independent classical theory
sumed, according to the first image one keeps from
of crystals plasticity, which involves a generalized
latent *‘strain” hardening tests results, to be comSchmid law plastic flow criterion-is
the hardening
posed of a diagonal tem~ (for self hardening) and a
on slip systems which are not actually active during
non diagonal one (for al1 cross hardening?).
plastic flow. Although for a single active system
But in the most realistic assumption, already suggested by the hardening anisotropy variation with
(stage f of the uniaxial test stress-strain curve and
single slip in latent hardening measurements, each
part of stage 11) the hardening rate can be appreciated
hardening modulus representing the cross interaction
from the resolved shear stress-resolved shear strain
between two particular systems, mostly depends on
curve slope, when multislip occurs only a rough
average hardening rate can he estimated for the the history of these two systems, and is therefore
simu1taneously active systems. However in both cases expected to vary, according to the whole single crystal
no direct measurements of the hardening on the history, in a specific way.
When one investigates the mechanisms of single
latent systems are allowed except for lower bounds
crystal hardening at a microstructural
scale, the
without any great interest.
internal dislocation densities and the short range
The well known latent hardening tests (successive
different single slip activation in axially loaded single interactions dislocatiop strengths (intersection processes) appear as the main hardening parameters,
crystals) have been the first attempt at a methodical
together sometimes with the interactions between the
study of the inactive systems behavicrur and indeed
they have revealed a general anisotropy tendancy in dislocations and the lattice structure (interatomic
p&en&I, Peierls Force). Ccmsidcring now the disthe sense of a hi&er latent than self hardening
locations contact interactions as fundamental hardimportant at the very beginning of plastic strain but
ening processes, the assumed biunivocity between slip
remaining so far this plastic strain results in predominant single slip [1,2]. The problem lies with the rates and hardening rate (the critica) resolved shear
stress rate on each system) does not appear coninterpretation of these results: usually, authors intervenient anymore: while slip depends on the mobile
ested in single crystal behaviour consider the Iatent
hardening test results as a good description of the dislocations, created from the beginning of the p&tic
single crystal hardening law which is supposed to be flow and irrespective of whether they are still in the
crystal or have disappeared or rearranged (say by
well enough represented with a unique hardening
iinnihila’tioti
or recbmhination processes), hardening
ratio (latent to self hardening) taken as a constant
depends on the interactions between all the internal
and a material characteristic at a given temperature.
dis1ocations at a given instant, c&m m&k or not,
If this was true, in any testing situation the hardi.e. the dkbc&xts
which i3fe actually in the crystal
ening slope would always be lower in single slip than
structure.
in multislip and no case ofpreferred multislip in axial
Jf glide can be related to the mobile dislocation
testing would therefore be expected according to any
densities (with an Orowan type law), the non-mobile
minimum plastic work criterion.
The first addressed question here is concerned with dislocations loop$ densities, prevalent as obstacle
densities on inactive systems, are not expected to be
a law of hardening anisotropy that varies according
to the parameters desczibing the actual single crystrd so glide dependent as the mobile ones, but, since they
multiply by’source activation under stress, they have
plastic state.
to be, at least partially, dependent on the resolved
The selection of these parameters raises the second
shear stresses on the slip systems.
question which is concerned with the idea of strain
This dependency, that one expects from such
hardening: hardening is referred to as strain hardmicrostructural
analysis, could be interpreted at the
ening because of the widely admitted assumption that
phenomenotogicai scale as a double orientation and
all hardening parameters can he neghtcd compared
to the plastic strain, when non viscous materials arc loading process hardening effect to add to the sptcific
considered (metallic crystals, low temperature range, plastic strain effect.
This is the second point we shall be concerned with:
quasi static deformation). At the slip systems scale,
the discussion, still within the rate independent thethe most commonly used phenomenological hardory frame, of a hardening law for single crystals
ening law form
which depends upon both strain and load conditions
Tk+(y+ff=~h*f$f
co (including orientation) through the assumption of a
significant effect of the resolved shear stress rate
tWhart the coplanar latent systems are assumed squally distribution in addition to that of the slip rates. This
hardened than the primary one, say hJ’= P when g and seems reasonably realistic with regard to previous
I are copIanar systems, this assumption is referred to as
analyses of experimental investigations in axisyrn~
fomt hardening.
fMem the loops whiih have a mall free path kngth metric multislip tests on f,c.c. single crystals [3].
Now, the rate independent approach of the single
compared with those able to move large distacxes in
their slip plane.
crystals plastic behaviour still encounters a funda-
FRANCIOSI:CONCI~-R~ OFI L&TE~
APPD STRAIN HARDENING
1603
___-
Al
cu
.
0
1
2
3
AQ
Cilrl%Zn
4
ZI
5
Y %
Fig. 1. Experimental curves of the latent hardening ratios for Al, Cu. Cu 4x2~1 and Ag in tension.
mental difficulty in the research of solution uniqueness in strain at the ScaIe of the siip systems: in any
situation of load where the n number of the simultaneously potential systems-according
to the generalized Schmid law-is larger than the required one to
satisfy to the prescribed strain boundary conditions,
one has to decide of a selection criterion to ensure
uniqueness. While the question of what criterion has
to be chosen is still debated, attempts to eliminate the
unde~~ination
problem have been derived tbrough
the so called rate dependent approach of single
crystals plasticity which assumes, at the macroscopic
scale of the slip systems, that the singularity of the
single crystals yield surface is a feature of the non
realistic concept of critical r&solved shear stresses at
which systems pass from inactivity to activity: in
other words, the introduction of a (even very weak)
rate dependency in the single cry&& plastic flow
description, more realistic than a strictly rate independency assumption, suffices to round the yield surface and so to eliminate the undetermination problem.
In the third part, a comparison is therefore made
between the present developments according to the
rate independent
approach and the assumptions
either implicit or not involved, at the microstructural
scale of the dislocations
densities, in the rate
dependent theory.
From that comparison which underlines both the
similarities and some fundamental differences between the two formalisms, the conclusion summarizes
what are the most ensured characteristics of the single
crystals hardening anisotropy with regard to material
dependency and evolution with load.
LATENT HARDE~G
On Fig. 1 are plotted latent hardening ratios
(LHRs the ratio of the critical resolved shear stress on
tNot to be confused with the rate dependent hardening
model discussed in part 3.
a secondary (latent) system 7: to that on the primary
one 7,” for different f.c.c. single crystals, say Al, Cu,
Cwto/,Zn, Ag, at an equivalent temperature {T/T,)
of about 0.2: as already mentioned in the introductory remarks, the hardening anisotropy can be
important in single slip, with higher latent than self
hardening, but with several groups of latent systems
and also a large variation with the primary slip
amplitude. This has been discussed in detail in a
previous work [2]. Now, in multislip situations, analysis of the experimental results through the mte
independent theory has to be supported by a lot of
theoretical developments as shown in recent papers
[3,4]: for axisymmetric tensile or compressive teats,
the preferred active slip system combination (when
the strain rate is assigned) is the one minimizing the
incremental plastic work 6*(ip when isotropic hardening is assumed on the n s~ul~~~ly
potential
systems (i.e. an isotropic hardening moduli submatrix
I h 1” for the considered axis).
So far as this physical minimum work condition
applies (as analyzed in a recent thesis [4]), this
conclusion seems u priori contradictory with the
single slip hardening anisotropy in terms of classical
strain dependent harder&#,
but no more when
considering an orientation
dependent hardening
through the applied resolved shear stresses distribution(r’,...,
7 3. In single slip, the active system is the
most stressed but the less hardened, and the hardening anisotropy decreases whiie the applied resolved
shear stresses on the latent systems increase more
rapidly than on the primary.
In axisymmetric situations, a partially isotropic
hardening would be expected for the n equally
stressed systems, while for the 1”v-nlatent ones, the
hardening wouid be more important according to the
smaller resolved shear stresses value: this agrees very
well with the results of Washburn and Murty [S] from
latent hardening tests with a prestrain in tension
along a (111) axis. The measured hardening aniso-
FRANCIOSl: CONCEPTS OF LATENT AND STRAIN HARDENING
1604
tropy in such case is much smaller than after a single
glide prestrain, as will be discussed in the following.
Indeed, this hardening anisotropy never appears to
be very important in the numerical sense: with the
exception of the small strain region of the nonsymmetric axis orientations corresponding to the
orientations showing the larger single slip initial
stage, the hardening anisotropy in terms of the LHRs
remains somewhere between unity and 1S. However
the following shows that the consequences of the
large initial hardening anisotropy (between 2 and 4
near IO-’ strain according to the material) can be
fundamen~l at larger strains.
Let’s suppose, to simplify, a microstructural hardening law-valid whatever the formalism involves or
not a rate dependency-of
the form
/N
\ Ii2
(corresponding to a first order Taylor’s series deveIopment of the r:’ in terms of the p’) where 1u 1is the
d&cation
interaction coefiicient matrix and p’ the
internal dislocation density on f systems. In initial
single slip, as long as pp3pic po, one can write
S
f,
5=
asPprfp, + &us-
(
a SP rj2
awpP~po+ZgP-aw
(31
>
and the maxima of the LHR curves, Fig. 1. correspond to an approximate value of the (asp/a”)‘~
coefficient ratios giving the 1Q1matrix anisotropy for
each considered material. Such an anisotropy for the
interaction coefficients is in accord with calculations
of the short range interactions strengths for the
different possible dislocations pairs geometry [6,71
which are generally admitted as the prevalent ones
[S]. In such a microstructural approach, the ]a I
matrix appears material dependent at a given temperature. From the experimental results, four 4
coefficients are enough for f.c.c. crystals to represent
the main interactions types: a, >a2>al >o, respectively associated with dislocations interactions
leading to sessile junction formation, glissile juno
tions, no junctions, and with self hardening. Figure 2
reports the LHRs maxima vs the parameter E/pb
where E is the material stacking fault energy (from
Coulomb’s recent measurements [9]), fl the shear
modulus and b the Burgers vector: except in the very
small energy domain, the lower the stacking fault
energy (i.e. the wider the dislocation diss~ation)
the
higher the la f matrix anisotropy.
This is related to very small plastic strains and then
the hardening anisotropy decreases with the increase
of the ps. Now, Fig. 3, on the contrary, is a large
strain feature: it represents the (001) texture orientation proportion, compared with the (111) one, in
axially loaded f.c.c. polycrystals (from English and
tA, B. C, D respecttvely represent the (TI 1). (I I I), (TTI),
(ITI) slip planes and 1, 2, 3, 4, 5. 6 the lOI11, lOTI I,
lIOI[. }TOIl, ITIOj. Ii101 slip directions.
‘NI
-q
-
-.
--_
-___--•
1
-\__
--.
----.
I
CuAgCu
10
4Zn
Al
E/
@b
Fig. 2. Latent hardening ratios maxima vs the stacking fault
energy (E/~) for f.c.c. crystals.
Chin [IO]) as a function of Ejpb. The similarity of
Figs 2 and 3 cannot be purely accidental. One can
interpret this correlation in the following manner:
first consider the copper texture type domain, say
from silver to aluminium. According to the different
interaction types between dislocations, for strongly
non symmetrical load orientations the latent systems
are differently classikd from one material to another
according to their respective hardening, i.e. their
reape4Siveactual critical resolved shear stresses: since
their respective activation domains change, as soon as
a small additional stress (say neighbouring grains
effects in polycrystals) increases the applied resolved
shear stress, the most probably a&able latent systems, must differ for different materials. On Fig. 4
one shows in a f.c.c. reference triangle-for (110)
(111) systems--the secondary potentially activable
systems for different la I matrix anisotropy amplitudes illustrating the aluminium and copper cases.
The Schmid and Boas convention is here used to
denote the slip systems$ for a weakly anisotropic I a 1
matrix the most probably secondary system in tension is the so called conjugate system which, added to
the primary one, rotates the tensile axis towards a
( 111) orientation. When I u 1 is more anisotropic, an
orientation
domain appears (extending with increasing Ia I anisotropy) where the most probable
secondary systems, say A2 and A3 added to the
primary BQ, will rotate the tensile axis towards the
(001) orientation.
100
I.
A9
Fig. 3. Proportion of <MO> texture orientation for f.c.c.
polycrystals in tension vs the stacking fault energy (E&h)
(from [IO]).
FRANCIOSI:C~NCWTS
C)F LATENT AND STRAIN
WRDENING
1605
junctions formation [Is]: moreover as the stacking
fault energy decreases (from the silver value), the twin
wi& aIs decreases while plastic strain in muItip~e
twki (several simbltaneous twin systems activated)
occurs more and more easily [Is].
These observations would be in accord with the
assumption of a decreasing hardening anisotropy
between plastic strain mechanisms when the stacking
fault energy decreases from the silver value in the
brass type texture domain, as it decreases in the
copper domain when the stacking fault energy in-
t
3
UiRsMAX
t
3
creases.
Returning to the copper type texture domain, one
has to test the microstructural hardening law (2) in
ax~~rnet~c situations: if for n equally stressed
systems one assumes equal dislmtion densities (say
p’ = p Vk(1, nf when the applied resolved shear
stresses are equal (t’= T), one can write
while for the remaining latent N -n systems
IP
V,c(n + 1, N).
t
3
Resrduol
LHRs
t
3
(5)
Then, the maximum hardening anisotropy, given by
the ratio
Fig. 4. The most activable latent systems aam’ding
to the
load axis orientationand the materiaIl-mdming
anisotropy.
These initial differences of active system combinations at very small strains, even only from such a
rapid analysis, ~11 be expected to have an irreversible
consequence on the single crystal hkrge strain response, and the comparison between Ag. 3 and 2
strongly asserts this argument.
In the low stacldng fault energy domain, below
silver, corresponding to the brass type texture domain, the hardening anisotropy decrease would have
to be interpreted in terms of an increasingly independant behaviour of the partial dislocations, now separated by a wide stacking fault ribbon, in order to
remain consistent with the above microstructural
interpretation. If the partials now move separately,
the plastic strain mechanisms to consider are the
( 112) (111) systems-whether they glide or twinrather than the <1lO> {111) slip systems. Compared
with the literature, this approach is more consistent
with the brass type texture analysis based upon the
consideration of the (112) {11I) systems as possibte
plastic strain mechanisms [I 1,121, than with other
inte~re~tio~
((001) slip 1131cross slip 1141.. . ).
Indeed, the nucleation of twins has been experimentally related with dislocation dissociation and
tFor a (1 I I > axis prestrainand with the &j values from
Fig. 1,oneobtains~LlOforAland
=UOforCu,for
maximum hardening of the systems of the non prestrainedplane. The measuredvalue from Washburnand
Murty’swork for Cu is = 1.25 for (111) axis prestrain.
is much less important than in highly u~~e~~
orientations, i.e. than in single glide, in complete
accord with the results of Washburn and Murty [S].t
Since the hardening orientation dependency is expressed in terms of the applied resolved shear stress
distribution, the above analysis should still hold for
ones, at least
other Ioading processes, say multi~
so far as they are monotonic.
A confrontation with partial plane strain compression (channel die compression) and total plane
Fig. 5. S shape of the exponential law (19) vs r/r,.
1606
FRANCIOSI: CONCEPTS OF LATENT AND STRAIN HARDENING
strain compression tests on aluminium single crystals
from Driver et 01. 116, !7] has shown that the isotropic hardening assumption allowed good predictions of the observed behaviour. Since the initial
hardening was assumed isotropic, al! the experimental situations,
even not geometrically symmetrical, corresponded to symmetrical situations in
the sense that n systems were equally stressed (i.e. the
same applied resolved shear stresses) and the good
accord between observations and predictions for
isotropic hardening of the potential systems is a solid
support of the present loading process dependent
hardening assumption, at least within the rate independent theory frame.
As a conclusion for this first part, the latent
hardening concept for single crystals, should be more
understood as resulting in the microstructua! !a 1
matrix global anisotropy, which depends on the
material stacking fault energy, than in one (or even
more) phenomenologica! parameter(s) directly associated with the !h 1 hardening moduli matrix, which
varies continuously according to both the relative
rotations of the lattice and the loading frame and the
loading process.
The question which now justifies the second part of
the paper is how to reconcile the physical hardening
law (2) with a more appropriate one at the slip system
scale, in a form which is not strictly strain dependent
according to the previous developments on dislocation densities? We start from the apriori assumption that the hardening law depends upon both the
slip and the applied resolved shear stress, but still is
rate independent.
STRAIN AND LOAD PROCESS DEPENDENT
HARDENING LAW
The present aim is not to entirely define the set of
hardening parameters at the siip system scale but to
show how the double dependency with slips and
resolved shear stresses allows one to formulate a
phenomenologica! hardening law whose anisotropy
variation is consistent with the fundamental experimental results reported in part one. Looking for a
hardening law of the form
r! = r: (y ‘, X’)
(7)
where the X1 are restricted to the T’ applied resolved
shear stresses one can assume an incrementa! law
tlf mb snd nx are respectively the unit vectors along the slip
direction and along the normal slip plane of the g
system one has
Rj =
1/2(frqrf + mpp).
those sometimes referred to as dynamic or piled up dislocations densities-
SThese densities are approximately
partly allowed to decrease when the stresses are relaxed
or reversed-and
are enough to describe the main
hardening feature at small strain in single slip [a].
involving two instantaneous
ces !A 1 and tB/ so that
ff = f (#j’
Ia I
hardening moduli matri-
+ @i’).
(8)
Since the applied resolved shear stresses depend
partially upon the slip rates by the relations
where the latter part
(CafY>
I
of the third term represents the variations of the
geometrical Rt factorst while the nonprescribed
stress rate terms kaj also depend on thr: slip rates, one
can finally write [Sj in the form (see Appendix 2)
where the I h I moduli, which are still the strain
dehardening moduli, are strongly orientation
pendent, and where the SI terms only depend on the
prescribed c&. stress rate terms and vanish when they
are assigned to be zero as is the case in all the
experimental situations discussed here.
The important orientation
dependency of the
strain hardening modu!! must not hide rhe load
process dependency-through
the effect of the non
prescribed stress rate terms &--and particularly the
microstructural causes contained in the I A I and I B I
modu!! matrices, w!+richcan now !X interpreted from
the preliminary remarks on the dislocation densities.
Assume that on each system, the dislocations density can be separated into a mobile part and a
nonmobile one, say p’ = pi, + pj. A second order
Tayior’s series deveIopment of the square of each
critical resolved shear stress with respect to both the
p,,, and pf is
&f’
rp =ri+dp:P”
1 ; w
a=tg2
dp;P;+jjjr&Pkp!k
a2#
+2 --+ptlpj+
aP”,aP,
d2r”
--+p:p:.
8Pr aP/
(11)
The first order terms represent the direct hardening
effect of the internal dislocation densities on the
motion of the g mobile dislocations. Assuming a
weakly variable mobile dislocations flux during plastic flow (a steady state where annihilation compensates dislocation creation on a given system), the firstorder main effect would be the nonmobile dislocation
effect. This corresponds to the 1st order law given in
(2) where p’ = pil. The second order terms represent
1607
FRANCIOSI: CONCEPTS OF LATENT AND STRAIN HARDENING
the hardening effect of the interaction products
created in the crystal. Since no interactions (say by
contact) are allowed between nonmobile dislocation
densities, the only terms left represent the interactions
products from mobile-nonmobile
dislocations pairs
and the interactions products from mobile-mobile
dislocations pairs, a multislip characteristic. This
leads to a second order microstructural incremental
hardening law
where the creation probability factor Khk is assumed
zero for the nonmobil~nonmobile
interaction (say
pp;t-pj)
and where the il coefficient (,I > 0) represents
an eventually greater creation probability for the
mobile-mobile
dislocations
pairs than for the
mobile-nonmobile
ones.
One therefore obtains the following second rank
Taylor’s series equivalency
+ K”‘m
which still has to be compatible with the form (2). For
this reason, the dislocation densities in (2) have now
to be understood as the sum of both the loop
densities created from sources activated under the
applied resolved shear stresses and of the interaction
product (let us call them junctions) densities from
mobile-nonmobile
or mobil+mobile
dislocations
pairs of two different systems. The former terms can
be associated with the direct effect of the nonmobile
dislocations, and the latter to the indirect but prevalent effect of the mobile ones. (Notice that they can
also be interpreted as a representation of the cell
substructure built up by the inleracting dislocations.)
One can then write (2) in the more explicit form
7:
=
/lb
5
I-I
a”@:
+
pj)‘”
(13)
where the summation is extended to all the systems
on which interactions products or junctions can be
created (on (001) {lOO} systems for the Lomer
Cottrell sessile junctions
in f.c.c. crystals for
example).
Assume now that the junctions density created on
a given 1 system from an interacting (h, k) systems
pair is proportional to the interacting dislocation
densities product: summing up on all the (h, k) pairs,
one can write
when identifying
the coefficients
=(@)2~~u*‘P(l+~)
crystal, say which do not disappear from their slip
system by annihilation or interaction pnwxsses. One so
obtains dislocation density variation laws which are, as
(19), characterized by the S shape shown Fig. 5.
h
k
7
F
a27xl
c
aP:aP:
=
(pb)’
ad Kihk.
(16)
Assuming all the Kihk factors either equal to 0 when
the (/I, k) systems pair does not lead to any junction
formation, or to a unique K value for any (h, k) pair
leading to junction formation, this modified hardening law still only depends on the Ia I dislocation
interaction coefficient matrix (now including interaction coefficients between glissile dislocations and
sessile junctions, say for f.c.c. a fifth coefficient associated with all the Lomer Cottrell sessile locks when
assumed all geometrically equivalent).
These coefficients can be estimated through the
usual latent hardening tests.
Such a microstructural hardening law requires a
knowledge of the laws governing the mobile and nonmobile dislocation densities.
Instead of the usual Orowan law
i’ = p;w
(17)
where V is the dislocations velocity, assuming either
a steady state or a jerky dislocations motion, one can
use the simple linear relation
&,=A?’
TAssuming this probability equivalent with the proportion
of dislocations sources able to emit freely an identical
number of loops at a given T/Z~ratio, it gives approximately the ratio p/p,,
where p_
is a saturation
density on each system, roughly equal to the maximum
of emitted loops which remain “as grown” on the
(15)
)1
where
A=&
(18)
between the mobile dislocation density variation and
the slip rate on a given system, where b is the Burgers
vector and L the average dislocations free path.
The nonmobile dislocation densities are prevalent
in the inactive systems. They seem to mainly increase
before the macroscopic yield shear stress is reached
(so far rg < 79 and to saturate when the considered
system becomes macroscopically active.
Therefore any variation law which depends on the
rg/rf
ratio so as to represent an initial rapid density
increase with this ratio and saturation when rK/7f = 1
seems relevant enough with the available data.
Derived as explained in notet from Kocks statistical analysis of the dislocations motion through pene-
FRANCIOSI: CONCEWS OF LATENT AND STRAfN HARDENlNG
Fig. 6, Siulsted stress-straincnrves for f.c.c. crystals(in
tension) from the present anaiysis.
trable obstacies 1191au exponential law on the form
has been sucoessfSIy used in computed simulation of
single crystals behaviour in tension, on the here
developed bases. The simulation principle is briefly
reported in Appendix 2.
Figure 6 reports the simulated stress strain curves
for various teusile axis orientations. It shows that
with only a small number of parameters, fixed for a
given material at a giveu temperature, such a microstructural hardeniug faw (which, at the phenomeu~
logical scale of the slip systems, can be expressed as
a function of both slip rates and applied resolved
shear stresses rates), represents reasonably well most
of the main features of single crystal hardening within
the rate indepeudent tbcoretical frame. The “three
stages like” O(C)curves for some o~~n~tio~ in Fig,
6, as the parabolic ones for high symmetry axes are
only a result of the assumed orientation and therefore
rotation dependency of the crystal hardening anisotropy through [lo] as developed in Appendix 1. For
the former ones (noninitially parabolic) the strain
mode remains single slip for all the reported domain
what supports the opinion that stage I and stage If
are not ffin~en~y
different and have not to be
analyzed separately.
The experimental tensib curves for the same orientations as in Fig. 6 are plotted in Fig. 7 for copper
single crystals at room temperature. *Theseexperimental curves are issued from a previous study of
f.c.c. single crystals reported in [3].
Various other computed simulation results, concerned with hardening anisotropy variation, dislocation density vacations, and rotations, have been
reported in detail in f4]_They will be submitted for
put&cation in a specific paper.
The fohowing part now compares the present
discussion with the “macroscopic” rate dependent
approach of the singfe crystals plastic behaviour-as
Rxperimentzd
stmss+&& curves for Eopper single
cqstais in tension.
presented in [18J--in order to check the main similarities and differences at the ~~s~~t~al
scale of
the dislocations densities.
While with the Schmid law assumption a selective
criterion is required in singular situations (vertices of
the siuguiar yield surface) to determine the slip rates
~s~bution, in the rate dependent approach the slip
rates are given, for any g system, by
(201
from an Orowan type law on the usual form (17) and
with the dislocations velocity
This relations set undermeans that all the systems of
the structure are active as soon as the crystal is
loaded-as in (IO) with the effect of $-but since the
m exponent is small compared with unity there are
only signi&.ant amounts of slip on the systems where
rr is cfose enough to r$. The ?a parameter represents
a maximum slip rates assumed to be equal on geometricalfy equivalent systems.
The rate dependency ~s~ption (20) rounds the
single crystats yield surface at the vertices and is, for
that reason, an appreciate simplifying for polycrystals
behaviour modelling [20f.
FRANCIOSI: CONC!~
-r---G-
OF L&I-E!& AND STRAtlrf HARDENING
This still satisfactorily describes the Ih I matrix as
material characteristic through the 1a 1 matrix of the
&tocation
contact interaction coefficients (II”’ decreases as r$ increases and, although Iu 1 is symmetrical, one has for nonsymmetrical
situations,
hc’ E ax’/‘lrfdifferent from /z” E nk/z:).
Since in the classical strain hardening law (1) the
shear strain rates are no more unknown parameters,
the problem to solve is purely elastic, i.e. one only has
to solve a set of equations on the form
---\
Fig. 8. Dislocations storage mechanisms: from elastic interactions (A) dy* x p’-+dp” I # h; from contact interactions,
jogs, kinks (B) or junctions Q creation, dyh x ph+dp’
I#h#k.
This macroscopic
dependency
(20), microscopically assumes a dislocations storage on each
system, which is only related with the shear strain on
this system, what impliciteiy considers elastic interactions between mobile dislocations and obstacles to
be predominant, in opposition with one of the starting assumptions of this paper based upon contact
interactions predominancy (Fig. 8). As a matter of
fact, the calculated matrix of elastic Mcients
for
f.c.c crystals 1211 is basicially contrary to the la I
matrix here involved and concludes to a predominant
self hardening what conflicts with latent hardening
results.
If one stiil assumes the ~~0~~
of contact
interactions between dislocations, one can interpret
the relations (20) following the present ana@%: from
part two, (20) assumes all the dislocations densities
involved in the hardening processes to be (more or
less) mobile and therefore mainly shear strain dependent.
Relating (20) with the incremental strain hardening
law (1) it comes for each g system
1609
$7 = CF;+ ,ry,j&
(24)
where 1A 1 is the elastic moduli matrix), in which the
plastic rates are always fully prescribed.
Indeed, such an assumption of prescribed slip rates
can be made in any description starting from the
microstructural
law (2) where, on one hand, the
dislocation density variations are expressed in terms
of macroscopic flow parameters, while on the other
hand, the hardening law is still assumed to be written
on the classical form (I), instead of using the Schmid
law conditions. So it is, for instance for the present
developements which separately consider the mobile
and the non-mobile dislocations, and if the hardening
law takes the form given in (8): a simple variation law
for the non-mobile dislocations in accord with the
shape shown Fig. 5 is
zl I/m
I
with
P =Anax -/
0 TC
mc.1.
(25)
Assuming, to simplify, that the non mobile dislocations are predominant in (Q--i.e. neglecting the
terms I B I 11;I what is correct at small strain-the
A#’
moduli are very close to the expressions
(26)
which have evident similarities with the above developments on the rate dependency assumption and
suggests the possible introduction of another rate
dependency on the form
(27)
where the Ih 1N macroscopic hardening matrix still
has to he physically interpreted, but now remains the
only unknown part of the hardening law. Then, a
comparison with the incremental form of the microstructural hardening law (2)
with which the slip rates would also be completely
prescribed, using the P complementary equations
i$ = M+q&,
+ i Qhfh
h-l
immdiately shows that with _3= Cb (what assumes
@ = @,,,and therefore all the dislocations in (22) to be
mobile) one obtains
(28)
to express the P unknown 6$ in the ir involved in
(27).
Even more complicated, this would remain formaily similar for different density variation laws such
as, for example (13, 18, 19) instead of the simple law
(22, 23) or even (27).
So, whatever is, at the ~~~~ctu~i
level, the
description of the evolution of the dislocation densi-
1610
FRANCIOSI: CONCEPTS OF LATENT AND STRAIN HARDENJNG
Fig. 9. Deviationfrom the classicalSchmidlaw of equalCRSSon any equivalentsystemof an as grown
~ornogcn~u~~crystal: experimentalvslucs from (14 (dots) and caIcuiatedvalues from the present
analysis(surfact).
ties, the addition of a rate dependency assumption
allows to eliminate the und~~~ation
problem
associated with the inequations system of the generalized S&mid faw plastic Bow criterion: this assump
tion 5ubstitutes to this inequation set, a system of 12
strict equakties giving a single admissible slip rates
~s~bution, and therefore rounding the single cry&t1
yield surface.
Now, while in the rate independent approach
di&rent assumptiona were tested for the hardening
matrix form to fit with experimental data, in the rate
dependent theory, different rate dependency assumptions can be tested: the so called rate dependent
theory defined in (20) is one of the simplest forms one
can think about. Now, a lot of other ones-such as
the possibility expressed in {27)-m& be relevant
too, and perhaps more relevant, so far they satisfactoriiy describe the microstructure evolution with
strain and loading.
As an example, the analysis of the latent hardening
tests results reported in part one in terms of dislocation muItipli~tio~s on the different systems are
shown in Appendix 3 to remain q~i~tively unchanged under the rate dependent assumption 120)or
even 127):we only have here to say “predominant”
single slip since the concept of inactivated systems is
no more m~~n~ull.
Now, the main remaining dif%rence between the
macroscopic rate independent @&mid law) and rate
dependent (20) approaches is the folIowing in the
former approach, since a sehxtive criterion has to be
added to the Schmid law flow conditions to solve
undet~~~n~ situations, &heset of ~~omin~t~y
active systems is not necessarily equivalent with the
set of the predominantly stressed ones. On the contrary, so it is in the latter approach. Compared with
experimentaf data on symmetrically loaded singte
crystais 131,the rate independent approach as developed in part 2 gives better correlations with observed
behaviour. Even if this batter correlation is due to the
~~~~
~~~~~t~)
to real&e ideal experimenti
situations of toad, this would mean that such a rate
independent approach better describes reality with
~avoidable deviations from the ideal situation.
Whatever this diffe~n~ is academic4 or not when
one is concerned with realistic loadings (never perf&y sketch)
of a poIycrystal g&r for example, the fundarnentai results of the present work also
apply for the rate dependent approaches.
These results are summarized now in the conclusion,
CONTUSION
This work first underlines the important fact that
even. not high numerically, except at the onset of
plastic strain or at small strain in rate dependent
theories, the hardening anisotropy of the single crystai, due to the anisotropic variation of the total
dislocation densities, can be expected to have au
irreversible consequence on its larger strain behaviour, and then on the large strain behaviour of
polycrystais, the more, the lower is the material
stacking fautt energy.
This con&&on suggested from the comparison
with textures anisotropy on po~y~s~~s loaded in
tension, of single crystals latent hardening test results,
would have to be confirmed by spec%c experiments,
that we actually prepare.
The present developments at the microstructural
seafe of the dislocations densities show that, in the
rate independent approach, a restriction of single
crystals hardening to strict plastic strain hardening is
excessive and difficult to conciliate with the simplest
FRANCIOSI:
CONCEPTS OF LATENT AND STRAIN HARDENING
microstructural
analysis.
Conservely,
the introduction of a toading process and orientation de-
pendent hardening through the applied resolved
shear stress rates is able enough to represent the main
observed features of single crystals plastic behaviour,
the best when distinction is made at the dislocations
densities level, between mobile ones, mainly related
with slip and therefore mainly responsible for the
strain hardening, and the nonmobile ones that participate to the hardening without participating (in a
noticeable manner) in the plastic straining.
The present description of specific hardening
effects of mobile dislocations leads to a second order
hardening law which appears more accurate than the
‘“classical ones” in multislip situations.
Incidentally, the load dependent hardening implies,
in the rate independent formalism, that one never can
really experimentally reach the yield surface of an
unloaded single crystal: as soon as a nonzero load (or
strain) is prescribed to the crystal, the load process
specific hardening increases the critical resolved shear
stresses of the different systems, and differently according to the load orientation. When the applied
stress state reaches the yield surface, it has already
moved away from its initial form and position.
Compared with the classical Schmid law of an identical (o~en~tion independent) initial critical resolved
shear stress on equivatent systems for an homogenous crystal, the present assumption gives a better
agreement with the measured initial resolved shear
stresses with regard to the orientation [22] as shown
Fig. 9.
Such a “load sensible” critical yield surface, initially vanishing in front of the surface that one can
associate with the current applied stress state reduces
the conceptual difference about yield that exists with
the rate dependent approach where the usual yield
surface only appears as an asymptotic limit of the
current flow stress and always remains beyond the
current applied stress state associated surface.
More generally, the main conclusions of this paper
(dislocations hardening mechanisms and load effects)
appear also valid for the usual rate dependent theory
when it is assumed at the macroscopical ievel that one
considers the diskrcations present in the crystal to
be-more
or less mobile--rather
than a pr&ominancy of elastic interactions
between them.
The here developed notion of additional load
hardening to represent some of the microstructural
hardening mechanisms must have consequences on
unloading, changes in loading paths, reverse loading
and so forth. But as in nonmonoto~c
loading the
distocation density variation laws are expected to be
more complex, it would be unjustified to here underline other apparent implications of this “strain and
stress” hardening law without further analysis.
REFERENCES
I. P. J. Jackson and J. S. Basinski, Can. J. Pbys. V4g
(1967).
1611
2. P. Franciosi, M. Bcrveiller and A. Zaoui, ~cm meroll.
zq p 273 (1980).
3. p. Frahciosi &nd A. .&oiti, Acla mefull. 30, p 1627,2141
(t982).
4. P. Franc&i,
Thesis, Paris (1984).
5. J. Washburn and J. Murty, Can. J. Whys. V45, p 523
(1967).
6. J. D. Baird and B. Gale. Nerd. Plrys. Lub. V257 (1964).
7. G. Shoeck and R. Frydman, Plysica status solidi (b) 53,
661 (1972).
8. G. Saada, Acra mernll. V8, p 841 (1960).
9. P. Coulomb, Scripru meruli. 15, p 769 (1981).
10. A. T. English and G. Y. Chin, Acta rnerull. 13, p 1013
(1965).
11. S. R. Goodman and H. Hu, Trans. met&. Sot.
A.I.M.E. 242, p 88 (1968).
12. 1. L. Dillamcre, E. Butler and D. Green, Metul Sci. J.
2, p 161 (1968).
13. F. Haessner, 2. Metall. K 54, p 98 (1963).
14. R. E. Smallman and D. Green Actn met&. 12, p 145
(1964).
15. J. Vergnol, Thesis 3 12, Paris (1980).
16. J. H. Driver, H. Skalli and M. Winterberger, M.E.S.
Reo. Metall. pp. 241, 295 (1983).
17. A. Skalli, Thesis, Grenobfe (1984).
18. J. W. Hutchinson, Proc. R. Sot. Lond 348, p lOl(1976).
19. U. F. Kocks, Phil. Mug. 13, p 541 (1966).
20. U. F. Kocks and G. R. Canova, ICOTOM 7,
Amsterdam (1984).
21. J. Zarka, J. Mech. Phys. Solia’s 20, 179 (1972).
22. 3. Jaoul and 1. Bricot, Reu. Metall SZ, p 629 (1955).
APPENDIX I
Expression of the strain hardening m&Ii kfl from the strain
and load process dependent hardening Iaw
Introducing (9) into (8) one obtains
(Al.l)
+ i qxj’j$’
i-1
>
Then, from p of the m strict equalities +s = if which have
to be satisfied by the m active systems according to the
generalized S&mid law plastic flow criterion, one expresses
the unknown stress rates in terms of the slip rates, by
inversion of the system
(A1.2)
This allows to express the applied resolved shear stress rates
in the form
which tinally gives the hardening law expression type
FRANCIOSI: CONCEPTS OF LATENT AND STRAfN HARDENING
1612
with, for the strain hardening moduli
A-!
All the parameters involved in this simulation are, in
addition to T,,, 10 1, p. A and b already encountered:
k-l
\
I
Cc
ai,
B’%$-IX;+
A"
+A”
(AIS)
v-1
I)
and for the “loading process specific hardening”
h-1
J/
The load process hardening term here only depends on
the assigned stress rate tensor terms, both because of the
assumption of a nem
elastic &rain and because the p
assigned @la&c) strain rate conditions would appear if
introduced with the hardening law strain dependent part.
If the elastic strain is not neglected, the load proass
hardening terms depend on both the assigned stress rates
and the complementary assigned total strain rate terms.
APPENDIX 2
Simulation priwip~ef0rjk.c single crystalrplarlic behavknu
inaxirrlload
In addition to the 12 (11 I){ I IO> easy glide systems and
to the [a 1 matrix of dislocations interactions coe%&nts
whose anisotropy amplitude is related to the stacking fault
energy folowing Fig. 2, one represents the initial Frank
network of the crystal by a weak anisotropic r, CRSS value,
at which the dislocations multiplication is initiated (activation of the largest sources.)
Before the macroscopic yield point, in a “preplastic”
domain, the rotations are neglected (the loading frame keeps
an invariant crytallographic orientation) and tbe CRSS
increase according to (19) and (2): the exponential laws for
each p(f) make the CRSS to increase slower than the
applied resolved shear stresses z(i), so that the S&mid law
plastic flow condition z(l) = r,(Z) can be realized, after few
elastic step% on one or several systems. Since no microslip
is associated with the first dislocations multiplications, the
macroscopic yield point corresponds. for initial single slip,
to the maximum hardening anisotropy of Fig. I giving the
/a 1 matrix one.
Initial multislip happens for symmetrical initial loading
orientations. The present hardening law gives an isotropic
hardening for equally stressed systems and when several
combinations of slip are admissible, simulation fits the
experimentally observed behaviours when selecting tbe active mode as the one which minimizes the quantity Au&~,.
In the plastic domain, one calculates for each step, the
Ih 1 matrix terms following (13) to (16), appendix 1 and
using expressions (18) and (19).
The Rt variations (onIy Rf, are here required) giving the
Ict,i (here }++I) matrices in (9) and (1.1) are cakualted
assuming a zero totat rotation so that the iat& rotation
(w’) is opposite to the p&tic one (co”). [Since the prescribed
stresses are here prescribed to be zero, $ is zero in (IO).]
To calculate the 1h 1 matrix terms, all the possibilities of
junctions creation from (h,k) pairs are stored in a IJ1
matrix so that
J(h,k)=Oor/
1 being the system where junctions from the (h, k) pair are
created (I is not necessarily an easy glide system, for
example, the Lomer Cottrel
locks are created on
~i~]<I~)
systems).
With (IS) and (19). the h(g, I) expression for each aL&
step is given in (1.5).
-the elastic moduli matrix }&I relating Au and At’
-p_,
in (19) the maximum of loops stored on each
system before to macroscopic activation
-/3, in (19) too, the distribution of the loops sources
lengths near the average value
-&in (14), (15), (16). the fiction ofjunctions created by
two populations of dislocations h and k. mis fraction
is assumed identical for all (h.k) pairs.]
p_, /I and k are adjustable parameters. Convenient order
values are respectively
‘), Sand I/3.
All necesaq ~fo~a~o~
‘& included in part two,
appendix I and appendix 2. For more details, the reader will
report to 141.
APPENDIX
3
Ana&se$ of the latenthardeningtestsresults,throughthe rate
drpnrrknt tlvwy
(a) From the “classical” rate dependent assumption given
in (20) and with the 1h 1 expression from (23) the resolved
&ear stress-resolved shear strain curves correspond with
the expressions
IP
with
Ifc’= @f~)*o” = cte. (2.1)
The latent hardening ratio curves are therefore defined by
So far the yL are negligible (7%<24),
the LHRS increase
towards (HSp/Hp~‘~ as fifs dt increases.
As for the discussion in part one, the LHRS maxima
correspond with an approximate value of the ratio
(asr/ap?‘fl.
Now, the LHRS decrease is still due to the rotation
associated with strain, say the relative variations of the r*,
w&h increases [in (2) and (3)) the secondary dislocations
densities.
while this increase of dislocations densities on the latent
systems is not associated with significant slip in the rate
independent assumption, here the assumed biunivocity between dislocations and slip, directly relates the LHRS decrease to the secondary slip increase.
(b) From the here derived rate dependent assumption as
given in (17). the discussion remains identical. Since one can
write (27) on the form
for a tensile test, the variation on the slip rates is more
sensible to the rotation (if ci/a is identified to ito# cte, the
usual rate dependent assumption (20) would correspond
here to ill=: 0). These different assumptions lead to different
behaviour descriptions more or less realistic: for example, if
the load is reversed, in the usual (a) case the slip rates
decrease to zero remaining positive until the complete
unload of the crystal. On the contrary, in the (b) case the
stip rates still decrease to zero but after becoming negative
when the load sign is reversed, which seems more physically
correct. However this discussion is out of the present paper.