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Model for instabilities during plastic deformation at constant cross-head velocity
J. Grilhé, N. Junqua, F. Tranchant, J. Vergnol
To cite this version:
J. Grilhé, N. Junqua, F. Tranchant, J. Vergnol. Model for instabilities during plastic de- formation at constant cross-head velocity. Journal de Physique, 1984, 45 (5), pp.939-943.
�10.1051/jphys:01984004505093900�. �jpa-00209827�
Model for instabilities during plastic deformation at constant cross-head velocity
J. Grilhé, N. Junqua, F. Tranchant and J. Vergnol
Laboratoire de Métallurgie Physique, Faculté des Sciences, 40, avenue du Recteur Pineau, 86022 Poitiers, France
(Reçu le 13 juillet 1983, révisé le ler décembre, accepte le 19 janvier 1984 )
Résumé. 2014 On attribue aux dislocations mobiles une vitesse finie; par conséquent, un déphasage a lieu entre le temps où les boucles sont créées et le temps où la déformation est réalisée. Cette hypothèse appliquée à la relation de Mecking et Lücke permet d’expliquer l’apparition d’instabilités sur les courbes effort-deformation. Le rôle de la dureté de la machine, les processus de formation des boucles, le déphasage apparaissent dans le critère de stabilité.
Abstract. 2014 We consider the case of mobile dislocations; with a finite velocity there is then a phase displacement
between the time of loop nucleation and the time at which the strain is recorded With this hypothesis applied
in an equation analogous to the Mecking and Lücke relation, we can explain the appearance of instabilities on the strain-stress curves, the influence of stiffness of the loading system, the formation processes of dislocation loops,
and the phase displacement which appear in the criterion of stability are studied.
Classification
Physics Abstracts
46.30J - 62.20F
1. Introduction
Instabilities on the stress-strain curves of metals and
alloys (Portevin-Le Chatelier effect, twinning, thermo-
mechanical effect) are frequently observed and have been widely studied (see for example Johnston [1]
concerning the yield drops on the stress-strain curves
of LiF). Recent theoretical progress dealing with
instabilities in physical systems (catastrophe theory,
bifurcation theory, non-linear phenomena) lead howe-
ver to a new approach and to a better understanding
of these questions, Kocks [2].
For the Portevin-Le Chatelier effect, Anantha-
krisna and Sahoo [3] postulate a non-differential system for the rate of change in the densities of three types of dislocations : respectively mobile, pinned,
and those interacting with the Cottrell atmosphere.
These systems of equations are similar to the Orego-
nator model for the oscillatory chemical reactions of Zhabotinski [4] and yield a good description of the
strain jumps during creep deformation.
Kubin, Estrin and Spiesser [5] start from a set of
two non-linear differential equations coupling the temperature and the flow stress, to explain the low
temperature thermomechanical instability of metals.
Using bifurcation theory, they point out that periodic
solutions can be obtained in some areas of the phase
space.
Using a rheological model similar with that of Simon, Caisso, Guillot and Violan [6], Penning [7],
Demirski and Komnik [8] describe the influence of the stiffness of the loading system on the initiation of instabilities on the stress-strain curves.
In the following treatment, we use the Mecking and
Lucke [9] relation i = b pL but we introduce a phase displacement between the emission of a dislocation (or of a twin) and the resulting deformation. Then we
show that, close to the stable solution, periodic solu-
tions can appear. A bifurcation point is characterized and analysed, and a stability criterion is defined in a
simple way.
2. Equation for the change in stress.
2. 1 MECKING AND LucKE RELATION. -- The imposed
strain rate s is the sum of the plastic strain rate ip of
the specimen and of the elastic strain rate i, of the
combined sample and loading system (with a stiffness M) :
The plastic strain rate may be written :
where V is the volume of the sample (the cross-section
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004505093900
940
of which is supposed to remain constant), b the com- ponent of Burgers vector along the tensile axis, the dislocation existing over the area f(t). This relation
can be further detailed in various way, depending on
the dislocation mechanism which is operative in the crystal.
When the dislocation density p(t) and mean velo- city v(t) vary slowly and in a monotonous way, equa- tion (1) becomes :
Such a form applies when plastic deformation is
governed by Peierls’ forces.
The deformation may also be controlled by the
emission of dislocation loops (from Frank-Read sources) or by emission of a collective set of loops (a twin for example).
If n(t) is the number of loops arising at t in the unit volume and during unit time, relation (1) becomes :
We call S the mean area swept by the loops and we
suppose that S remains constant during times which
are long enough compared with the period of instabi- lities. In fact, S depends on the instantaneous density
of the forest and thus on the previous strain history
of the sample. So we consider that S varies slowly.
If L is the mean free-path of a dislocation we can
define the rate of creation of dislocation as :
Thus, we get the relation given by Mecking and Lucke :
which illustrates the influence of a fast variation in dislocation density occurring in some types of deforma- tion.
To establish relations (3) and (4) we have assumed that the area S is instantaneously swept by each
dislocation as soon as it is emitted. Now, leaving aside
this last hypothesis, we show below that with finite values of dislocation velocities a time phase shift
takes place between the emission of a loop and the crystal deformation consequently induced.
This time phase shift can be related with instabilities
on the stress-strain curves.
2.2 DETERMINATION OF THE EVOLUTION EQUATION. - With a finite dislocation velocity, the area swept by a loop nucleated at a time t = 0 is a function S(t) which depends on the mechanism considered (slip, twinn- ing...) and on the state of the crystal, through the density of obstacles which can hinder or block dislo- cation motion. After the flight-time T’, the mobile dislocation gets pinned or reaches the free surface of the sample having covered a constant area S(i’) = S
since it was emitted. Then only loops generated at a
time (t - t’), with 0 t’ T’, will contribute to the deformation at a time t and we can write from equa- tion (3) :
The number of loops nucleated at a time t is a function
of time through the applied stress. Thus, the strain rate is given by the general relation :
We have not considered the influence of temperature, germination sites, on the number of nucleated loops.
In a tensile test with constant strain rate 8, a stationary
solution of equation (5) is a = constant = co and then :
Given the strain-rate, n( Qo) and co can be considered
as remaining constant only during periods shorter
than the duration of the tensile test This observation enables relations (5) and (6) to be used as the starting point for discussing the stability of the solution 0" o.
Equation (5) becomes simpler if we assume that :
where S is a constant equal to S(r’), b(t - 1:) is a Dirac
distribution and 1: is the first moment of the S(t)
curve :
This approximation for §(t) amounts to replacing S(t) (Fig. 1), by a step function (Heaviside function).
Thus, appears as the phase displacement between
the time of loop nucleation and the time at which the mean strain is recorded This means that the loops
sweep up an area 5’(r’) at an infinite velocity, but with
a phase displacement with respect to their nucleation time. The equation for evolution of the stress is now :
and the previous solution (6) is always valid.
Fig. 1. - Approximation for S(t) and S(t).
3. Stability of the steady-state solution t1 = t1 O.
This stability is investigated by a linearization of the function n(O’) close to value ao :
with b 6 t = Q t - Q . a =
with ðO"(t)
()
= a(t) - 0"0’ Then, with aOa),ffl
=MbS ::)
0’1equation (8) becomes :
combining with equation (6) we write :
or, with the above expression of 6 a(t) :
The general solution is a linear combination of solu- tions such as :
where À,i and coi must obey the following relations :
Or :
Depending on values of coi (solutions of (I I a)), Ai is
either positive or negative and 6 a is or is not damped
The solution a = constant is stable when all the A,
are negative. The solutions of equation (Ilb) are graphically given by the intersection of the curves
y, (x) and Y2(X) :
In figure 2, we have also represented Y3(X) which gives the sign and values of A, :
The following conclusions are deduced from figure 2 :
- when the slope of y 1 is larger than 2/n, i.e. when
Ta 7r/Z all the values of (JJ verifying (lib) correspond
to negative values of Â.. The perturbation 6 a is damped
and the solution a is stable.
- when n/2 Ta 5 n/2, only one solution OJ yields A > 0. All other fluctuations are damped and
then the first instability takes place. Every time the
increase of Ta is 2 ’It, an additional solution appears,
corresponding with a positive value Â.i. The number k
of solutions coi corresponding with one instability
Fig. 2. - Graphic resolution of equations (lla) and (I I b).
.0 "B ’A J- -. -.
is given by :
A succession of instabilities with a time periodic
character appears (Hopf type bifurcation (10)) and the perturbation 6 a is the sum of oscillations with increas-
ing amplitude.
We point out that with a linearized function for
n( (f), one determines only the criterion for instabilities and not the periodic solution for the evolution of stress.
Discussion.
The first instability arises as soon as Ta becomes
larger than n/2, i.e. from equation (9) when :
We notice that this condition is most easily fulfilled by hard-loading systems. Moreover, the influence of strain-rate i = bSn(u) depends on the curve n(c),
as can be seen from figure 3.
Fig. 3. - Two possible evolutions of n( (J) vs. r : a) Above
a critical value (J c’
an
is constant for any value of ij bS.(J ao
b) With thermal activation,
an
increases with (Jo and thus with ëjbS.J)
942
For example, if n(a) is a linear function (Fig. 3a)
an
does not depend on the strain rate. Every valueTa- y
i ao
Qo depending on the strain rate ëi is characterized
by the same behaviour relative to the stability. In the
second example p (Fig. 3b
an
increases with ë. TheouQo
inequality (12), governing the appearance of the insta-
bilities, can be verified for the highest values of i or of
the steady solution o-o.
4. Solutions when Ft(o’) is a discontinuous function.
Such a situation arises with some mechanisms (emis-
sion of twins, unlocking of dislocations) which need a
critical stress ac for operation (Fig. 4). For the values
of EjbS between nl and n2, the solution a = cc is unstable (the stability may be examined for infinite values of an/a 6 in the previous calculations). Owing to
the experimental evidence of a critical stress associated with deformation instabilities (twinning, P.L.C. effect)
we have developed the unstable solution for a function
n(a) corresponding to figure 5. We assume that, when
the stress level reaches a critical value ar ,, loops are
nucleated at a time ti, with a constant rate n(t) = no, The relation (8) can be written :
Fig. 4. - Discontinuous function n(a).
Fig. 5. - Simple model of the discontinuous function n( u).
with :
Up to t = T + tj, we have n[u(t - T)l = 0; from
relation (13), we obtain :
From t = T + t1 onwards, u(t - T) > a,, and n( 6) _
no ; the relation (13) gives :
Then a(t) = (C - D) t + K is a linear function
decreasing with t if C D or :
The stress level reaches a = ar after a period t2, but no loop is nucleated during the consecutive
period ’to Then this process is repeated and we observe
a sequence of yield drops characterized by an ampli-
tude A a and a period T :
The shape of the yield drops depends upon the slopes
of the successive linear parts (Fig. 7).
Fig. 6. - a(t) curve for n(Q) curve of figure 5.
Fig. 7. - Two shapes of the yield drops. a) C D - C or
2 80 bS(i’) no i.e. low strain rate, b) C > D - C or
2 80 > bS(T’) no i.e. high strain rate.
5. Conclusion.
Many types of plastic strain instabilities may be
explained as follows :
- the applied stress gives rise, during an unit period, to a definite number of elementary processes
(nucleation of twins, emission or unlocking of dislo- cations) according to a relation n( (J) ;
- each elementary process results in a plastic
relaxation of the strained crystal (bS for example,
for a dislocation with Burgers vector b sweeping
an area S during its motion) ;
- a mean period describes the phase displace-
ment between the plastic relaxation and the initiation of an elementary process;
- we have found a criterion of stability for the
solution a = constant which may be simply written :
where M is the stiffness of the loading system. As long
as this criterion is obeyed, stress-strain curves are
smooth and plastic deformation agrees with the
Mecking and Liicke relation :
As soon as this criterion is no longer fulfilled, periodic instabilities appear on the stress-strain curves.
We point out the physical interest of the elementary
processes which start from a critical stress onward.
A simple illustration has been developed to display
the shape of yield drops for unstable solutions.
References
[1] JOHNSTON, W. G., J. Appl. Phys. 33 n° 9 (1962) 2716.
[2] KOCKS, U. F., Prog. Mat. Sci., Chalmers Anniversary
volume (1981), 185.
[3] ANANTHAKRISHNA, G. and SAHOO, D., J. Phys. D.
14 (1981) 2081.
[4] NICOLIS, G. and PRIGOGINE, I. Self-organization in non- equilibrium systems (New York, Wiley) (1977).
[5] KUBIN, L. P., ESTRIN, Y. and SPIESSER, Ph. Res. Mecha- nica (1983). To be published.
[6] SIMON, J., CAISSO, J., GUILLOT, J. and VIOLAN, P.
Mémoires scientifiques, Rev. Met. LXI n° 12
(1964).
[7] PENNING, P., Acta Met. 20 (1972) 1169.
[8] DEMIRSKI, V. V. and KOMNIK, S. N., Acta Met. 30
(1982), 2227.
[9] MECKING, H., LUCKE, K., Scripta Met. 4 (1970), 427.
[10] IOOSS, G., JOSEPH, D. D., Elementary Stability and Bifurcation Theory (Springer Verlag) (1980), p. 123.
