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EFFECTS OF INTERFACES ON THE PLASTIC
BEHAVIOUR OF METALLIC AGGREGATES
C. Rey, P. Mussot, A. Vroux, A. Zaoui
To cite this version:
C. Rey, P. Mussot, A. Vroux, A. Zaoui. EFFECTS OF INTERFACES ON THE PLASTIC
BE-HAVIOUR OF METALLIC AGGREGATES. Journal de Physique Colloques, 1985, 46 (C4),
pp.C4-645-C4-550. �10.1051/jphyscol:1985470�. �jpa-00224725�
JOURNAL
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PHYSIQUE
Colloque
C4,
suppl6ment au n04, Tome 46, avril 1985 page C4-645EFFECTS OF INTERFACES ON THE P L A S T I C BEHAVIOUR OF METALLIC AGGREGATES C. Rey, P. Mussot, A.M. Vroux and A. Zaoui
Laboratoire PMTM,CNRS, Avenue J . B . Cle'ment, 93430 VilZetaneuse, France
Abstract
-
The mechanical behaviour of tricrystals exhibiting three concur- rent plane grain boundaries is analyzed by an analyticalcalculation of the internal stresses. A correlation between grain boundary sliding and the plastic strain field in the grains is established.
INTRODUCTION
Grain boundaries and junctions of grain boundaries play an important part in the mechanical behaviour of polycrystals. On one hand they induce elastical and plasti- cal incompatibilities of deformation hence, internal stresses, and on the other hand they act as an obstacle to the propagation of plastic slip systems, leading to the formation of inhomogeneities of the plastic deformation inside the grains. Moreover, at high temperature, it is well known that Grain Boundary Sliding (GBS) is a speci- fic mode of deformation of polycrystals which can initiate cavities and cracks. The polycrystalline models /1,2/ generally used to describe the mechanical behaviour of aggregates take into account the relative crystallographical orientation of the grains but omit the interfaces and junctions own effect.
The specific effect of grain boundaries were previously investigated through well defined structures such as bicrystals / 3 , 4 / . As a second step towards the descrip- tion of the polycrystal behaviour, the tricrystal case (three crossing grain bounda- ries) is hereafter analyzed. Our aim is to report observations of slip patterns carried out on Cu and Zn tricrystals plastically deformed under tensile test at room temperature : a new kind of intragranular interface
-
bound to the triple node - could be observed in addition to the intragranular interface bound to the grain boundary (already studied in the bicrystal case). In case of zinc tricrystals, conditions of activation of GBS were analyzed. Measures of GBS /5,6/ as well as the plastic strain on both sides of the sliding boundaries were carried out thanks to fiducial grids in order to establish a correlation between GBS and intergranular pl'astic incompatibilities.To obtain more quantitative informations from the observations, analytical calcu- lations of internal stresses (by means of the continuous theory of dislocations) are given in case of tricrystals, with grains assumed to be uniformly plastically deformed. Different phenomena, such as inhomogeneities of plastic deformation and formation of new intergranular interfaces able to relax the hereabove calculated internal stresses are then discussed. In case of grain boundary sliding it is shown that the occurence of GBS modifies the classical superficial compatibility condi- tions (thus the involved internal stresses) for a plane grain boundary, as soon as the GBS is non uniform.
EXPERIMENTAL PROCEDURE
Tricrystals were obtained from initially 99,99 % and 99,9999 % pure Cu and Zn
JOURNAL DE PHYSIQUE
respectively, thanks to Bridgman vertical growth method under secondary vacuum. Three conical cavities are machined in a cylindrical graphite crucible to house the initial seeds.
3
The tested parallelepipedic samples (about 50 x 3 x 15 mm ) were cut out from the initially grown crystals by spark cutting, then etched to remove the eventually damaged superficial layers. After mechanical and electrolytical polishings, the samples were annealed under argon at 1200 K for copper and 650 K for Zinc. The specimen orientations were checked by the backwards reflexion Laue technique. All tensile tests were performed at room temperature, allowing grain boundary sliding activation in case of Zn (T = 0 , 4 TM) but restricting diffusion and disloca- tions climb mechanisms. No grain boundary sliding was observed in copper specimens.
-4 -1 -6 -1
The axial macroscopical strain rates were 10 s and 10 s for copper and zinc samples respectively. The use of fiducial grids with 14 um steps allowed accurate evaluation of GBS and total strain on both sides of the grains boundaries. The samples were observed at different deformation stages through an optical and a scanning electron microscope.
EXPERIMENTAL RESULTS
Previous papers / 3 , 4 / were devoted to nearly compatible aluminium bicrystals and incompatible copper bicrystals. It was established that, at the beginning of the plastic deformation, each bicrystal component showed an inhomogeneity of plastic glide : different slip systems were activated in different areas (hereafter called domains) of a same grain. As the deformation increased, these heterogeneities either vanished or not (and at the same time the resulting bicrystals hardening was either weaker or stronger). A model was proposed which allowed us to forecast all the main experimental observations : the nature and the localization of the different acti- vated slip systems inside each grain, the relative stability of the slip inhomoge- neities as well as their resulting hardening effect. The formation of the intragra- nular interface; (hereafter called A interface) separating these domains depends on pure geometrical conditions, such as the relative orientation of the primary slip systems and the grain boundary : for some orientations the grain boundary can act as an obstacle to the progression of the primary slip system so that a secondary one must be activated near the grain boundary (domains). One may expect that these previously reported results will stand in the case of tricrystals.
Fig. 1 sums up the most important observations carried out on plastically deformed tricrystals (Cu an Zn). Fig. 2 and 3 give an example of the slip patterns obtained (after tensile tests) near the junction on Cu and Zn tricrystals respectively. Far from the junction the grains were uniformly plastically deformed according to the primary slip systems, except near some grain boundaries where large domains limited by intragranular interfaces similar to the bicrystals ones (interface A). But near the triple node, in a 0,30 mm radius range (fig. 2 and 3) the observed slip patterns were very inhomogeneous and some extra intragranular interface (named B) appeared in a given grain as an extension of the third grain boundary. These interfaces vanished far from the triple node but grew with increasing deformation. It must be pointed out that in Zinc tricrystals exhibiting only three coplanar basal slip systems, the two kinds of interfaces A and B could be distinguished as in fig. 3.
In copper tricrystals measurements of the lattice rotation in a same grain, at two stages of deformation by Kossel technique
/7/
showed that lattice rotation was greater far from the junction in grains with no B interface- and was close to zero near the junction in the grains where B interfaces were observed. These observations led to assume that formation of B intragranular interfaces are activated to lower the internal stresses and to reduce the lattice rotation near the triple node in order to avoid cracks. It must be pointed out that these processes were observed in crystals presenting a sufficient number of plastic slip systems.F i g 1 : P l a s t i c a l l y deformed t r i c r y s t a l . The i n t r a g r a n u l a r i n t e r f a c e s A and B
bound t o t h e g r a i n boundary and t o t h e j u n c t i o n r e s p e c t i v e l y a r e pointed by ar- rows. F i g 2 : Copper t r i c r y s t a l f o r a n a x i a l macroscoaic s t r a i n of 5 %. I n t e r f a c e B i s i n d i c a t e d but i n t e r f a c e A i s mask lmm I
F i g 4 : Axes system used t o c a l c u l a t e F i g 3 : Zinc t r i c r y s t a l , f o r an a x i a l t h e i n t e r n a l s t r e s s e s .
macroscopic s t r a i n of 5 %.
I n t e r f a c e s A , B and g r a i n boundaries
C4-648 J O U R N A L D E PHYSIQUE
Grain boundary sliding was studied in Zinc bicrystal and tricrystal samples at room temperature (0,4 T ) by fiducial qrids / 6 / . The results can be summed up as
follows : M
-
no GBS could be observed in bicrystals with plane grain boundary for any orienta- tion of the tensile axis ;-
GBS was observed along one or several boundaries in tricrystals right from the beginning of the macroscopic deformation.Note that G.B.S. had not necessarily the same amplitude in the two faces of the samples, this phenomenon being due to the existence of domains which (as in bicrys- tal cases) may not appear on the two faces of the same grain. In the next section it will be shown that GBS amplitude is bound to the incompatibility of the total strain, thus is bound to the kind of the slip systems activated near the grain boun- dary.
It must be pointed out that in all case GBS amplitude is non uniform near the triple node. Furthermore, measures of some total strain components on each side of a grain boundary show that a non uniform grain boundary sliding increases the plastic incompatibility and thus modifies the internal stresses state.
INTERNAL STRESSES CALCULATIONS IN THE TRICRYSTALS CASE
Let us consider (fig. 4) an infinite crystal which exhibit three (or more) concur
+ +i
rent boundaries. Let OZ be the axis parallel to the junction and oy the normal to the i plane grain boundary (see fig.
4).
Let us assume that the crystal is submitted to an external stress field so that each grain (k) (assumed to be elastically isotropi~) undergoes an uniform plastic strain fieldIn a system of axes bound to the i interface which separates the k and k+l grains respectively, the plastic strain field can be written as follows :
cpi = ai
+
bi Y(B-
ei)
Y(r)- - -
(1)i
where a and bi are symmetrical tensors such as ai
-
-
-
=-
and bi- -
= E ~ ( ~-
~-
E ~ ( ~ ) ~ )The upper index refers to the relevant interface. +i
8. is the angle between oy and the macroscopic reference axes XYZ.
Any point p in the XY axes is determined by the angle 8 and the distance r to the node.
y(8-Oi) and Y(r) are the Heaveside function :
idem for Y(r). (The term Y(r) is not necessary in expression (1) but it had the advantage to simplify the calculations).
The internal stress field is analytically calculated from the continuous theory of dislocations 181 in cylindrical coordinates. In this case b is only a function of 8. The internal stress field is given for the i interface, by the following expres- sions :
i I-1 0 = - -
I
.
2 (bSr+
~b:~) 26 sin $i+
~ i n 2 @ ~ log -ee
~IT(~-v) i i i 2 z 26(bZZ+
vbrr)-
"be, (2 log Ri+
1) 1 oi =-
b:zl- 6 sin@
- C O S + ~
( T
- log
7 ez i i -u
o r -
-
z:b1
6 cosOi- sin4
i(f
+
log-)
R~1
1
u
,
,i = - -
-
(b:r+
~b:~) ( ~ 0 ~ 2 4 ~ log&
+
6sin2@.)r 2n(l-v) R1
+
(b:r+
b:Z) sin 4. '11 <with @i = 8-Bi, 6 = T sgn@
-
@i, RL is a constanti
1-1 is the shear modulus and v the Poisson's ratio.
The global solution in cartesian coordinates in any point of the tricrystal can be easily deducedi from (2). The global solution fulfills the condition div q = 0. and we have Cbkl = 0.
i
These solutions are found to be close to Kleman's /9/ ones for magnetic domains. In the XYZ axes, the internal stress field depend on
e
and log r. This last term means a concentration of stresses near the node. In metals with numerous slip systems, it can be assumed that the observed phenomena such as activation of slip systems different from the primary ones, formation of new interfaces can relax these inter- nal stresses and avoid cracks.An other point to be clarified is the existence of correlations between GBS and the superficial plastic incompatibility which derives from the difference between the components of the strain tensors in the adjacent grains. On one hand several authors /10,11/ have mentionned it and on other hand other authors 1121 have denied it. We have shown that these correlations exist by writing the conditions of compatibility and by solvin the obtained equations. Let us consider in a given medium a displace- ment field
d($
which undergoes a sliding discontinuity accross a surface S, enclos-ing the v o l u s V. Wscan write everywhere in the medium :
a.cn
-
=u(?)
+
w($) AO(V) ( 3 )+
where U and W are continuous vectorial function of
'rC;
6'(V) is a characteristic function of V so that 6"(V) is either one or zero according as '?lies withinor outside V respectively. The sliding character of the displacement discontinuity W on S yields the conditionTi?)
.;f((r3 = 0 ifT e
S.;I2lij)
is the unit outward normal to S at?
In a case of a plane boundary S may be taken as the plane y = 0. According to a small strain formulation and using rectangular cartesian coordinates the strain field is given by
E(X,Y,Z) = a(x,y,z) + b(x,y,z) Y(y)
+
c(x,z) 6(y)C4-650
JOURNAL
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where
2 ,
and5
are symetrical tensor c is the GBS strain tensor-
Y(y) is the Heaviside function and 6(y) indicates the Dirac function on S with the extra condition (no decohesion) : c~~(x,z) = 0.
The compatibility conditions
eijk
Elmn
cjn = 0 can be solved(Eijk denotes the permutation tensor).,
mThe results means that GBS strain tensor 2 is correlated with the displacement discontinuity W, according to the relations.
-where the terms between brackets in (5) indicate an arbitrary rigid motion with a rotation vector (O,q,O) normal to the interface. W1 and W are bound to the kind of activated slip systems near the grain boundary. 3
This result clearly shows that the involved strain components of GBS (namely here c12 and cg2) are bound with strain discontinuities b 11, b13, b33 which are concerned by compatibility conditions provided that GBS is not uniform.
CONCLUSION
In order to get a better understanding of mechanical behaviour of polycrystals, we have calculated the internal stresses in a crystal presenting three concurrent grain boundaries and superficial plastic incompatibilities of deformation. We have paid attention to the physical processus able to relax these internal stresses : forma- tion of a new kind of intragranular interface bound to the triple node. We have also shown that correlations can exist between grain boundary sliding and plastic strain on both sides of the grain boundary as soon as the GBS is not uniform. Consequently a given non uniform GBS is a source of superficial incompatibility and internal stresses which can be relaxed by an adequate plastic strain field in the grains and conversely.
REFERENCES
/1/ KRONER, E., Acta Metal12 (1961) 155.
/ 2 / BERVEILLER, M. and ZAOUI, A., Journal de MGcanique 19, 2 (1980) 343. /3/ REY, C. and ZAOUI, A., Acta Metall.
8
(1980) 687./4/ REY, C. and ZAOUI, A., Acta Metall.
2
(1982) 523./5/ MUSSOT, P., Speciality thesis (1983), Universits Paris VI (France).
/6/ MUSSOT, P., REY, C. and ZAOUI, A., 37th International Meeting of the SociBt6 de Chimie Physique, edited by P. Lacombe, Paris (France), 1983.
171 REY, C. and VROUX, A.M., First Risd International Symposium on Metallurgy and Materials Science, sept. 1984.
/8/ KRONER, E., Theorie der Versetzungen und Eingenspannungen, Berlin, Springer Verlag (1958).
/ 9 / KLEMAN, M., J. Appl. Phys.
65
(1974) 1377./lo/ RAJ, R. and ASHBY, M.F., Metal. Trans. 2 (1971) 1113. Ill/ LANGDON, T.G., Metals Forum
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