A MODEL FOR THE FATIGUE SATURATION RANGE OF ALUMINIUM FROM DYNAMIC BEHAVIOUR AND INTERACTIONS OF DISLOCATIONS

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A MODEL FOR THE FATIGUE SATURATION RANGE OF ALUMINIUM FROM DYNAMIC

BEHAVIOUR AND INTERACTIONS OF DISLOCATIONS

R. Fougeres, J. Chicois, A. Hamel, A. Vincent

To cite this version:

R. Fougeres, J. Chicois, A. Hamel, A. Vincent. A MODEL FOR THE FATIGUE SATURATION RANGE OF ALUMINIUM FROM DYNAMIC BEHAVIOUR AND INTERACTIONS OF DISLOCA- TIONS. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-155-C8-160. �10.1051/jphyscol:1987820�.

�jpa-00227124�

A MODEL FOR THE FATIGUE SATURATION RANGE OF ALUMINIUM FROM DYNAMIC BEHAVIOUR AND INTERACTIONS OF DISLOCATIONS

R . FOUGERES, J . C H I C O I S , A . HAMEL and A . VINCENT'

G E M P P M , CNRS UA-341, INSA Lyon, F-69621 Villeurbanne Cedex, France

' ~ a b o r a t o i r e d'ultrasons et d e Traitement du Signal, INSA Lyon, F-69621 V i l l e u r b a m e Cedex, France

RE=. Le but de cet article est de presenter un nodtle de fatigue applicable dans le cas du domaine de la saturation cyclique. Aprks avoir rappel& des donntes exp6rimentales rtcentes concernant la mobilit6 des dislocations au cours de la fatigue de l'alurainium 5bl polycristallin, n o w dt?veloppom un nodtle qui s'inspire d u "modble compositeu, oh la "phase dure" et la "phase molleu sont respectivenent les parois et l'inttrieur des cellules de dislocations. Le modble propost3 fournit des informations sur le comportement dymmique des dislocations au c o w s de la fatigue. Les donnbes obtenues sont en bon accord avec celles deduites des nesures d'attenuation ultrasonore ef f ectutes pendant 1' essai de fatigue lui-mkme.

ABSTRACT. The aim of this paper is to propose a model for the fatigue saturation range of metals. First, recent experimental results obtained on 5N polycrystalline aluminium are succinctly presented. The main part of this paper is devoted to the development of a fatigue model of "composite" type, where hard a d aoft phases are cell valls and cell interiors, respectively. Data concerned with the dynamic behaviour of dislocations during the fatigue are deduced from the model. They are in good concordance with those obtained with ultrasonic attenuation measurements during the fatigue test itself.

ItfmDxm.

Since fifteen years, numerous theoretical and experimental studies have been carried out on the fatiaue behaviour of Dure metals (see for exan~le /1/ and / 2 / ) . Several models have-been proposed in-order to describe mechanical properties of fatigued materials fron the structure of dislocations defined by TEM observations. Among these models the "composite model", with "hard" and

"soft" phases corresponding to crystal regions with respectively high and low dislocation density, appears as being the nost suitable nodel /3/,/4/,/5/.

However, the conposite nodel has not been fitted, up to now. to include precise mechanisms of dislocation -crystal defect interactions and to give informations about the dynamic behaviour of dislocations during fatigue test. The aim of this paper is to propose a new model for the behaviour of pure aluminiun (5N) cycled in the saturation r q e at room temperature. It is based on the idea of composite rmaterials including nicroscopic aspects which have recently been experimentally determined by TEN in situ experinents and ultrasonic attenuation measurements / 6 / .

EXPER=& GROUND.

The experimental results mentioned above can be summarized as follows.

According to the value of the fatigue strain amplitude, it was shown /6/that two fatigue ranges can be defined in the 5N polycrystalline aluminium : at low strain amplitude the interactions between dislocations and point defects

(probably impurity atoms) play the main role,whereas at high strain amplitude

(

Et

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987820

C8-156 JOURNAL DE PHYSIQUE

amlysed as variations of the mobile dislocation density

.

Results of TEM in situ experilaents can be summarized as follows ij cell walls can pe considered as both sinks and sources for mobile dislocations. Horeover, d u r m g TEIi experiments, some dislocations were observed crossing the cell valls very easily.

ii) the dislocation movement s e e m to be controlled by interaction mechanism that are located in the surroundings of the cell walls, where the dislocation curvature is generally very small : these nechanislps lead probably to a local internal stress field of strong value. In addition, without loading of the thin foil, numerous curvated dislocations c8.n be observed quite near the cell walls: therefore that suggests the presence of internal stresses, after unloadirq of the thin foil.

According to the previous composite model / 3 / , internal stresses of opposite polarities appear in the cell wall and inside the cell, in the case of unloaded fatigue samples. This particular internal stress field should lead to

a s m e t r i c a l behaviour like in BAUSCHMGER's phenomenon. In order to verify such a behaviour, particular tests were carried out, after fatigue at strain amplitude of 1.5~10'3 in the saturation range, corresponding to a stabilized fatigue loop. Details of cyclic conditions are given in reference /6/. The results are reported in figure 1 : after a stop in the fatigue process at a given position in the fatigue loop and unloading of the sample at zero stress, an asymmetrical behaviour is observed according as the reloading is carried out in traction or in compression (compare O"Au and O'B" loadings in the figure I). The degree of this asymmetrical behaviour is depending on the stop position. However, for a given position as well in traction as in compression, characterized by a stress value approximatively equal to 0.5 a, (6, being the maximal stress on the fatigue loop), a symmetrical behaviour is observed ( c o m p a r e 0 ~ ~ ~ 0 ~ 0 ~ l o a d i n g s in figure 1). In addition, f rols one side to the other side of this special position the direction of the asymmetrical behaviour is reversed.

In conclusion, it appears that an important part of the fatigue stress is probably due to internal stresses which are required to assume the coapatibility of the deformation between cell walls and the inside of dislocation cells.

a) j&pothesis.The model proposed here is based on the 'composite model'. The cycled sample is assused to be composed of two phases : the "hard"

phase which is ascribed to cell valls and the "softn one corresponding to the dislocation cell interior. Recent Pleasurements of KIKUCHI line displacements in pure aluminium suggest effectively that cell walls can be compared to a second phase /7/.Effectively, the displacesent of K I K K A I lines from one side to the other side of the w l l exhibits a discontinuous variation that corroborates this hypothesis. In addition,at each point of a slip plane,it is assumed that a dislocation b e c o ~ e s mobile if the resolved shear stress Z reaches a value given by :

Z

=

1r1 this relation, Zo is due to the point-defect dislocation intera.ction, that corresponds to the thermal stress component Zp and

t,

attractive trees of the dislocation forest and to internal stresses which assume the compatibility between the hard phase and the soft one. As in the lWHRbBI1 3 model, the compatibility of the def oraation between two phases is supposed to be assumed by interface dislocations. Let us recall that subgrains of mixed nature have been observed by HMEL et al. /7/ in the sane fatigued materials. Therefore, two networks of infinite parallel dislocations are considered at each of the two interfaces of a cell wall (fig. 2). In addition, conpatibility dislocations at each interface of a cell w l l are of opposite sign, like in S H B Y ' s model /8/ for materials containing hard particles. On the other hand we consider that the asymmetrical behaviour described in figure I is due to the internal stress!i&which is equal to zero at F and G points of the figure 1 and which is reversed according to the position in the fatigue loop i.e. in traction or in compression. At F and G points, the level of the fatigue stress is m i n l y attributed to the effect of forest attractive trees ( ~ $ 7 . ) .

Finally, in order to take into account the fact that dislocation movements are locally controlled in the surroundings of the cell wall let us postulate that at a given polnt of the fatigue loop, the fatigue stress is given by the maximum value of the expression (1) :i.e. the highest stress value that a nobile dislocation has to overcome in this region of the glide plane.

k)

the 2 c c internal stress field to which an edge dislocation (MI is submitted

when it. comes near a cell wall from the inside of the ceil: The sign of dislocations in the interface is chosen suchthattheir stress field acts against the ll dislocation novelaent touards the cell wall. Details of stress value computations are given in reference 1 9 / . '2he stress field of the figure 3 corresponds to a position of the glide plane suchthatits intersection with the infinite network of interface dislocations is exactly at the half distance between two successive dislocations of the dislocation network (fig. 2). This particular position of the glide plane leads to a minimal value of the stress field and it will be the only one considered in this paper. The stress field defined in figure 3 exhibits long range internal stresses with a pronounced

m a x b u m value near the interface : the smaller is the distance between two successive dislocations of the infinite network at the interface, more pronounced this maximun is (fig. 3) .Since the stress field around dislocations is of elastic nature the effect of a cell wall can be obtained by summing the stress field of two interfaces with dislocations of opposite signs.

As a result, figure 4 (a,curve) shows the stress field thus obtained in the particular case already used .in fig. 3 with two similar dislocation networks of opposite sign, exhibiting the sane size (distance Dl, between two successive dislocations in the network). It can be seen in figure 4(a,curve) that, inside the cell, high values of the stress field are mainly located near the cell wall in the surroundings of each interface. To penetrate in the cell wall the dislocation M has to overcome this stress field which acts along distances between 100 awl 200 0 what leads probably to an athema1 behaviour.

On the contrary, the stress field in the cell m l l has a n attractive action upon the M dislocation and probably coapensateslnore or less completely the effect of the high density of forest trees. Finally, because of this high internal stress value in the wall, a dislocation of opposite sign can leave more easily the cell m l l . Rowever, such dislocations have to escape the maximum stress value located inside the cell near the cell wall.

In order to obtain the ZCC stress field under the hypothesis of an infinit.8 solid it would be necessary to sum the effect of an infinite network of cell -11s like that defined in figure 2. In the case of a semi-infinite solid, i m g e forces, as a result of the surface effect, have to be introduced

C8-158 JOURNAL DE PHYSIQUE

in the stress field computations. Another way to estimate surface effects is to consider thutzccstresses are internal stresses that remain approxinatively at the sane value after unloading at zero applied stress. This hypothesis is probably not quite right because reverse microplasticity can occur during unloading. However, if it is assumed that the reverse microplasticity effect is small, compatibility stress have to verify the following relation :

(

cCip. . ^f! + ( r , , , , , ( r - { ) - ~

with (zcc )ma and (2cc)pa internal stresses in the cell and in the wall respectively, and f the volume fraction of cell walls.

In order that the internal stresq field reported in figurei@jsatisfy the relation (2). additional positive3tfesses have to be considered. It is assumed that this additional stress field is due to the effect of both other cells and isage forces of the seai-infinite solid here envisaged. In the case where the cell size is twenty times larger than the cell wall thickness (as TEn observations of dislocation cells show /7/)a srall increase in positive stress equal to approximtively 1x10-* G (G being the shear modulus) leads to a satisfactory agreement of the relation (2) with the stress field of figureqb) The discussion of the compatibility stress field effect on the dislocation behaviour in the surrolmdings of the cell wall. will not be reported in this paper. However, all observations carried out during in situ TEn experiments can be explained by this kind of internal stress fields /9/.

C ) ati e st ' the . lasti r e. If the point-

defect contrg:t:on i y neg1;:"t"d l?ef f ect:"PP"ioi",t-def",ct?af e generally very srall in 5N aluninium /6/) the fatigue stress a can be obtained from the relation (1) :

( 3 )

with

rp

According to the hypothesis described above and with the relation (3) it can be deduced that the fatigue stress at point F in the fatigue loop reported in figure 1 is - y = i/y.Zf ;indeed because of the symmetrical behaviour, Z c c

must be equal to zero. In order to promote a model for the fatigue behaviour in the plastic range between points F and A of the fatigue loop (fig. 1) the fatigue stress increase between F and A is ascribed to the increase in the compatibility internal stresses

z , ,

with

p

According to ASEBY's model for uncleformable inclusions in a plastic matrix, the dislocation density of the interface network (geometrically necessary dislocations) is given by :

where g p i s the plastic shear strain in the slip plane and L. the network size of the particle distribution. In a similar way, we postulate that for a dislocation cell structure, the dislocation density A ; , at the cell wall

intertace, is given by :

A:

,w

with L the cell size. \Y is a parameter such as \Y a 'Y ( 8 ) (k '4'11) in order to take into account the capacity for cell walls to be plasticail deformed (nainly by emission or absorption of dislocations). When Y = 0 or

6

walls are entire1 plastically def orned or remain uadef ornable respectively. Whenhalues are between 0 and I. cell walls are partially deformed. i. e. the corresponding dislocation displacement gives a -11 shear strain contribution smaller than the assigned shear strain gp. From relation3 ( 4 ) and (6). it can easily be shorn that the elementary increase of the Shear stress Z. dz. is as follows :

d t

=

p2

It has been shorn /9/ thanthe Yparameter can be deduced fron e x p e r k n t a l valu s of U = f (Ep) and2given by :

B

' 4 '

s

B

4 G = b < E - E s y ) (8)

EP

Both \Y parameter and doc ultrasonic attenuation are reported in figure 5. As it has been previously established this last parameter is directly proportional to the nobile dislocation density. Ihe evolution of the two parapeters (\Yawl hcd ) is very siailar.

As it was assumed theyparameter value is mainly dependent upon the dislocation emission or absorption by the cell #all. On the contrary

4%

variation is due to the mobile dislocations inside the cel1,because dislocation loop lengths of the cell wall are too snall to contribute strongly to the ultrasonic attenuation.

Thus, at point F, theyparameter is equal to zero because cell walls emit easily numerous dislocations in order to a s s m e the plastic deformation.

So, the ultrasonic attenuation increases strongly because the total density of aobile dislocations increases: then the increase of the 'i' value corresponds to a smaller number of created dislocations because plastic deformation is mainly due to the extension of dislocation loop inside the cell. W e n mobile dislocations reach the opposite wall of the dislocation cell. they can penetrate in this cell wall. Therefore, the V paraaeter value aust decrease as it can be effectively seen in figure 5. So, the ultrasonic attenuation decreases ,also slightly, the number of mobile dislocations inside the cell becoming slightly less numerous.

Thus, the model here proposed,allows us to deduce data about dynamic behaviour of dislocations from mechanical properties. These data are in good agreement vith those obtained fron ultrasonic attenuation measurements.

P E r n r n E S .

I/ H.=I and R.IbA#C, Proc. 2 d RISQ Int. Synp. b t a l . m t e r . Sci. ,p 87 (1981).

2/ H . l E H R A B I Proc. of the 7th Int. conference on the s t r e q t h of metals and alloys-flontreal-Pergamon Press,p 1216 (1985).

3/ H.fI[K;HRABI , Acta Hetall. 31,p 1367 (1983).

4/ 0 . B. PEDERSEW and A. T. WINTER, Acta fletall. 30,p 711 (1982).

5/ H.HUGlRABI, T.UNGAR,W.KIENLE and H.VILKENS, Phil. Hag. A, Vol. 63, N06, p 793, (1986).

6 / J. CHICOIS,R.FOUGERES,G. GUICIK)N,A.Hann and A. VINCENT, Acta Metall. 34,p 2167

, (1986).

7/ A. EAUEL,G. TEDLLET,C. ESNOW and R. FOUGERES to publish Scripta fletall. (1987).

81 H. F. ASHBY, Phil. Hag. 21,p 399 (1970).

9/ J. CHICOIS, These Lyon (1987).

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Fiqure 2 : Diagram of a cell wall uith two networks of infinite parallel dislocations at each interface.

Straight edge dislocation

.

n"

and

- .

__

Figure 5 : Variation of the ultrasonic attenuation &I and of the 'f'parameter as a function of fatigue strain in traction or in conpression inside the plastic range from F and G points respectively. Note the similar variation of these two parameters.

Fatigue test conditions are those defined in figure 1.

Figure 3 : Ratio between the shear stress and the shear reodulus in a glide plane. Case of a single interface with a dislocation network Figure 1 : Fatigue loop obtained W e r parallel to a M dislocation

total strain control condition8 in the f isure 2). BURCEi?Sivector

~~=tEf

saturation range of 5N polycrystalline interface dislocations parallel to y

aluminium cycled at E Q / z = ~ . ~ x I O - ~ at axis (fig. 2). b = 0.29 nm, 8 angle room temperature and with a test (fig. 2) = 3S0. Effect of network size frequency of 0 . 2 Hz.

Successive particular loadings in ^p~

order to disclose a n I s m e t r i c a l behaviour along the fatigue loop : stop on the fatigue loop (point P), unloading to zero stress and reloading

traction or in compression (6f or O'A' or O'B'.

.

sanple is cycled for 1 0 cycles. Figure 4 : Ratio between the shear stress and the shear modulus in the case of the two interfaces of a cell wall exhibiting dislocations of opposite signs (a and b see in the text.) Characteristics of each interface network are similar to figure 3.

1