Pergamon

PII: S0020-7403(97)00056-8

Int. J. Mech. Sci. Vol. 40, Nos. 2 3, pp. 305-311, 1998 (Q 1997 Published by Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0020-7403/98 $19.00 + 0.130

HIGH-STRAIN-RATE SUPERPLASTICITY OF PARTICULATE REINFORCED ALUMINIUM MATRIX COMPOSITES

K.C. C H A N and B.Q. H A N Department of Manufacturing Engineering, The Hong Kong Polytechnic University, Hung Horn,

Hong Kong

Abstract In recent years, there have been a lot of research efforts in studying high-strain-rate superplasticity of aluminum matrix composites. However, the superplastic deformation behavior of the composites has not been fully understood. Superplastic deformation mechanism in aluminum matrix composites was considered to be grain boundary sliding and interfacial sliding. In this paper, the model proposed recently by the authors to explain the superplastic deformation of a particulate reinforced aluminium matrix composite at temperatures above and below its solidus temperatures is further developed. The theoretical predictions are in agreement with the published experimental findings. © 1997 Published by Elsevier Science Ltd.

Keywords: high strain rates, metal matrix composites, superplasticity.

NOTATION

cr applied stress ao threshold stress ~ons strain rate due to grain boundary sliding ~l strain rate due to interfacial sliding ~t total strain rate G shear modulus Dr interfacial diffusivity Dgb grain boundary diffusivity 11 length of the pile-up in front of the reinforcement dm matrix grain size d, particle size of reinforcement Vr volume fraction of reinforcement h climb distance T absolute temperature T" solidus temperature b Burgers vector k Boltzmann's constant w binding energy between a solute atom and a boundary dislocation Co solute concentration corresponding to w = 0 2 distance between solute atoms on a dislocation line

INTRODUCTION

Di scon t inuous ly reinforced meta l ma t r ix compos i tes ( M M C s ) have been successfully manufac tu red t h rough p o w d e r meta l lu rgy t echno logy in the last decade. This k ind of M M C s is a t t rac t ive for m a n y s t ruc tura l app l i ca t ions because of their high specific s t rength and modu lus of elasticity. However , in general , these mate r i a l s have relat ively low r o o m t empera tu re duct i l i ty which means tha t to cer ta in extent they are no t easy to be shaped. Even at e levated tempera tures , they n o r m a l l y show only

l imi ted tensile ducti l i ty. Recently, a n u m b e r of researchers [ 1 - 7 ] have r epor t ed tha t some d i scon t inuous ly reinforced

a l u m i n u m M M C s could behave superp las t ica l ly when tested at high s t ra in rates under the r ight condi t ions . These f indings are significant since one of the m a j o r shor tcomings of conven t iona l superp las t ic forming process is tha t the forming rate is too low. The d iscovery of superplas t ic capab i l i ty of some a l u m i n u m - b a s e d M M C s at high s t ra in rates thus gives these mate r ia l s a be t te r o p p o r t u n i t y for indus t r ia l appl ica t ions . Al though, there are so m a n y exper iments on this k ind of superplas t ic i ty , de fo rma t ion mechan i sm on h igh-s t ra in- ra te superp las t ic i ty of M M C s has no t been

305

306 K.C. Chan and B. Q. Han

understood clearly. Recently, the authors [8] have attempted to develop a theoretical model to explain the deformation behavior. In their model, the constitutive equations for superplastic deformation of a SiC particulate reinforced aluminum composite at temperature below and above its solidus temperature were developed under the consideration of the accommodation process for dislocation movements. In this paper, a model taking into consideration of threshold stress will be further developed.

THEORETICAL ANALYSIS ON HIGH-STRAIN-RATE SUPERPLASTIC DEFORMATION

The present model is an extension of the model previously proposed by the authors. Some essential parts of the model are still outlined in this paper to keep it self-consistent. Nieh et al. [9] have pointed out that the presence of a liquid phase in the material was a prerequisite for the MMCs to behave superplastically. But, in fact in many cases, the fraction of liquid could not be too high since superplastic forming of MMCs is normally undertaken at temperatures close to or just slightly above the solidus temperature of the composite. In these cases solid "bridges" would exist between solid particles. Koike et al. [10] had obtained some direct evidence to suggest that partial melting had occurred at the reinforcement/matrix interfaces when the composite was tested at a temperature slightly above its solidus temperature.

Figure 1 shows an array of matrix and reinforcement phase in a particulate reinforced MMC material under a optimum superplastic forming condition. The primary deformation mechanisms in superplastic forming of MMCs are considered to be grain boundary sliding (GBS) and interfacial sliding (IS). However, the movement of grain boundary dislocations (GBDs) is very often blocked by obstacles, such as triple points, reinforcements or grain boundary ledges. In order to pile-up and climb, dislocations are required to overcome the threshold stress, ao, that is similar to the friction force during the sliding. The effective applied stress, will be used to produce the climb of the leading dislocation of the pile-up. The rate of sliding is, therefore, controlled by the rate at which dislocations are removed by climbs at the head of the pile-up.

Now, considering the case that triple points are filled with liquid, GBDs between grains will disappear very rapidly as happens between grains 1 and 2 and grains 2 and 3 of Fig. 1. Under such condition, the strain rate contributed by GBDs will be very high and, therefore, can be neglected in the calculation of the total strain rate. On the other hand, GBDs will pile up if GBDs encounter solid grain boundary and particle interfaces as occurs between grain 3 and grain 4. The stress of the pile-up at the reinforcement apl, is given by

apx = 2 l l ( t r -- a o ) 2 / 3 G b . (1)

Since, the climb velocity (vet) of a dissociated leading dislocation is controlled by the rate of diffusion of vacancies which are associated with the climb process, vc~ is given by

Ds[- _ _ _ fap~ baXi J = /expt - 1 (2)

As in high-strain-rate superplastic forming, k T >> apl b 3, and ll is assumed to be (d, . - d . ) /2 , hence,

211D1(a - ao)2b 3 (din - d,)Ds(a - cro)Zb 3 = (3)

vd = 3hx G b k T 3hx G b k T

The local sliding rate of grain boundaries, ~1, becomes

(din - - d . ) D 1 ( a - - 6 o ) 2 b 3

fj~ = 3 h 2 G k T (4)

GBS rate kGBs = f ~ l / ( d x ~ ) , gives

(d,, - d , ) D d a - Cro)2 b 3 (5)

~aBs = 3 x / ~ h 2 d , , G k T

Now, take a different situation where a reinforcement particle is located at a triple point where liquid is present. In such a condition, GBS will lead to IS. Subsequently, dislocations produced by interfacial sliding will move across the grain and pile up at grain boundaries (as in grain 5 of Fig. 1).

High-strain-rate superplasticity of aluminum matrix composites 307

® SiC particle liquid phase dislocation grain

Fig. 1. A typical array of matrix and reinforcements in a superplastic MMC at the temperature slightly higher than the solidus temperature.

Similarly, the following relationship is established:

(2d., - dr)Dob(a - O'o)2b 3

~IS = 3x/~h2LGkV (6)

For random distribution of reinforcements, L = fdr and the volume factor, f = [re/(3 xfl2vr)] 1/3. The total strain rate, ~t can, therefore, be determined by the relationship

~t = ~GBS -~ ~IS, (7)

According to Nabarro [11], the shortest distance, r, between two dislocation lines is 2.5b when T ~ T,,. In assuming that hi ~ h 2 ~ 2r, the total strain rate, ~,, thus becomes

+ 1 1)bDgb (a~TG°)2 (8)

where p is the ratio of the matrix grain size to the particle size of the reinforcement. Test temperature was found to be one of the most influential parameters in governing the

superplastic behavior of MMCs. In most cases, high-strain-rate superplastic deformation would take place at temperatures between 0.93 and 1.02 of T ' . Although the maximum elongation occurs at the temperatures of slightly higher than or near to the solidus temperature, elongation was still above 200% at test temperatures between 0.93 T£, and T~, [12]. This really implies that aluminum alloy composites will behave superplastically without the presence of liquid at grain boundaries or interfaces. In the case, GBS and IS are assumed to be the primary deformation mechanisms, and a typical array of matrix and reinforcement of a particulate reinforced composite was shown in Fig. 2.

Referring to Fig. 2, four possible situations for dislocation movements are considered. The first case is that dislocations generated at triple point grain boundaries will pile up along grain boundaries and then move up to another grain boundary, as happens between grains 1 and 6. The second option is to allow the dislocations generated pile up along the grain boundary before climbing up at a matrix/SiC interface (grains 2 and 3). The third situation is that the dislocations generated at a matrix/SiC interface will pile up along grain boundaries and climb at a triple point (grains 3 and 4). The last case is that dislocations generate at a matrix/SiC interface will pile up along a glide plane then climb at a grain boundary (grain 5).

For the first situation of dislocations pile-up, the length of the pile-up, L1, is equal to d,,/2, and

since the strain rate, is equal to iq/(d,,,f3), therefore,

Dgb b3

~1 - 3x /5 -~ 6 k T t ~ - ~o) ~. (9)

308 K. C. Chan and B. Q. Han

0 SiC particle

-1_ dislocation

grain

Fig. 2. A typical array of dislocations in a particulate reinforced MMC when deformed below the solidus temperature.

In the second case, the length of the pile-up in front of the reinforcement is assumed to be equal to

( d i n - d,)/2. So, the strain rate e2 = ~i2/(dmx//-~) can be expressed as

(dr, - d,)D~b 3 , e2 = ~ - - - x ~ , , h 2 ~ t a ao):. (10)

w

In the third case, again the length of the pile-up in front of a triple grain boundary point is assumed

to be equal to (dr, - d,)/2. Thus, the strain rate ea = ~23/(dmv/~) becomes

(d, - d,)Dobb 3, ~3 = V x / / - ~ ~ [ O" -- 0"0) 2. (11)

In the fourth case, the length of the pile-up in front of the grain boundary is assumed to be equal to (din - 0.5d,). The strain rate k, = ~4/(L,x/~), therefore, becomes

(2dm - d,)Dobb a, k4= - ~ L ~ (t7 - Oo)2. (12)

The total strain rate, ~t, can therefore be determined using the grain boundary sliding rate ~ns and the interfacial sliding rate ~is by the relationship of

et = Z~Bs + Z~1s. (13)

The total number of grain boundaries in the first, second and third case of the pile-up is assumed to be N1, N2 and Na, respectively. Now, the total strain rate, ~t, can be determined using the sum of the above four independent strain rates by the relationship

• N 1 N 2 . N 3 .

i.e.

b a FNx Dob

e' = 3,/TGkrL g +

N2 (1 - 1/p)DI Na (1 - Up)D# + U h~ N h~

(14)

(15)

+ (2P~h~)D°b](~-ao)2 .

According to Friedel [13], saturation occurs at T < T m only if ln(h/b) > 2.5. Under the test temperature, assuming hi ~ h2 ~ ha ,~ h4 ,~ h ,~ 121); N1 = N2 = Na = 1/3N, the total strain rate is given by

1

6-k-f k,=43@~[(~--~+~)(2p-1)bDob+~(1-i b ~ l ( ~ - ~°)2 (16)

High-strain-rate superplasticity of aluminum matrix composites 309

where B = x/3Cow/bb~G. It is temperature.

which is the constitutive equation describing the stress-strain relationship of superplastic partic- ulate reinforced composites deformed at temperatures below the solidus temperature.

Presently, there are some theories which could explain the origin of the threshold stress in superplastic materials, such as (a) Orowen stress [14], (b) addition dislocation line length due to climb [15], (c) detachment of the dislocation from dispersion particle [16], (d) internal stress associated with subgrain size [17], (e) load transfer to the stiffer reinforcement [17], (f) the restricted mobility of the grain boundaries owing to the presence of dispersoids [18], and (g) the interaction between dislocation and impurity atom segregation at grain boundaries [19, 20]. However, the strong temperature dependence of the threshold stress in superplastic MMCs cannot be fully explained by theories (a), (b), (c) and (e). On the other hand, theory (d) is also unable to explain the phenomenon because subgrains are not likely to form at the superplastic temperature and theory (f) cannot fully explain the relationship between the threshold stress and the matrix grain size. The theory of the interaction between dislocation and impurity atom segregation at grain boundaries proposed by Mohamed [20] seems to be able to give a better explanation to the origin of the threshold stress in superplastic MMCs, though it is still unable to fully explain the phenomenon.

According to Mohamed [20], distribution of solute around the boundary dislocation as a func- tion of temperature is approximately similar to that of solute atoms around a lattice dislocation. The following equation can be introduced [13]:

bb ( Iwl 2 = ~ o e X p t - ~-~7. (17)

The normalized threshold stress, ao/G, resulting from the pinning of boundary dislocation by solute atoms may be approximately given by

oo flwl' --~- = B expt~--~), (18)

concluded that the threshold stress increases with decrease of

The threshold stress, ao, is in fact the stress required to separate a boundary dislocation from solute atoms during the superplastic deformation. The threshold stress does not vary with applied stress. In order to obtain the threshold stress, the following empirical equation can be also used [20]:

A D G b ( b ~ V ( a - a ° ~ " (19) = t - - - - d - - )

Assuming m = (l/n), the equation can be written as

e '~ = C(a - ao), (20)

1 [ADGbCb~V] m C = - G L k T \ d J .J "

where

By extrapolating the best straight line of the double linear plot of e m vs a, the threshold stress can be determined.

RESULTS AND DISCUSSION

The composite material used in this study was a commercially available A1-Li based alloy (8090) reinforced with 17 vol% SiC particles. Superplastic tests were carried out under temperatures from 773 to 873 K and initial strain rates from 10 -4 to 10-1 s- 1. A maximum elongation of 300% was obtained at the strain rate of 1.83 x 10- ~ s- 1 at 848 K, which is slightly higher than the solidus temperature, i.e. 833 K. In high-strain-rate range between 10 -2 and 10- i s -~, a relatively high m value of 0.37-0.53 was found. The activation energy for high-strain-rate superplastic deformation is determined to be 342 KJ mol- 1 K - 1 and the interfacial diffusivity at the matrix/SiC interfaces,

310 K.C. Chan and B. Q. Han

DI, was calculated to be 1 x 10 -4exp( - 342000/RT)m 3 s-1. Furthermore, Dobb equals to 2.5 x 10-14 exp(10348.3/T)m 3 s-a [-21]. By assuming m (= 1/n) being equal to 0.5 and extrapolating the best straight line of the double linear plot of ~0.5 vs a as shown in Fig. 3, the threshold stress was determined to be 0.5 and 2.0 MPa at 848 and 823 K respectively.

Putting the above data into Eqns (8) and (16), the constitutive relationships relating strain rate and flow stress for 8090-SiCp composite material at temperatures just above (848 K) and below (823 K) its solidus temperature were determined and are listed as follows.

6 = 4.9392 x 10-3(a - ao) 2, (8A)

kt = 8.0335 x 10-4(a -- a0) 2. (16A)

Figure 4 shows a comparison between the theoretical predictions according to Eqns (8A) and (16A), and the experimental results, and a reasonable agreement was observed. Although in this paper it is shown to be able to determine the threshold stress by using the phenomenological equation of k" vs a, a theoretical model that can fully explain the origin of the threshold stress is required. Obviously, further research is required to predict the threshold stress in superplastic composites.

0.70

0,60

(NI --. 0.50 flJ

m 0.40 Or"

0.30 CO

L 4-J to 0.20

0.10

0.00 0.0 40.0

I ~ l I I I I

• •

• = • o

• = =

/~ / / / , / ~ * 823 K

/ • 798 K • o • o o /:/ob<./~ o 773 K

e " t " i I I I i I 5.0 10.0 15.0 20.0 25.0 30 .0 35.0

S t r e s s , M P a

Fig. 3. Determination of threshold stress•

102

O) Q_ T-

101

• Experimental Oa~a at 823 K

..... . Theore t ica l Data at 823 K

• Exper imenta l Data a t 8 4 8 K

-~- - Theore t ica l Data at 848 K • , . .41."

L p " ? m "

(-~ "" 7 /

0 -'"

/ /

• / /

/

1 0 0 , i i i i i x l l _ l ~ t i 1 l l l l l _ i t i L I h l l l I , l l , , I

10-4 10-3 1052 10L1 100

StraLn RaZe, S -~

Fig. 4. Comparison between the theoretical results and the experimental results of the 8090-SiCp composite•

High-strain-rate superplasticity of aluminum matrix composites 311

CONCLUSIONS

The model with the considera t ion of threshold stress has been developed to explain high-strain- rate superplastic behavior of part iculate reinforced MMCs. Superplastic deformat ion mechanism operated in part iculate reinforced a l u m i n u m alloy composites is considered to be GBS and IS when tested at temperatures near to the solidus temperature of the composite. The superplastic const i tut- ive equat ions for part iculate reinforced a l u m i n u m alloy composites were developed for the temper- atures above and below the solidus temperature of the composite. A reasonable agreement has been obta ined between the theoretical predict ion and the experimental findings. The model was found to be able to describe the superplastic deformat ion behavior of MMCs.

Acknowledgements--This research is support by the Hong Kong Polytechnic University grant under the code no. 350/396.

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