Formability of a high-strain-rate superplastic Al�/4.4Cu�/1.5Mg/21SiCW composite under biaxial tension

K.C. Chan *, G.Q. Tong

Department of Manufacturing Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 12 October 2001; received in revised form 20 February 2002

Abstract

The cavitation behavior and forming limits of a high-strain-rate superplastic 21 vol.% SiC whisker-reinforced Al�/4.4Cu�/1.5Mg

(Al�/4.4Cu�/1.5Mg/21SiCW) under biaxial stress states were investigated in this paper. The composite sheet was bulged using dies

with aspect ratios of 1:1, 4:3 and 2:1 at the constant applied stress of 4 MPa and at the optimal temperature of 793 K determined

from superplastic tensile tests. The thickness distributions of bulged diaphragms were measured at different strain levels. For

diaphragms deformed equibiaxially, a good agreement between experimental thickness distributions and the theoretical predictions

of Cornfield and Johnson (Int. J. Mech. Sci. 12 (1970) 479) was observed at fractional heights of the deformed diaphragms ranging

from 0.4 to 1.0. The cavitation behavior of the composite under biaxial tension was compared with that of uniaxial tension. It was

found that at a similar effective strain, the amount of cavities obtained under equibiaxial tension is slightly greater than that under

uniaxial tension, and the cavity growth rate parameter under uniaxial tension was also slightly larger than that of uniaxial tension.

The influence of stress state on cavity growth rate was discussed. Limit strains of Al�/4.4Cu�/1.5Mg/21SiCW at different stress ratios

were predicted based on a plastic damage model recently developed for superplastic materials (Chan and Chow, Int. J. Mech. Sci.,

submitted). The trend of the prediction was in good agreement with the experimental findings.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Aluminum-based composite; Superplasticity; Cavitation; Limit strains

1. Introduction

The discovery of high strain rate superplasticity

(HSRS) which is defined as the ability of a material to

achieve high tensile ductility at strain rates larger than

10�2 s�1 is significant because one of the major

disadvantages in the field of conventional superplasticity

is the low forming rate, typically in the range of 10�5�/

10�3 s�1. Since HSRS was originally reported in a 20

vol.% SiC whisker-reinforced Al2124 (Al2124/20SiCW)

[1] in 1984, many other materials, such as mechanically

alloyed aluminum alloys and its composites [2], magne-

sium-based composites [3] and zirconium or scandium

improved aluminum alloys [4], had demonstrated to

exhibit HSRS. HSRS materials are phenomenally asso-

ciated with grain sizes much finer than those conven-

tional superplastic monolithic materials. Furthermore,

the grain size of HSRS materials is relatively stable

during deformation as a result of large volume fraction

of ceramic whiskers/particles preventing excessive ma-

trix grain growth or dispersion of extremely fine

intermetallic particles pinning grain boundaries effec-

tively. One of the significant features of HSRS in

discontinuously reinforced metal matrix composites

(MMCs) is their optimal deformation temperatures

being close to or slightly above the solidus temperatures

of the composites [1,3]. Deformation models [5,6] for

HSRS MMCs have been successfully developed to

explain the experimental findings. The relatively high

forming rate of HSRS makes superplastic forming

technology more possible in commercial applications.

Near net-shape superplastic forming of sheet materials

has been well-documented for metal alloys [7�/10]. In

particular, gas pressure forming of superplastic materi-

als offers the specific technological advantage in a cost

effective and weight saving manner of producing net-

* Corresponding author. Tel.: �852-2766-4981; fax: �852-2362-

5267

E-mail address: mfkcchan@polyu.edu.hk (K.C. Chan).

Materials Science and Engineering A340 (2003) 49�/57

www.elsevier.com/locate/msea

0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 1 5 4 - 5

shape articles. Superplastic forming of Al-MMCs has

been explored in Al7475/SiCP [11], Al6061/20SiCW [12],

formability of Al2009/SiCW [13]. It was reported that an

Al6061/20SiCW hemispherical dome had been success-fully gas pressure formed in 17 seconds at the applied

flow stress of 4 MPa and at 873K [12]. Two main factors

influencing the formability of superplastic materials and

the quality of formed articles are thickness uniformity

and cavitation behavior. Assuming a spherical dome

profile, Cornfield and Johnson [8] have proposed an

analytical model to predict thickness variations in

superplastic bulging of domes, and demonstrated thatthe thickness non-uniformity increases as m value

decreases. Their predicted values are in good agreement

with the experimental findings. Yousif et al. [14] have

further reported that it is precise enough to assume an

elliptical profile in cold forming of sheet materials using

elliptical dies.

It is well established that most superplastic materials

undergo cavitation during deformation, and that thepresence of cavities may lead either to premature failure

or to degradation of the mechanical properties of post-

formed components. However, research work on the

cavitation behavior of HSRS Al-MMCs are at present

limited to Al6061/20Si3N4P [15], IN9021/15SiCP [16] and

Al2124/20Si3N4W(P) [17] and Al6061/20SiCW [18].

Furthermore, the cavitation behavior of Al-MMCs

deformed under complex stress states, which are com-monly observed in industrial processes, has drawn less

research attention. It has been demonstrated that cavity

growth is primarily controlled by the plastic flow of the

matrix rather than diffusion mechanisms. Some investi-

gations have been carried out for conventional super-

plastic aluminum alloys to examine the effect of stress

state on cavitation. It is interesting however to note that

some of the findings are conflicting [19,20]. More workneeds to be done to examine the effect of stress state on

cavitation of Al-MMCs.

There is also relatively less research work to examine

and predict limit strains of superplastic materials under

biaxial tension. Ding et al. [21] have developed a

simplified instability criterion based on the Marciniak

and Kucznski (M�/K) model [22] which assumes the

existence of an initial imperfection in the sheet whichwill develop into a groove. Based on the damage

mechanics approach and the M�/K model, Chan and

Tong [23] have developed an analytical model to predict

limit strains of a HSRS Al6061/20SiCW under equibiax-

ial tension. Chow and Chan [24] have recently extended

the damage model to predict the formability of super-

plastic Al5083 alloy under other biaxial stress states.

This model will be applied in the present paper, and thepredictions will be compared with the experimental

findings. The aim of this paper is hence to investigate

the thickness distribution and cavity behavior of a 21

vol.% SiC whisker-reinforced Al�/4.4Cu�/1.5Mg (Al�/

4.4Cu�/1.5Mg/21SiCW), and to utilize the damage model

[24] to predict the limit strains under biaxial tension.

2. Materials and experimental procedures

The Al�/4.4Cu�/1.5Mg/21SiCW composite sheet sup-plied by Advanced Composite Corp. was investigated in

present study. A typical scanning electron microscope

(SEM) micrograph of the composite is shown in Fig. 1,

indicating that the distribution of SiCW is reasonably

homogeneous. The solidus temperature of the composite

determined by a Du Pont 9900 differential scanning

calorimeter (DSC) was found to be �/792 K. The DSC

test was started at room temperature and ended at �/

873 K with a constant heating rate of 10 K min�1. It has

been reported by the authors [25] that the composite

exhibits a strain rate sensitivity exponent (m value) of

�/0.35 at the strain rate regime of 10�2�/100 s�1 and

at the temperature range of 723�/818 K. A maximum

elongation of about 335% was obtained by superplastic

tensile tests at the initial strain rate of 0.17 s�1 and at

the temperature of 793K which is defined as optimaltemperature. The corresponding flow stress is �/4 MPa.

It has also been reported that Al�/4.4Cu�/1.5Mg/21SiCW

composite exhibits the same m value of �/0.35 under

both uniaxial and equibiaxial tension [26].

Square Al�/4.4Cu�/1.5Mg/21SiCW discs of dimensions

65 mm�/65 mm�/2.28 mm were machined and photo-

printed with grid circles of 1.8115 mm in diameter.

Based on the tensile test results of the composite, HSRSgas pressure forming was performed in a fully compu-

terized tester [13] using dies with aspect ratio of 1:1, 4:3

and 2:1 at the constant applied stress of 4 MPa and at

the optimal temperature of 793 K. The surface strain

distributions of the gas pressure formed diaphragms

were obtained by measuring the grid circles under an

Fig. 1. SEM micrograph of as-received Al�/4.4Cu�/1.5Mg/21SiCW.

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/5750

optical microscopy, whereas the thickness strain of the

diaphragms was determined by using a ball point gauge.

Specimens were cut from the gauge area of the tensile

samples and from the top of the formed diaphragms.The volume fraction of cavities was determined by

measuring the specimen density in water with the

corresponding undeformed regions as the reference

density. After polishing, the specimens were also exam-

ined using a Leica Q500 image-analyzing system to

determine the volume fraction of cavity. Since the

volume fraction of cavities may be over-estimated by

the image analyzing technique due to the presence ofsecond phases, the measured volume fraction was

compared with that obtained by the density measure-

ment to ensure that errors are no more than 10%.

3. Experimental results

3.1. Strain distributions of bulged diaphragms

Fig. 2 shows the schematic diagram of a HSRS bulged

elliptical specimen. The die has an aperture of 2b in

length and 2a in width. The major and minor axes are

represented by OY and OX respectively. Based on theassumption that the major and minor profiles of the

diaphragm are approximately in circularity, the radii of

the curvature of the membrane in planes XOZ and YOZ

are described by rx �/(a2�/h2)/2h and ry �/(b2�/h2)/2h ,

respectively. For a circular bulging, i.e. a�/b , rx equals

to ry .

The thickness strain distributions of the composite

diaphragm bulged to the pole heights of 13.22 and 20.00mm using a die with aspect ratio of 1:1 are shown in Fig.

3. Based on the theoretical analysis of Cornfield and

Johnson [8], a computer program was written to predict

the thickness strain variations of superplastically bulged

domes. The theoretical thickness strains are also shown

in Fig. 3. The experimental results show a good

agreement with the theoretical values in the fractional

height (h /h0) ranging from 0.4 to 1.0. A relatively large

difference between the experimental and theoretical

results, however, occurs in the fractional height rangingfrom 0.0 to 0.4. The higher the dome height, the larger

the difference. One of the main reasons for the difference

is considered to be the fact that the equibiaxial tension

assumption used in the modelling work may not be valid

for most regions of the diaphragm [8]. In reality, the

equibiaxial tensile stress state only exists at the top of

dome. The dome edge is under the plane strain state and

most areas of the dome are subjected to biaxial tensilestresses. Moreover, the equilibrium equations of the

membrane theory which may also not be valid for the

region close to the edge, and materials which are allowed

to flow from the clamped portion into the die cavity are

other reasons for the discrepancy.

At pole heights of 12.99 and 9.32 mm, the strain

distributions of the diaphragm deformed in dies with

aspect ratios of 4:3 and 2:1 are shown in Figs. 4 and 5,respectively. It should be noted that the strain gradient

in the minor axis is much greater than that in the major

axis, and that the strain gradient observed in both axes

increases with increasing die aspect ratio. It is also

interesting to point out from Figs. 4 and 5 that the

volume strain, a o�/oX�/oY�/ot , does not equal to zero

and is inconsistent with the constant-volume principle.

This is considered to relate to the cavitation behavior ofthe HSRS Al-MMCs sheet during bulging, which

includes: (a) void nucleation; (b) cavity growth; and (c)

cavity aggregation and failure.

3.2. Cavitation behavior

Variation of cavity volume fraction with true strain of

the Al�/4.4Cu�/1.5Mg/21SiCW composite bulged equi-

biaxially at s�/4 PMa and stretched uniaxially at the

initial strain rate o0 of 0.17 s�1 [25] is shown in Fig. 6. Itis obvious that the amount of cavities increases with

increasing strain. Similar phenomenon has been re-

ported for some conventional superplastic metals andFig. 2. Schematic diagram of a HSRS bulged diaphragm.

Fig. 3. Comparisons between experimental thickness strain distribu-

tions (triangles and squares) with the theoretical distributions [8] (solid

lines) using m�0.35.

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/57 51

HSRS Al-MMCs [15�/19]. It reveals that, in general, the

amount of cavity resulting from equibiaxial tension is

slightly greater than that from uniaxial one. The

relationship between cavity volume fraction and strain

clearly illustrates an exponential relationship, indicating

that cavity growth is essentially plastic-controlled. Stow-

ell [27] proposed an analytical model to predict thevolume of cavities CV

CV�CV0exp(ho) (1)

where CV0

is the volume of cavities at zero strain, h is

the dimensionless cavity growth rate parameters, whichvaries according to material, grain size, strain rate and

temperature. CV0

and h values for the composite at

different stress ratios under biaxial tension are also

illustrated in Table 1. The cavity growth parameter

increases from 2.0 to 2.2 when the stress ratio increasing

from 0.6667 to 1.0. It is shown that a slightly larger h

value is observed under biaxial tension than that of

uniaxial one. The phenomenon is consistent with thereports of Pilling and Ridley [19]. They have reported

that the h values of Al7475 and Supal 220 are higher

under equibiaxial tension than under uniaxial tension.

For comparison purpose, the cavity behavior of

Al7475 alloy and Al6061/20Si3N4P deformed at theiroptimum conditions is also shown in Fig. 6. The h

values for HSRS Al�/4.4Cu�/1.5Mg/21SiCW, HSRS

Al6061/20Si3N4P [15] and Al7475 [15] under uniaxial

tension, as shown in Table 1, are 1.9, 0.5 and 3.6,

Fig. 4. Strain distributions on the (a) major and (b) minor axis for Al�/

4.4Cu�/1.5Mg/21SiCW diaphragm bulged in a die with aspect ratio of

4:3 at s�4 MPa and at 793 K.Fig. 5. Strain distributions on the (a) major and (b) minor axis for Al�/

4.4Cu�/1.5Mg/21SiCW diaphragm bulged in a die with aspect ratio of

2:1 at s�4 MPa and at 793 K.

Fig. 6. Variation of volume of cavity with true local strain for Al�/

4.4Cu�/1.5Mg/21SiCW deformed at 793K.

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/5752

respectively. A higher h value means that cavity is more

sensitive to plastic strain. It is noted that the testing

temperature of 788 K for Al7475 is lower than its solidus

temperature, thus the alloy contains no liquid phase

during deformation. The optimum deformation tem-

peratures are, however, close to/or slight higher than the

partial melting temperatures both for HSRS Al�/4.4Cu�/

1.5Mg/21SiCW and for HSRS Al6061/20Si3N4P. It is

likely that the role of liquid phase in the composite is to

effectively relax the stress concentration caused by grain

boundary/interfacial sliding and thereby to limit the

growth of cavity. The experimental findings on Al6061/

Si3N4P illustrating that a smaller h value is obtained at a

temperature (833 K), which is slightly above its partial

melting temperature, than at below it (818 K) [28],

further illustrate the effect of liquid phase.

Metallographic examinations of the specimens were

also performed to reveal the morphology and distribu-

tion of cavities. Fig. 7 shows the cavity morphologies of

the specimens deformed to strains of (a) 0.11 and (b)

2.00, (c) 0.27 and (d) 1.97, and (e) 0.18 and (f) 1.90 using

dies different aspect ratios. The results confirm that the

samples deformed to a higher strain level contain larger

cavities. It is found that voids are preferentially nu-

cleated at the matrix/reinforcement interfaces. Large

cavities with diameter of about 5 mm can be observed in

the heavily deformed sample, these cavities are the result

of interlinkage. The interlinkage is probably assisted by

the presence of liquid phases at the reinforcement/matrix

interfaces.

In addition to void nucleating at the interfaces

preferentially, it is reported that the number of cavities

for HSRS Al-MMCs tends to increase with increasing

strain [15,17,18], indicating that this is a continuous void

nucleation and cavity growth process. Whereas, there is

a decrease in the total number of voids per unit volume

or area with increasing strain in Al7475 as a result of

large cavity growth rate and cavity coalescence [29]. It is

considered that void nucleation, which occurs much

more easily at the interface as a result of weak bonding

force between the liquid phase and the solid, is thereason for the difference.

3.3. Forming limits

Fig. 8 shows the samples bulged to failure at the

constant applied stress of 4 MPa and at 793 K using dies

with aspect ratios of (a) 1:1, (b) 4:3 and (d) 2:1. The

experimental limit strains are reported in Fig. 9. It isshown that the major limit strains increase when the

stress ratio decreases. The shape of the curve is different

from that of conventional forming limit curve [22]. This

discrepancy is considered to relate to their differences in

deformation behavior. In conventional sheet metal

forming, strain hardening phenomenon is observed

and it plays an important role in neck formation, while

in superplastic deformation, a high m value representsthe material possesses a high capacity to resistance

necking duration deformation and cavity failure is the

dominant failure mechanism of the material.

4. Discussion

4.1. Influence of stress states on cavity behavior

For strain controlled cavity growth phenomenon, the

parameter of cavity growth rate, h, has been shown

from theoretical considerations as [30]

Table 1

CV0

(%) and h values for superplastic Al-MMCs and aluminum alloys

Materials T (K) /o (s�1) s (MPa) Stress m CV0

h Ref.

Al4.4Cu1.5Mg/21SiCW 793 0.21a �/ UTb 0.35 0.17 1.9 [25]

4c BTb 0.13 2.2 This work

4d BT 0.16 2.0 This work

4e BT 0.15 2.0 This work

Al6061/20Si3N4P 833 2.0 8 UT 0.50 0.27 0.5 [15]

Al7475 788 5.0�10�4 5 UT 0.50 0.01 3.6 [15]

Al6061/20SiCW 873 0.17a �/ UT 0.34 0.28 1.2 [18]

2 BT 0.31 1.6 [18]

Al7475 788 1.17�10�3 UT 0.53 �/ 2.4 [19]

BT �/ 5.6 [19]

Supral 220 733 1.17�10�3 UT 0.55 �/ 2.7 [19]

BT �/ 3.2 [19]

a Experiments conducted at constant displacement rate.b UT and BT means uniaxial and biaxial tension.c Stress ratio 1.0, i.e. die aspect ratio 1:1.d Stress ratio 0.8293, i.e. die aspect ratio 4:3.e Stress ratio 0.6667, i.e. die aspect ratio 2:1.

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/57 53

h�ks

2(1�0:932m�0:432m2)1=m (2)

where m is the strain rate sensitivity exponent, and ks is

a constant depending on the stress states and the extent

of grain boundary sliding which is given by,

ks

3�

s lm

se

(3)

where se is the von Mises equivalent stress, s lm is the

mean stress local to the grain boundary. Without grain

boundary sliding, ks is given by

Fig. 7. Cavity morphologies for Al�/4.4Cu�/1.5Mg/21SiCW deformed to equivalent strains of (a) 0.11 and (b) 2.00 using a 1:1 die, (c) 0.27 and (d)

1.97 using a 4:3 die, and (e) 0.18 and (f) 1.90 using a 2:1 die at s�4 PMa and at 793 K, respectively (sheet thickness*/vertical).

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/5754

ks(0)

3�

slm(0)

se

�s�

m

se

(4)

where slm is the applied (remotely) mean stress. For

freely sliding grains, ks is given by [31]

ks(100)

3�

slm(100)

se

�1

2

�s�

1

se

�s�

m

se

�(5)

/s�1 is the applied (remotely) maximum principle stress.

Since in superplastic deformation, approximately 50% if

the strain is derived directly from grain boundary

sliding/interfacial sliding, Eq. (3) is modified as follows:

ks(50)

3�

slm(50)

se

�1

2

�sl

m(0)

se

�sl

m(100)

se

�

�1

2

�ks(0)

3�

ks(100)

3

��

1

4

�3s�

m

se

�s�

1

se

�(6)

Table 2 illustrated the predicted and experimental

parameters of cavity growth rate for the composite. It is

shown that a higher h value is predicted under

equibiaxial tension than under uniaxial tension which

is in agreement with the experimental findings. Forstrain controlled cavity growth, the local mean stress

plays a major role in cavity growth behavior. Since the

local mean stress is proportional to the overall mean

stress, the h value should be higher under equibiaxial

tension than under uniaxial tension.

4.2. Comparison the experimental forming limits with the

model prediction

Recently, Chow and Chan have developed a limit

strain model based on the M�/K model and the damage

Fig. 8. Al�/4.4Cu�/1.5Mg/21SiCW diaphragms formed using dies with

aspect ratios of (a) 1:1, (b) 4:3 and (c) 2:1 at 793 K and s�4 MPa.

Fig. 9. Comparison between predicted and experimental forming

limits of HSRS Al�/4.4Cu�/1.5Mg/21SiCW assuming f0�0.999.

Table 2

Comparison between predicted and experimental h values under

different stress states

Stress ratio, b 0 0.6667 0.8293 1.0

ks (50) 1.5 2.27 2.29 2.25

Equivalent stress, se s1 0.8819s1 0.9265s1 s1 (s2)

Mean stress, sm s1/3 0.5556s1 0.6098s1 2s1/3

Theoretical h 1.5 2.26 2.28 2.24

Experimental h 1.9 2.0 2.0 2.2

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/57 55

mechanics approach [24] for a superplastic Al5083 alloy.

In this paper, this model is applied to predict the limit

strains of the HSRS composite. In their model, an initial

geometrical defect is assumed to exist and the inhomo-

geneity caused by an inhomogeneous distribution of

cavities is considered. The geometry of the total

inhomogeneity is shown in Fig. 10 and the initial

equivalent M�/K inhomogeneity index, denoted by F0

becomes

F0�f0

(1 � DB0)

(1 � DA0)(7)

where DB0 and DA0 denote the initial volume fraction ofcavities inside (region B) and outside (region A) the

groove respectively, and f0 (�/tB0/tA0) is the initial

inhomogeneity index due to the geometrical defect as

proposed by MK. In order to determine the limit strain,

the following equation has to be solved.ffiffiffiffiffiffiffiffiffiffiffiffiffi1 � B

py[t]mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � By[t] 2p

�f0

� 1 � DB0exp[ht]

1 � DA0exp[hx]exp

�Cx

�gt

0

fA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�By[t]2

q�Dy[t] gdt

�(8)

where

y�doA

doB

; x� oA; t� oB; A�

ffiffiffi3

p

2;

B�3a2

4(1 � a� a2); C�

ffiffiffi3

p

2

1 � affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � a� a2

p ;

D�

ffiffiffi3

p

4

affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � a� a2

p

For a set of values of m , h; DB0, DA0 and f0, Eq. (8)

can be solved numerically by the Runge�/Kutta method

for a given strain ratio, a , under the initial condition,

x�/0, t�/0 and y0, and the latter can be determined by

the following equation

ffiffiffiffiffiffiffiffiffiffiffiffi1�B

pym

0 �f 0

1 � DB0

1 � DA0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�By2

0

q(9)

The value of y will decrease as x and t increase, and

when the value of y equals to zero, the strain value of x

is regarded as the limit strain.

In this paper, the experimental initial cavity volume

fraction and the cavity growth rate parameters obtainedFig. 10. Geometry of initial inhomogeneity [24].

Fig. 11. Predicted limit strains for Al�/4.4Cu�/1.5Mg/21SiCW for

different f0 and DBo/Dao (dashed line*/experimental results).

K.C. Chan, G.Q. Tong / Materials Science and Engineering A340 (2003) 49�/5756

under different stress states are compared with the

prediction. Fig. 11 shows the predicted limit strains of

the HSRS Al�/4.4Cu�/1.5Mg/21SiCW in biaxial tension

for different f0 and DB0/DA0. f0 is allowed to vary from 1to 0.99 and DB0/DA0 varies from 1 to 2. It is considered

that in practice, f0 ranging from 0.995 to 0.999 is

regarded as reasonable for real aluminum or Al�/

MMC sheets. The values of DBo/DAo which range

from 1 to 2 is also considered to be reasonable in Al-

MMCs as the phenomenon of inhomogeneous distribu-

tion of voids can be easily observed in SEM, though it is

not intended in this paper to determine quantitativelythe DB0/DA0 values by measuring the areas of cavities in

different regions. It is worth to mention that in the

modeling works of Tai [32], the same range has been

used for soft aluminum. The predicted limit strain is also

found to increase with increasing f0 and decrease with

increasing DB0/DA0, and it is successful to predict limit

strains of the MMC sheet under biaxial tension without

assuming any initial geometrical defect. In case of f0

being 1, the best coincidence between predicted strains

and the experimental findings is obtained when DB0/DA0

lies between 1.3 and 1.8. If a small value of initial

geometrical inhomogeneity factor is assumed, the dif-

ference between DB0 and DA0 is reduced in order to

match the predicted limit strains with the experimental

findings. A comparison between the experimental and

theoretical forming limits of the composite is alsoillustrated in Fig. 9. The trend of the prediction is in

good agreement with the experimental findings.

5. Conclusions

In this paper, the thickness distribution, the cavitation

behavior and forming limits of a high-strain-rate super-

plastic Al�/4.4Cu�/1.5Mg/21SiCW sheet were investi-gated under biaxial stress states. For diaphragms

deformed equibiaxially, a good agreement between the

experimental thickness distributions and the predictions

based on the analytical model of Cornfield and Johnson

was observed at the fractional heights (h /h0) ranging

from 0.4 to 1.0. It is considered that the relatively large

discrepancy occurred at the fractional heights lying

between 0 and 0.4 is due to the oversimplified assump-tions used in the analytical model. It is found that the

parameter of cavity growth rate of the composite is

dependent on stress state, and that it increases with

increasing stress ratio, which is consistent with the

theoretical values. The parameter of cavity growth rate

of 2.2 obtained under equibiaxial tension is shown to be

slightly higher than that of 1.9 under uniaxial tension.

Forming limits of high-strain-rate superplastic Al�/

4.4Cu�/1.5Mg/21SiCW at different stress ratios were

predicted based on a plastic damage model recently

developed for superplastic materials [24]. The trend of

the prediction was in good agreement with the experi-

mental findings.

Acknowledgements

This work described in this paper was supported by a

grant from the Research Grant Council of The HongKong Special Administration (project no. PolyU5162/

99E). Support from The Hong Kong Polytechnic

University is also acknowledged.

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