The deformation of particle reinforced metal matrix composites during temperature cycling

Acta metall, mater. Vol. 38, No. 12, pp. 253%2552, 1990 Printed in Great Britain. All rights reserved

0956-7151/90 $3.00 + 0.00 Copyright © 1990 Pergamon Press plc

THE D E F O R M A T I O N OF PARTICLE REINFORCED METAL MATRIX COMPOSITES D U R I N G TEMPERATURE CYCLING S. M. PICKARDI" and B. DERBY Department of Materials, University of Oxford, Parks Road, Oxford OXI 3PH, England (Received 17 January 1990; in revised form 16 May 1990)

Ahstruet--Superplasticity during temperature cycling of particle reinforced metal matrix composites has been studied over a range of reinforcement sizes and volume fractions. Above a critical volume and thermal cycle amplitude, the mean strain per cycle is proportional to stress and approximately proportional to cycle amplitude. For a given thermal cycle the constant of proportionality with respect to stress increases with reinforcement fraction to a maximum at around 30%; it then decreases with further increase in reinforcement. Transmission electron microscopy revealed no characteristic dislocation substructure; even after 90% strain the material was indistinguishable from its undeformed state. The experimental results confirm an internal plastic flow model for the phenomenon rather than an enhanced creep. A model of the process derived from the Ldvy-Von Mises equations predicts both the effect of thermal cycle amplitude and MMC microstructure on the enhanced creep rate. Rrsmnr--La superplasticit6 pendant le cyclage thermique de composites fi matrice mrtallique renforcrs par de particules est 6tudire pour une gamme de tailles du renfort et de fraction volumique. Au dessus d'une fraction volumique critique et d'une certaine amplitude du cycle thermique, la drformation moyenne par cycle est proportionnelle fi la contrainte et approximativement proportionnelle fi l'amplitude du cycle. Pour un cycle thermique donnr, la constante de proportionnalit6 par rapport ~ la contrainte croit avec la fraction du renfort jusqu'fi un maximum autour de 30%, elle drcroft ensuite pour une augmentation ultrrieure du renfort. La microscopic 61ectronique en transmission ne rrvrle aucune sous-structure de dislocations caractrristique, m~me aprrs 90% de drformation la mat~riau ne peut ~tre distingu~ de l'rtat non drformr. Les rrsultats exprrimentaux confirment un modrle d'rcoulement plastique interne pour le phrnomrne plut6t qu'un renforcement du fluage. Un modrle du processus drriv6 des 6quations de Levy et Von Mises prrvoit fi la fois l'effet d'amplitude du cycle thermique et la microstructure du composite pour des vitesses de ftuage renforcres.

Zusammenfassung--Das superplastische Verhalten w/ihrend des Temperaturzykelns einer teilchenverstfirkten Verbundmaterials mit Metallmatrix wird in Abh/ingigkeit von TeilchengrrBe und Volumanteil untersucht. Oberhalb eines kritischen Volumanteiles und einer thermischen Amplitude ist die mittlere Dehnung pro Zyklus proportional zur Zyklusamplitude. Bei einem gegebenen thermischen Zyklus nimmt die Proportionalit/itskonstante bez/iglich der Spannung mit dem Volumanteil der Teilchen bis zu einem Maximum bei etwa 30% zu. Danach sinkt sie mit weiterer Zunahme des Volumanteils. Im Elektronenmikroskop beobachtet man keine charakteristische Versetzungssubstruktur; auch oberhalb yon 90% Dehnung lieB sich das Material nicht von seinem unverformten Zustand unterscheiden. Die experimentellen Ergebnisse best/itigen f/Jr diese Erscheinung ein Modell des inneren FlieBens, und nicht des beschleunigten Kriechens. Ein von den Gleichungen von Lrvy-Von Mises abgeleitetes Modell dieses Prozesses sagt sowohl den EinfluB der thermischen Zyklusamplitude als auch der MMC-Mikrostruktur auf die beschleunigte Kriechrate voraus.

1. INTRODUCTION M e t a l m a t r i x composites ( M M C s ) are engineering materials in which a h a r d ceramic c o m p o n e n t is dispersed in a ductile metal m a t r i x to produce high specific s t r e n g t h a n d stiffness. The ceramic comp o n e n t can be in a n u m b e r o f shapes, e.g. long fibres, short fibres, whiskers a n d particles. Several ceramic c o m p o s i t i o n s have been utilised in M M C s a n d a representative list is s h o w n in Table 1 a l o n g with some light metal matrices. All f a b r i c a t i o n techniques 1"Present address: Department of Materials, Univesity of California, Santa Barbara, CA 93106, U.S.A.

developed to m a n u f a c t u r e M M C s operate at elevated t e m p e r a t u r e s either just a b o v e or just below the melting p o i n t of the matrix. O n cooling to r o o m t e m p e r a t u r e after fabrication, the difference between the coefficients of t h e r m a l expansion (CTE) of rei n f o r c e m e n t a n d matrix (Table 1) will generate considerable internal strains. This creates associated internal stresses at the interphase interfaces which m a y exceed the yield point, causing plastic flow a n d dislocation generation. I n t e r n a l stresses have been m e a s u r e d in M M C s by the use of n e u t r o n diffractions [1, 2] a n d calculated from differences between compressive a n d tensile mechanical properties [3]. Dislocation generation has been observed directly by in situ

2537

2538

PICKARD and DERBY:

DEFORMATION OF METAL MATRIX COMPOSITES

Table 1. Properties of metal matrices and typical ceramic reinforcements used in metal matrix composites Elastic Material

AI Mg Ti SiC (pure) SiC fibre (Nicalon) AI203 (pure) Al2OflSiO2 fibre (Saffil)

modulus (GPa) 70 45 108 440 180 360 103

Coefficient of thermal

structure (e.g. volume fraction and particle size of reinforcement) were made to determine their effect on the deformation process.

expansion (K ~× 10 6) 23.1 25.7 8.4 4.8 3.8 8.1 5.4

experiments in the transmission electron microscope [4] and inferred from the dependence of composite yield strength on quench conditions [5, 6]. During controlled thermal cycling severe microstructural damage is seen to accumulate in both continuous and discontinuous reinforced MMCs. In continuous MMCs there appear to be two dominant mechanisms. One is an elasto-plastic "shakedown" followed by "ratcheting" [7, 8] which leads to a permanent deformation of the composite and possible separation of fibres and matrix after thermal cycling. The other is the generation of interfacial voids during cycling and subsequent debonding of the interface [9]. The thermal cycling behaviour of discontinuous reinforced MMCs has been less well charaeterised, but dimensional changes have been observed under zero load conditions [10]. However, when discontinuously reinforced MMCs are thermally cycled while under small applied loads a very different behaviour is observed. Under these conditions the MMC undergoes an accelerated creep deformation and can extend under tension to strains of several hundred percent without fracture [11]. This superplastic behaviour occurs with a creep stress exponent close to 1, as would be expected for conventional superplasticity. Superplasticity, or anomalously high creep rates, during thermal cycling have been observed in both SiC whisker [1 l, 12] and SiC particulate [13, 14] reinforced MMCs. Enhanced deformation or superplastic behaviour during thermal cycling has also been observed in polycrystalline materials which have anisotropic CTE [15, 16], in materials cycled through a phase change [17, 18], and in ~-uranium during irradiation with neutrons [16, 19]. In all these cases substantial internal stresses are generated. There have been a number of mechanism models put forward to explain these superplastic phenomena [15, 17-22]; a critical review of these proposed mechanisms will be the subject of a companion paper [23]. In this paper we present a study of this superplastic effect in a model MMC of SiC particles reinforcing a commercially pure aluminium matrix. In order to generate information useful for deciding which mechanism model is most appropriate for thermal cycle deformation of MMCs' a series of experiments were carried out under different cycling conditions. In addition, significant variations in M M C micro-

2. EXPERIMENTAL The material chosen for this study was in the form of an extruded rod of commercially pure A1/SiC composite containing either 10 or 2.3#m mean diameter reinforcement particles in 5-40% volume fractions (Vp). The manufacturing route was that of powder metallurgy, with mixing of metal and reinforcement before hot pressing and extrusion into rods at 550°C. Dumb-bell test specimens of 3 mm diameter and 6 m m parallel gauge length were machined from the composite parallel to the extrusion direction. Some of the testpieces were mechanically polished to a mirror like surface finish using 3 to 1/4 # m diamond pastes so that surfaces could be examined optically after thermal cycling. A thermal cycling rig was specially constructed for the study. Rapid heating and cooling was supplied by forcing hot air through a low thermal mass furnace chamber [see Fig. l(a)]. Controlled heating rates of over 200°C/min were possible. The temperature and cycle duration were controlled by a combination of timing switches and a PID temperature controller. This enabled the temperature to be rapidly driven to a set point and then dwell for a pre-set time under temperature control before the cooling part of the cycle came into operation. A typical temperature cycle used in this study is shown in Fig. l(b). Forced air cooling was used to provide cooling rates similar (a]

~

---Thermocoupie

. ~

Heate~ ~'I ForcedDraught

LOAD 5OO

E

~o

o

(b)

/ I

2

I

I

4 6 Time(minutes)

I

8

I

10

Fig. I. (a) Schematic view of the thermal cycling furnace used. (b) Representative thermal cycle from the furnace.

PICKARD and DERBY: DEFORMATION OF METAL MATRIX COMPOSITES to those of heating, although the cooling rate was not actively controlled. The thermocouple used to monitor and control temperature was placed in the drilled out end portion of the testpiece and was positioned as close as possible to the gauge length. For calibration purposes some testpieces were drilled through the centre so that an additional thermocouple could be placed in the middle of the testpiece to measure the temperature. In all cases the temperature in the middle corresponded closely with that at the shoulder. Isothermal creep tests were performed on the same rig to allow comparison with thermal cycling deformation rates. In view of the small quantity of composite available, the tests comprised of incremental increases in stress during one series of cycles. This procedure appeared valid in light of the total absence of a primary creep phase with the phenomenon [14], unlike the behaviour of MMCs during isothermal creep. Figure 2 shows the behaviour of the MMC specimen under thermal cycling when changes in load occurred. The consequent change in stress is accompanied by an instantaneous increase in strain rate with no transient greater creep rate. The decrease in strain rate on load drop is equally sudden with no period of zero creep or diminished transient creep rate. In spite of this, the specimens were allowed to extend by strains of 10-15% at a given stress before the stress was increased to a higher level and another deformation rate recorded. The strain rate was monitored by a pair of extensometers accurate to + 2 pm. The measured strains were corrected for thermal expansion of the rig in real time using a microcomputer system. TEM examination of the composite was undertaken using thin foils produced by electropolishing of transverse and longitudinal testpiece sections. This was thought to be the most suitable technique to preserve any delicate substructure introduced by the deformation process. An electrolyte containing 20% nitric acid in methanol and a 2-jet electropolishing machine at a temperature of - 3 0 ° C was used. Ion

2539

milling of the perforated foils was sometimes used as a second stage of preparation after initial examination to study the particle/matrix interface more closely. A dual gun " G A T A N " ion mill was used at 4 kV accelerating voltage and a gun current of 50 mA. All the thin foils were examined on a JEOL 200CX microscope operating at 200 kV. 3. MECHANICAL TESTING The LVDT gauges attached to the extensometers produce a constantly changing signal during each thermal cycle. Consequently, the deformation of the material must be characterised by measuring specimen length at a fixed point within each cycle. Thus any strain rate quoted was calculated by measuring the strain per thermal cycle and dividing by the cycle time. Figure 3 shows the response to loading during thermal cycling of MMCs fabricated with 2.3 #m SiC particulate reinforcement. Three mean volume fractions of particles (lip) were used: 20, 30 and 40%. All three MMCs were tested under two thermal cycles with 7min period of 130-450°C and 130-350°C temperature range. The 20% V~ material was also tested with a 80-200°C cycle. The results are presented in the form of log (strain rate) against log (stress). For superplastic behaviour to occur, i.e. very large extensions without necking prior to failure, conventional mechanical arguments require the behaviour to show a strain rate sensitivity exponent (m) given by

~(~)m

(1)

with m equal to 1. Although in practise values of m from 0.4 to 1.0 can still result in significant extensions before failure. From Fig. 3 it can be seen that m (determined using a logarithmic interpolation function) is close to 1 with all three volume fractions of the MMC at the largest temperature cycle. However, at the 130-350°C cycle m is consistently smaller for all volume fractions and the 40% Vp material 9 -

18

8 __(b)

-(a)

I

16 14 6 ~

10

~=30MPa

/

.~///

"J

"- 4

6 ~

4

2 1

2

.~-I 0

12

I

I

I

I

I

I

1

24

36

48

60

"i'2

84

96

Time (x 10 3 s)

0

5

10

15

20

25

30

35

Time (x 10 z s)

Fig. 2. (a) Behaviour of 20% Vp, 2.3 ~ m SiC material thermally cycled between 130 and 350°C under load. There is no primary creep at t = 0, nor are there transient enhanced or reduced creep rates after a stress increase or a stress drop. (b) The behaviour of the same material as in (a) under isothermal creep conditions at 350°C, with 34.5 MPa stress.

2540

PICKARD and DERBY: DEFORMATION OF METAL MATRIX COMPOSITES Temperature) Cycle ) [] • O • +

130-450 (1) 130-450 (2) 130-350 (1) 130-350 (2) 80-200

(o)

2 0 % Vp

10-5 130-450 n=1.1 m=0.92 130-350 n=1.4 m=0.72 z:l= • 80-200 n=11.0 m=0.09 m O O • r'zm

A

(D

_=

"

10-6

+

O

0=

+

0•

,•

+

0 10-7

÷

. . . . . . . .

|

. . . . . . . .

10 Stress (MPa)

10 0 ITemperature Cycle J

I Temperature ) Cycle J

[] • O • o

[] 130-450 (1) •' 130-450 (2) • 130-350 3 0 % Vp

(b) 10-5

(e)

w

[]

130-450 n=l.0 m=l.00 130-350 n=2.4 m=0.42 A

,g

o

_=

4 0 % Vp

10"5.

130-450 n=1.3 m=0.72 130-350 n=1.6 m=0.63 [] a [] •

A

(D

10-6

130-450 (1) 130-450 (2) 130-350 (1) 130-350(2) 130-350 (3)

o

.E = f=

,-, ©

10-6.

,.a

j=

[]

O

[]

o O O

10-7

. . . . . . . .

i

10

. . . . . . . .

o

10-7

100

Stress (MPa)

. . . . . . . .

lb

. . . . . . .

;00

Stress (MPa)

Fig. 3. The behaviour of the 2.3/~m SiC particle reinforced MMC during thermal cycles and 130-450°C. Stress sensitivity (n) and strain rate sensitivity (m) exponents were determined by an interpolation function for 3 volume functions of reinforcement: (a) 20% Vp, (b) 30% lip and (c) 40% lip. shows a value of m ~ 0.4. Also at the cycle between the lowest limits (80-200°C), very low creep rates were measured and an exponent of under 0.1 was determined. A similar series of tests were carried out on the large SiC particle reinforced M M C which was available in 3 volume fractions, but over a lower range at: 5, 10 and 20%. These materials were only tested at one thermal cycle of 7 min period and 130-450°C temperature range with the results shown in Fig. 4. In this case both the 10 and 20% lip M M C s showed strain rate sensitivities close to 1, but the material of 5% Vp had a very low value of m. If the different

reinforcement sizes in the 20% Vp material are compared in Figs 3 and 4 it is noticeable that the larger particle sized material showed a consistently greater strain rate at all stresses during the 130--450°C thermal cycle. The 2.3 # m SiC particle reinforced M M C was also tested under isothermal conditions in the same equipment (Fig. 5). In this case a high stress exponent is measured with a constitutive relation of n = 15, i.e. g = A~r 15.

(2)

F r o m this a value of the strain rate sensitivity (m = l / n ) is determined as m = 0.07. Hence, with the

PICKARD and DERBY: (o)

DEFORMATION OF METAL MATRIX COMPOSITES

5% Vp

2541

10-5.

10-5 n=4.0 m=0.25 A

[] [][!

n=15.7

O 350 °C

n=15.0

(D

=t .E E

<>

[] 200 °C

0

10-6.

_=

10-6

[]

s*

[] []

•D

10-7 10-7

!

lO

........

1'O ....... Stress (MPa)

(b )

;oo

1 0 % Vp

n=0.73 m=1.37

o

[]

[] []

10-6. []

10-7 . . . . . . . .

I'0

. . . . . . .

;oo

Stress (MPa) (c)

2 0 % Vp

10-5.

D

n=0.92 m=1.09

Pi

•

,

rl ....

lOO

Stress (MPa)

Fig. 5. Isothermal creep rate as a function of stress for the 2.3#m, 20% Vp MMC. A stress exponent of n ~ 15 (d = A *a") was measured at both temperatures.

10-5.

,=_ E

. . . . . . . .

[]

[] [] O

10-6

10-7 ........

1'0 ....... ;oo Stress (MPa) Fig. 4. The behaviour of the 10/~m SiC particle reinforced MMC during thermal cycles of 7 rain duration and temperature range 130-450°C. Stress sensitivity (n) and strain rate sensitivity (m) exponents were determined by an interpolation function for 3 volume functions of reinforcement: (a) 5% V v, (b) 10% Vp, (c) 20% Vp.

exception of the 80-200°C thermal cycle, all creep tests during thermal cycling resulted in a considerable increase in m. Only the M M C s of volume fraction of 30% and below were found to deform to true superplastic extensions. It was only possible to extend to 150% strain in our test rig and these lower lip materials did so without fracture (Fig. 6). The 40% Vp material would only extend to about 60% strains before fracture occurred, but even this is much

greater than the strain to failure of the isothermal specimens. The various mechanism models suggested for this phenomenon predict a linear relation between strain rate and stress, i.e. rn = 1 in equation (1). In order to test some of the predictions of these models, the thermal cycle data for the materials which had values o f m close to 1 were replotted using a linear set of axes (Fig. 7), with the strain per individual cycle now plotted against stress. This can now be used to calculate a relation Ae = Ktr

(3)

where AE is the strain per cycle and K is a constant of proportionality with respect to stress. K was calculated from the data of Fig. 7 by using linear interpolation and the results are tabulated in Table 2. The variation of K with V p , and thermal cycle amplitude is very interesting. For the 2.3/~m SiC M M C the increase in A T (i.e. thermal cycle amplitude) leads to a closely corresponding change in AE (strain per cycle) for both the 20 and 30% lip material. On increasing A T, the cycle amplitude, from 130-350 ° to 130-450°C, AE increases by a factor of 1.8 at 20% lip and for the 30% Vp M M C the increase is by 1.7. Both the 20% Vp and 30% Vp M M C s have similar values of K but increasing the reinforcement

Fig, 6. 150% extensions in the 20 and 30% Vp, 2.3/~m SiC specimens after thermal cycling under load.

2542

PICKARD and DERBY:

DEFORMATION OF METAL MATRIX COMPOSITES

(a) from the same thermal cycle of 7 min between 130 and 450°C. All the mechanism models put forward to explain thermal cycle superplasticity [15, 17-22] rely on the temperature cycle to generate internal stresses because of local differences in CTE. Within M M C s these are caused by differences between the components of the material (Table 1). Hence we would expect details of the temperature cycle to affect the deformation behaviour and this is confirmed by the differences seen in Fig. 3 between tests carried out over various cycles. A systematic study was also carried out on the 20% Vp 2 . 3 / t m SiC reinforced M M C with an extensive investigation of the effect of thermal cycle amplitude and period on deformation. In Fig. 8 two experiments are illustrated; in Fig. 8(a) the top temperature of the cycle was held fixed at 350°C and a series of thermal cycles were carried out under the same load with different bottom temperatures. As the temperature interval (AT) increases, the strain per cycle (AE) also increases. Figure 8(b) shows a complementary experiment where the bottom temperature of the cycle was held at 130°C and A T varied by changing the top temperature. In both figures A T and AE seem to be almost linearly related and if an intercept is drawn a critical minimum temperature

lo-S. n Vp 20% = Vp 30% ,~, Vp 40% ~ ,",

5x10"6

_=

do

[]

.

0

10

2'0 Stress (MPa)

30

2'0

30

(b) 10-5

Vp 10% Vp 20%

D @

o

~1ff6 ' O

_¢

@

E 0

10

0

$ t m U (Mm~)

(c) 10"5. 'I o

=

(a)

Vp 20% Vp 30%

2x1 4>

5x10 "6

E

#J

[] 0

,,s

o

4)

10-4

i =

30 Stress (MPa)

E

Fig. 7. The data shown in Fig. 3 is replotted on a linear set of axes for the results where 0.7 < m < 1.4, for various volume fractions of reinforcement during thermal cycles of: (a) 130-450°C, 2.3 #m SiC MMC, (b) 130-450°C 10 #m SiC MMC and (c) 130-350°C, 2.3/~nv SiC MMC. fraction to 40% leads to a large decrease in K. Conversely with the larger SiC particle M M C , K increases by a factor of 2 with an increase in volume fraction from 10 to 20%. In addition K for the larger particle M M C at 20% reinforcement is approximately double that for the smaller reinforced material Table 2. Variation in gradient K (strain per cycle/stress) with reinforcementparameters and thermal cycle. Data taken from Fig. 7 Particle Thermal size lip cycle K (pm) (%) (°C) A~(MPa-') 2.3 20 130-450 1.21 x 10-4 2.3 30 130--450 1.23 X 10 -4 2.3 40 130~.50 5.17 x 10-s 2.3 20 130-350 6.88 x 10-s 2.3 30 130-350 7.27 x 10-s 10 10 130-450 1.21 x 10-4 10 20 130-450 2.46 x 10-4

/

m ¢0

in

AT" = 124°C ,

o

. ~

16o

200

360

400

AT (b) 2x10-3

/

.~

i

10.3'

i 0

AT* = 150"(3.

0

J ~

100

.

200

,

300

.

400

AT Fig. 8. The influence of thermal cycle magnitude (AT) on strain per cycle (AE) for 20% lip, 2.3pro SiC MMC. (a) t7 = 2.85 MPa with T ~ fixed at 350°C. (b) a = 9 MPa with T~. fixed at 130°C. In both (a) and (b) a linear intercept is drawn to give a critical temperature change (AT*) at AE = 0.

PICKARD and DERBY: DEFORMATION OF METAL MATRIX COMPOSITES

2543

"~ 3 x 10-3. a

-.~

~

2x10

[]

I;I

-3

10_3

0

0

28o

460

66o

8So

1000

PIGteeu Time (s)

Fig. 9. Variation in strain per cycle (AE) with increasing length of time at the high temperature plateau for the 20% lip, 2.3l~m SiC MMC. a = 9MPa, thermal cycle range 130-45ff'C. amplitude (AT*), in the range 130-150°C, can be estimated for the superplastic effect to occur. AE was found to be largely unaffected by the period of the thermal cycle. There was very little control available over the shape of the heating and cooling curves. Thus, only by reducing or extending the periods when the temperature was held constant could a limited control over cycle period be achieved. This was tested by varying the time held at the high temperature plateau of the cycle. Figure 9 shows the results of these experiments. Except for the very shortest cycle period, the strain per thermal cycle is effectively constant for all thermal cycle durations. The complete mechanical testing data are presented in Appendix A. 4. MICROSTRUCTURAL STUDY All MMC materials studied were fabricated by an identical powder route consisting of mixing A1 and SiC powders followed by hot pressing and extrusion. The 40% reinforced MMC underwent a series of hot rolling operations rather than extrusion. A typical microstructure of the small SiC particle material is shown in Fig. 10(a), the larger SiC particle material was always better homogenised [Fig. 10(b)] presumably because the metal and ceramic precursor

Fig. 10. Optical micrographs of MMC material as received before testing: (a) 2.3/tm SiC, 20% Vp; (b) 10 ~m SiC, 20%

zp.

particles were of the same order of size. Deformation during thermal cycling resulted in a gradual homogenisation of the microstructure. In Fig. 11 the microstructures of a 20% volume fraction 2.3 ~m SiC MMC specimen are shown prior to deformation and after an extension of 90%. A further experiment was carried out on a highly polished region of the surface of a test specimen. This area was examined under an optical microscope after various amounts of strain. In Fig. 12 a region of the microstructure of the specimen surface is shown after 50, 65 and 90% strain. A region in the centre can be seen where the interparticle spacing between closely packed SiC particles remains unchanged but a large region free of reinforcement deforms extensively. The surface of the 40% reinforced material showed extensive void

Fig. I 1. Microstructure of 20% Vp, 2.3/~m SiC MMC (a) as received, (b) after 90% strain during thermal cycling. The stress axis is horizontal in both cases.

2544

PICKARD and DERBY: DEFORMATION OF METAL MATRIX COMPOSITES

nucleation around the SiC particles which linked to generate final fracture after strains of around 60%. No such behaviour was seen in the 20 and 30% Vp materials at extensions up to 150%. The 40% reinforced MMC displayed a strain rate sensitivity (m) close to 1 during the 130-450°C cycle and no reason for its premature failure was determined. An extensive TEM investigation was carried out on the small SiC particle 20% Vp MMC after various strains. In its as received state, the composite shows a fine grain size of about 2 pm with a significant Fig. 13. TEM micrograph of 20% Vp, 2.3 pm SiC MMC in the as-received state. Note the moderately high dislocation density with sub-grain size similar to the interparticle spacing. dislocation density which is believed to be generated on cooling from its fabrication temperature (Fig. 13). Similar microstructure and dislocation densities in undeformed MMCs have been reported by other workers [4, 6]. After extensive plastic strain during thermal cycling, the dislocation microstructure showed very little change (Fig. 14). Large amounts of dislocation debris in the form of small dislocation loops were visible within the grains, indicating a continuing dislocation interaction during thermal cycling. No distinct substructure was seen which would be expected if dynamic recovery had occurred. Little substructure is expected if easy matrix flow conditions exist and the microstructure is reminiscent of elevated temperature deformation of pure aluminium. Careful observations of sub-grain size in the cycled material showed some sub-grain growth (Fig. 15) in regions depleted of SiC, indicating a possible pinning role of the SiC particles in other parts of the microstructure. Figures 13 and 15 were produced from specimens thinned by electropolishing; some other specimens were ion-milled to resolve the region close to the particle/matrix interface (shown in Figs 14 and 16). By this technique particles embedded deeply in the microstructure could be revealed. In Fig. 16(a) the interface region shows some evidence of poorly formed dislocation networks. Many of the dislocations near the interface appear bowed [Fig. 16(b)], indicating the presence of large residual stresses in these regions. No reaction layer was observed at the SiC/AI interface. 5. DISCUSSION

Fig. 12. Optical micrographs of polished surface of MMC test specimen at various extensions during thermal cycling. A common area is shown of a 20% Vp2.3 pm SiC specimen cycled 130-350°C after strains of: (a) 50%, (b) 65% and (c) 90%. In all 3 cases the stress axis is horizontal.

This study confirms the existence of an enhanced creep deformation during the thermal cycling of particulate reinforced metal matrix composites. At strain rates of 10-4-10 -6 s -j this can lead to superplastic extensions in excess of 150% with reinforcement volume fractions up to 30%. The creep rates reported in this study are lower than those reported for whisker reinforced MMCs [11, 12], but this is

PICKARD and DERBY:

DEFORMATION OF METAL MATRIX COMPOSITES

probably because of the shorter thermal cycles used in those studies, rather than any fundamental difference in the response of whisker and particulate reinforcements to thermal cycling. This is confirmed by the similar behaviour of whisker reinforced MMCs when tested parallel or perpendicular to the main whisker alignment direction [12]. To summarise, our findings concerning the mechanical response of particulate MMCs to thermal cycling under load indicate the following. The enhanced creep rate and large extensions prior to failure were found at all volume fractions above 5% reinforcement for deformation during the largest amplitude thermal cycle of 130~50°C. Under these conditions the strain per thermal cycle and calculated strain rate were approximately linear with applied stress. The coefficient of proportionality (K) between Ae and tr is seen to increase with increasing Vp to peak in the region between 20-30% and then fall away Undeformed (as received)

After

2545

llJm~ Fig. 15. Region of sub-grain growth in MMC microstructures after thermal cycling. with further increase. A similar increase in apparent strain rate with increasing reinforcement volume fraction from 10 to 20% has also been reported for the thermal cycling of whisker reinforced MMCs [12]. The strain per thermal cycle is almost independent of the cycle duration but is strongly dependent on the cycle amplitude. In fact from Fig. 8 there appears to be a critical temperature amplitude, in the region of 130-150°C, which must be exceeded for the effect to occur with the 20% Vp 2.3/~m SiC MMC. The microstructural study has produced intriguing results. Despite strong evidence for the generation of large dislocation populations in MMCs during rapid changes in temperature [4-6], the microstructure of these composites after deformation during thermal cycling shows comparatively low dislocation

9 0 % creep strain

Fig. 14. TEM micrograph of 20% lip, 2.3/~m SiC MMC microstructure, (a) before and (b) after 90% strain. In both micrographs an essentially similar substructure is seen.

Fig. 16. Dislocation debris near particle/matrix interfaces after 90% thermal cycling strain. (a) Some dislocation network formation near the interface. (b) Bowed dislocation loops very close to the interface.

2546

PICKARD and DERBY:

DEFORMATION OF METAL MATRIX COMPOSITES

densities. These were certainly no greater than those found prior to deformation, even after 90% plastic strain. This implies a full recovery of the deformation within each thermal cycle but none of the dislocation structures normally associated with a recovered matrix were observed. The mechanisms that have been proposed to explain the enhanced creep during thermal cycling are essentially based on one of two models. In the first series of mechanisms the internal plastic flow, generated by differences in thermal expansion coefficient after local yielding, is biased by an external stress to create a small strain increment per thermal cycle in the direction of the applied stress. The original paper of Anderson and Bishop [15] used the Lrvy-Von Mises equations to model this process. A slightly modified form put forward by Greenwood and Johnson [17] is most applicable to the current material, i.e. Ae =

5 AVax

(4)

6 Vrry

where A V~ V is the internal volume change associated with a temperature change which is accommodated by plastic flow. ax is the applied stress and try is the matrix yield stress. The second model was also first discussed by Anderson and Bishop [15]; in this the thermal expansion difference generates an internal elastic stress. This internal stress is of different sign but assumed identical magnitude on heating and cooling. If the matrix creeps with a power-law type relation, i.e. = A~"

(5)

then the internal stress (ai) will influence the creep rate, aiding it during one half cycle and impeding it on the other = 2 [(IG + a i l ) " - ( I G - ail)"].

(6/

This model has recently been developed further by Wu et al. [21] who use a hyperbolic sinh relation suited to high internal stress levels for the creep equation. This was used to show that, if G << ai, a linear relation existed between the enhanced strain rate and the applied stress / ~ //O- i "~n - 1 O-x

d =nK-~-~)

--E

(7)

where b is the matrix Burger's vector, E its elastic modulus, and /~ the matrix diffusion coefficient averaged over a thermal cycle. This averaging was used to account for the continually changing creep rate during the thermal cycle. Wu claimed a very good:fit with experimental results obtained by thermally cycling anisotropie metals [21]. Both equations (4) and (7) predict a linear relation between strain rate and applied stress for a given thermal cycle. However, the constants of proportionality are sufficiently different to allow a critical corn-

parison with our experimental mechanical testing results. From Fig. 9 we can see there is little effect of the duration of the thermal cycle on the strain increment per cycle. This is consistent with the predictions of equation (4) but can only be explained in terms of equation (7) if the internal stress has had time to fully relax. Internal stress relaxation was not considered by Wu et al. [21] but was investigated by Anderson and Bishop [15] who demonstrated that if complete relaxation of the internal stress occurs, the total strain per half cycle will be independent of relaxation mechanism and given by Ae = -~ In

(8)

where ai is the internal stress level at the onset of relaxation. Figure 9 is strong evidence for a full recovery of the internal stress if an internal stress augmented creep mechanism is correct. This recovery postulate is also supported by the TEM investigation which show very little dislocation substructure even after very large strains. In which case ai would be equivalent to the matrix yield stress ay. With complete internal stress relaxation necessary to explain Fig. 9, equations (4) and (8) should be compared. The major difference is now that in equation (8) the strain per cycle is related to the internal stress generated on thermal cycling, whereas in equation (4) the main variable (apart from the applied stress) is the mean internal plastic strain during each cycle. Our results show a clear relation between the thermal cycle amplitude and the strain per cycle at fixed stress (Fig. 8). We would expect both a i and AV/V to increase with increasing AT; a i should rise with increasing AT to a maximum once yield occurs and then remains essentially constant; AV/V represents plastic strain and should remain at zero until some critical AT, after which plastic flow of the matrix occurs then A V/V should rise with increasing AT. Figure 8 indicates a behaviour more akin to the plastic flow model described above rather than the internal stress model. The discussion so far has considered a rather simple model for the internal plastic strain mechanism [equation (4)]. This formulation has been most studied for the case of transformation superplasticity [17] when the deformation occurs at a practically constant temperature and all the terms within the equation can be considered constant. However, in the case where the internal plastic strain is generated by a difference in thermal expansion coefficients the change in local yield stress with temperature must also be considered. This variation in O'ywill influence the critical temperature increment at which internal plastic flow begins and change the predicted increment of external strain during each thermal cycle. The effect of a simple linear relation between temperature and yield stress is investigated in more detail in Appendix B and Appendix C.

2547

PICKARD and DERBY: DEFORMATION OF METAL MATRIX COMPOSITES In Appendix B the critical temperature change required to initiate internal plastic flow is calculated and tabulated in Table B1 for the 20% Vp MMC. These results predict that for the 130-450°C cycle plastic flow can occur during both halves of the thermal cycle while for the 130-350°C cycle it will only occur during the heating part. However, for the 80-200'C cycle investigated earlier no internal plasticity is predicted. This is in agreement with the data plotted in Fig. 3 where the smallest thermal cycle showed behaviour similar to that found during isothermal testing. Equation (4) is derived from the Lrvy-Von Mises equations of plastic flow where strain is related to the driving stress. The yield stress term O'y in this equation is normally expected to be strongly dependent on temperature. To account for this is the plastic flow rule models, equation (4) must be integrated over each thermal cycle and this is done in Appendix C to produce the following relation AE -- 3B*A~T° l/~(l - Vp)ax {(Ay 1+ AY2)} a0

2xlO-4-

//

o m o

F-

10-4.

.E

0

26o

0

(ro-(r, + Ar )) To - r z

I

300

400

AT

(b) 2x10-3

al

l o -a .¢

'

with AY, = l n \

(a)

/

[]

/

[]

if TI + AT~ < T2

o

o

1;o

2;0

a;o

400

AT

AYI = 0

if TI + A T* >~ T 2

and A Y2 = In \ To _ (7"2_ T* )//

if T l <

AY2=0

i f T ~ > / T2 - A T * (9)

T 2 --

Fig. 17. The results of equations (9) and (A6) are used to predict the critical temperature cycleamplitude for the onset of superplasticity in 20% Vp, 2.3 #m SiC MMC for the conditions of (a) Fig. 8(a) and (b) Fig. 8(b). Experimental results are superimposed.

AT*

where a 0 and TOdefine the temperature dependence of yield stress [ay = a 0 ( l - T/To) ]. Tl is the lower and T 2 the upper temperature of the cycle, and ATe' and AT* are the critical temperature increments for plastic flow on heating and cooling respectively, both derived in Appendix B. B* is fundamentally a geometric constant determined by the microstructure [15, 17], but in this case it wiU also include any further factors which may alter the yield strength of the matrix alloy once it is incorporated into a composite. Equation (9) was fit to the data of Fig. 8 and a value of B* = 1.34 gave the best result, this is shown in Fig. 17. There is a good fit with the experimental results derived from thermal cycles with a fixed Tmax of 350°C. A worse fit occurs at the highest temperature cycles when Tm,~ exceeded 350°C, this is not surprising since our simple model for the dependence of % on temperature only used data in the range 0-350°C (Fig. B1). Equation (9) still predicts a linear relation between a x and AE for a given temperature cycle above some minimum A T. It also predicts that AE should increase with increasing volume fraction of reinforcement to

a maximum value at Vp = 50% before falling again (i.e. K from Table 2 increases to a maximum at Vp = 50%). These predictions are in general, but not detailed, agreement with the experimental results reported earlier. If the value of B* computed for the model is used to predict likely values of K from Table 2 (Table 3), a deviation of theory from experiment swiftly occurs. However, a closer inspection indicates some systematic variations which suggest a further microstructural factor has yet to be considered. For example, in both temperature cycles Table 3. Calculated and measured values o f K for various M M C s during thermal cycling. Calculations carried out using equation (9) with B* = 1.34 Particle size (,am)

Vp (%)

Thermal cycle (°C)

2.3 2.3 2.3 2.3 2.3 2.3 2.3 10 10 10

20 30 40 20 30 40 20 5 10 20

130-450 130-450 130-450 130-350 130-350 130-350 80-200 130-450 130-450 130--450

aNo superplasticity.

K theoretical AE (MPa - I ) 1.47 2.20 2.94 2.61 3.91 5.22

x x x x x x 0 0 7.34 x 1.47 x

10 -4 10 -a 10 4 10 -5 10 5 10 5

10 -5 10 -4

K observed AE ( M e a -I ) 1.21 1.23 5.17 6.88 7.27

x x x x x

10 -4 10 4 10 5 10 -s 10 5

a 1.21 x 10 -4 2.46 x 10 -4

2548

PICKARD

and DERBY:

DEFORMATION

shown in Table 3 the predicted change in AE on an increase in Vp from 20 to 30% is different from the measured value by a similar factor. Hence we believe there to be two main explanations for the poor fit of the model. First we have only a poor model for the change in matrix yield stress for T > 350°C, and second there may be a change in matrix yield stress with reinforcement volume fraction which is so far unaccounted for. If we consider only the 130-450°C thermal cycle and assume that any changes in matrix yield strength governed by the MMC microstructure are equally affected by temperature, then some estimate of this influence of microstructure may be made. There has been much discussion in recent years as to the precise mechanisms of strengthening in metal matrix composites [5, 6, 26, 27] but there is agreement that the yield of a MMC is controlled by the onset of yield in the matrix alloy. Thus evidence that the MMC yield stress increases with decreasing SiC particle size [28,31] should be taken to indicate an increase in matrix yield strength. Such an increase could come from two sources. One mechanism comes from the observation that the matrix microstructure after thermomechanical treatment is controlled by the scale of reinforcement with grain size related to the mean interparticle spacing [6, 29]. This is in accordance with our earlier observations of microstructure in Section 3. The grain size then contributes to strengthening by a Hall-Perch type relation, i.e. try oc d - m , where d is the mean grain size. The second mechanism is the possibility of a constraint on matrix flow from the presence of undeformable particles. To model this constraint we consider a model geometry of two smooth plattens deforming an ideal rigid/plastic material. The constraint of such a geometry [24] leads to a magnification of the yield stress under plane strain conditions such that try* = try

0.75 +

(10)

where w is the width of the plattens and t their spacing. For MMCs this can be taken as an approximate guide to constraint with w taken as the mean particle diameter and t the mean particle spacing. Thus two potential mechanisms can be identified in which the reinforcement can influence the matrix yield strength. First, the mean inter-particle spacing can influence the matrix microstructure by controlling sub-grain sizes and the nucleation of recrystallisation [28]. The scale of the microstructure can control matrix yield in these single phase matrix materials by a Hall-Petch type effect. Second, the presence of rigid reinforcing particles will introduce some form of plastic constraint which can be modelled using a simple 2-dimensional planestrain analogue as given in equation (10). In order to quantify these effects the influence of particle volume fraction (Vp) and particle size (2a) must be

OF METAL

MATRIX

COMPOSITES

determined. We assume that the mean matrix grain size is given by the mean particle spacing. Using the terminology of Appendix B the mean particle spacing is given by 2 ( b - a ) where 2a is equivalent to the mean particle diameter, a and b are related to the volume fraction of reinforcement by ( a / b ) = ~ p . Thus the mean grain size will be given by d=2(b -a)=2a[(1/Vp)

x / 3 - 1]

(11)

and the mean aspect ratio used in equation (10) w

a

1

(12)

t - b - a - [(1/Vp) 113- 1]"

In order to use the modified matrix yield stress in the plastic flow model an absolute value is not needed. Instead it will be sufficient to determine the relative yield stresses of the matrices in the different volume fraction MMCs. Thus if one volume fraction and particle size is chosen for a reference matrix yield stress try, the relative strength of another M M C will be given by

tr;' tr--~y L a l ( l l V , ) , / 3 _ l ] [

+[(llVo),i 3

~

(13)

where the reference MMC, of matrix yield stress a~, has a volume fraction of reinforcement V0 and a mean particle diameter a0. The results of equation (13) predicting the relative matrix strengths of the MMCs used in this study are given in Table 4 with the 2.3 # m 20% Vp SiC reinforced material used in much of this study taken as the reference. These microstructural effects on matrix yield can be incorporated into the model to alter both AT* the critical temperature change for the onset of yield, and K the constant of proportionality between AE and ~r for a given cycle. Figure 18 shows the inclusion of these influences on try into equation (9) to predict the change in K with increasing volume fraction reinforcement for the 2.3 and 10#m particle size MMC. The good fit for both 2.3 and 10/~m M M C results is achieved with a new value of B * = 0.75. Now the predicted maximum in K at Vp ~ 30% is in good agreement with our experimental results. Table 4. Matrix yield stress (a*) is believed to be affected by both reinforcement volume fraction and particle size [6, 28, 31]. This is modelled using equation (13) with the yield stress of a 20% Vp 2.3 # m reinforcement MMC used as reference, i.e. a0 = 1.5gm, V0 = 20% and tr ° is the reference yield stress

Particle size (tim)

Vp (%)

",*t"7

2.3 2.3 2.3 2.3 2.3 10 10 10

10 20 30 40 50 I0 20 30

0.84 1 1.29 1.66 2.21 0.40 0.48 0.62

10

40

0.94

10

50

1.06

PICKARD and DERBY:

DEFORMATION OF METAL MATRIX COMPOSITES

5x10-4. 4x10-4. A

t.

[]

lOl~m Results

o

2.3 i~m Results

2.

3x10"4-

3.

2x10-4.

4. 5. 0'

'

0

2b

A

d0

do

100

vp% Fig. 18. Predicted variation in K, the strain per cycle/stress constant, as a function of volume fraction reinforcement and particle size using a modified yield stress. Experimental results are superimposed.

6. 7.

8. The L 6 v y - V o n Mises based model o f e q u a t i o n (9) is still in need o f further refinement. The most likely area u n a c c o u n t e d for, or at least poorly modelled, is the change in matrix yield strength with t e m p e r a t u r e a n d M M C microstructure. However, the modified model still predicts the general trends of t h e r m a l cycle b e h a v i o u r observed in this study. These trends c a n n o t be a c c o u n t e d for by the internal stress m e c h a n i s m models used in other studies [12, 21]. In s u m m a r y the b e h a v i o u r of M M C s loaded during t h e r m a l cycling can be described as follows. Superplastic d e f o r m a t i o n b e h a v i o u r is observed above some critical t e m p e r a t u r e cycle amplitude. There is an a p p r o x i m a t e l y linear relation between the increment of plastic strain per t h e r m a l cycle (AE) a n d the applied stress. The superplastic b e h a v i o u r occurs by m e c h a n i s m s which lead to n o discernible dislocation substructure n o r do they result in a noticeable increase in matrix dislocation density even up to very large strains. The m e a n superplastic flow rate at a given stress a n d thermal cycle increases with increasing reinforcement volume fraction up to a peak value in the region of 20~30% Vp, a n d then decreases with further increase in Vr. T h e superplastic flow rate also increases with increasing reinforcement particle size. This b e h a x i o u r can be approximately modelled using the a p p r o a c h of A n d e r s o n a n d Bishop [15], a d a p t e d to account for changes in yield stress within the thermal c3cle. Acknowledgements--We wish to thank Dr C. Brown and Mr A. Tarrant of BP Research Centre for the supply of the 2.3/am SiC reinforced MMC. We are also grateful to Dr E. A. Feest and J. Tweed of AE Technology Harwell for provision of powder consolidation and extrusion facilities and also Mr P. Mummery of Oxford University who manufactured the 10/am SiC reinforced MMC. This project was funded by the Science and Engineering Research Council, U.K.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

REFERENCES 1. A. J. Allen, M. Burke, M. T. Hutchings, A. D. Krawitz and C. G. Windsor, in Residual Stresses in Science

30. 31.

2549

and Technology (edited by E. Macherauch and V. Hauck), p. 151. DGM Int. Verlag., Oberursel F.R.G. (1987). P. J. Withers, D. Juul Jensen, H. Lilholt and W. M. Stobbs, Proc. ICCM VI & ECCM H (edited by F. L. Matthews, N. C. R. Buskell, J. M. Hodgkinson and J. Morton), p. 2255 (1987). R. J. Arsenault and M. Taya, Acta metall. 35, 651 (1987). M. Vogelsang, R. J. Arsenault and R. M. Fisher, Metall. Trans. 17A, 379 (1986). B. Derby and J. R. Walker, Scripta metall. 22, 539 (1988). F. J. Humphreys, Proc. 9th Riso Int. Syrup. Metall. Mater. Sci. (edited by S. I. Andersen, H. Lilholt and O. B. Pedersen), p. 51 (1988). R. Warren, L. O. K. Larsson, P. Ekst6m and T. Jansson, Proc. ICCM I V (edited by T. Hayashsi, K. Kawata and S. Umeckara), p. 1419 (1982). E. G. Wolff, B. K. Min and M. H. Kural, J. Mater. Sci. 20, 1141 (1985). T. Kyono, I. W. Hall, M. Taya and A. Kitamcera, Proc. 3rd U.S. Japan Cong. on Comp. Mater. (1985). W. G. Patterson and M. Taya, Proc. ICCM V (edited by W. C. Harrigan Jr, J. Strife and A. Dhingra), p. 53 (1985). M. Y. Wu and O. D. Sherby, Scripta metall. 18, 773 (1984). S. H. Hong, O. D. Sherby, A. P. Divecha, S. D. Karmarkar and B, A. MacDonald, J. comp. Mater. 22, 102 (1988). J. C. Le Flour and R. Locic6ro, Scripta metall. 21, 1071 (1987). S. M. Pickard and B. Derby, Proc. 9th Riso Int. Syrup. Metall. Mater. Sci. (edited by S. I. Andersen, H. Lilholt and O. B. Pedersen), p. 447 (1988). R. G. Anderson and J. F. W. Bishop, Inst. of Metals Syrup., Uranium and Graphite, paper 3, 17. Inst. of Metals, London (1962). R. C. Lobb, E. C. Sykes and R. H. Johnson, Metal Sci. J. 6, 33 (1972). G. W. Greenwood and R. H. Johnson, Proc. R. Soc. A 283, 403 (1965). R. A. Kot and V. Weiss, Metall. Trans. 1, 2685 (1970). A. C. Roberts and A. H. Cottrell, Phil. Mag. 1, 711 (1956). B. Derby, Scripta metall. 19, 703 (1985). M. Y. Wu, J. Wadsworth and O. D. Sherby, Metall. Trans. 18A, 451 (1987). G. S. Daehn and T. Oyama, Scripta metall. 19, 703 (1988). B. Derby and S. M. Pickard. To be published. R. Hill, The Mathematical Theory of Plasticity, Chap. V. Oxford Univ. Press (1953). H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps, p. 21. Pergamon Press, Oxford (1983). V. C. Nardone and K. M. Prewo, Scripta metall. 20, 43 (1986). M. Taya and R. J. Arsenault, Scripta metall. 21, 349 (1987). W. S. Miller and F. J. Humphreys, Fundamental Relationships Between Microstructures and Mechanical Properties of Metal Matrix Composites (edited by M. N. Gungar and P. K. Liaer), p. 517. TMS, Warrendale, Pa (1989). P. Jarry, W. Lou6 and J. Bouvaist, Proc. ICCM VI and ECCMII (edited by F. L. Mathews, N. R. C. Buskell, J. M. Hodgkinson and J. Morton), p. 2350. Elsevier, Amsterdam (1987). Metals Handbook, 9th edn, Vol. 2, p. 67 (1979). P. Mummery, private communication.

2550

P I C K A R D and DERBY:

D E F O R M A T I O N O F M E T A L M A T R I X COMPOSITES Cycle Particle TI T2 time Vp size Stress (°C) (°C) (s) (%) (#m) (MPa)

APPENDIX A Complete Thermal Cycle Creep Data Cycle Particle time Vp size Stress (°C) ('~C) (s) (%) (/~m) (MPa)

TI

Tz

130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 80 80 80 80 80 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130

450 450 450 450 450 450 450 450 450 450 450 450 450 350 350 350 350 350 350 350 350 350 350 350 200 200 200 200 200 450 450 450 450 450 450 450 350 350 350 350 350 350 350 350 350 350 450 450 450 450 450 450 450 450 450 350 350 350 350 350 350 350 350 350 450 450

420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 20 20

2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 10 I0

2.98 4.60 6.70 10.50 16.00 3.82 5.32 7.52 9.68 12.75 17.60 0.99 2.35 2.98 5.05 8.04 11.55 15.40 20.20 5.94 8.76 23.30 11.95 15.50 67.00 60.00 54.00 53.00 51.00 4.97 6.69 8.45 10.54 13.36 15.82 21.20 3.64 5.87 6.47 7.35 9.60 10.00 11.00 14.31 15.27 23.00 5.60 9.82 13.46 18.43 18.20 26.28 9.62 12.82 16.50 14.26 21.23 34.95 51.07 10.94 14.48 18.52 24.20 32.70 2.16 3.79

Strain per cycle

Strain rate (s)

3.98E-04 5.84E-04 7.77E-04 1.16E-03 2.00E-03 4.91E-04 7.31E-04 7.48E-04 1.16E-03 1.45E-03 2.13E-03 1.07E-04 1.72E-04 9.66E-05 2.04E-04 3.86E.04 6.30E-04 9.03E-04 1.24E-03 2.46E-04 4.89E-04 1.76E-03 6.85E-04 9.87E-04 1.68E-03 5.46E-04 2.52E-04 1.73E-04 5.88E-05 6.03E-03 2.34E-03 3.41E-03 5.51E-03 1.59E-03 2.45E-03 4.21E-03 3.40E-04 4.79E-04 8.90E-04 2.44E-03 1.lIE-03 1.29E-03 2.25E-03 7.31E-04 9.32E-04 1.85E-03 2.90E-04 5.08E-04 7.22E-04 9.24E-04 1.02E-03 1.33E-03 5.21E-04 7.35E-04 9.14E-04 5.38E-05 1.45E-04 4.87E-04 1.33E-03 3.57E-05 6.64E-05 1.02E-04 2.31E-04 4.62E-04 6.30E-04 1.04E-03

9.47E-07 1.39E-06 1.85E-06 2.76E-06 4.76E-06 1.17E-06 1.74E-06 1.78E-06 2.77E-06 3.46E-06 5.07E-06 2.55E-07 4.10E-07 2.30E-07 4.85E-07 9.20E-07 1.50E-06 2.15E-06 2.95E-06 5.85E.07 1.17E-06 4.18E-06 1.63E-06 2.35E-06 4.00E-06 1.30E-06 6.00E-07 4.13E-07 1.40E-07 1.25E-06 1.80E-06 2.30E-06 3.09E-06 3.78E-06 5.02E-06 8.88E-06 8.09E-07 1.14E-06 2.12E-06 4.56E-07 9.03E-07 8.55E-07 9.50E-07 1.74E-06 1.53E-06 3.20E-06 6.90E-07 1.21E-06 1.72E-06 2.20E-06 2.44E-06 3.17E-06 1.24E-06 1.75E-06 2.18E-06 1.28E-07 3.46E-07 1.16E-06 3.17E-06 8.5E-08 1.58E-07 2.43E-07 5.50E-07 1.10E-06 1.50E-06 2.47E-06

130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 120 140 160 175 200 225 130 130 130 130 130 130 130 130 130 130 130

450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 350 350 350 350 350 350 470 460 420 380 355 330 285 450 450 450 450

420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 420 925 450 170 25

20 20 20 20 10 10 10 10 5 5 5 5 5 5 5 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

10 10 10 10 10 10 I0 I0 10 10 I0 10 10 10 10 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3

5.40 7.10 9.75 14.63 1.66 2.45 3.42 4.72 7.67 9.34 10.53 11.70 12.57 15.45 14.78 2.85 2.85 2.85 2.85 2.85 2.85 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00

Strain per cycle

Strain rate (s)

1.52E-03 1.90E-03 2.36E-03 3.78E-03 3.31E-04 4.79E-04 6.01E-04 7.09E°04 1.03E-04 1.05E-04 2.27E-04 9.87E-04 3.23E-04 1.48E-03 1.02E-03 1.10E-04 8.00E-05 5.90E-05 4.60E-05 2.00E-05 7.50E-06 1.58E-03 1.36E-03 1.13E-03 8.40E-04 5.60E-04 3.00E-04 1.00E-04 2.62E-03 2.66E-03 2.68E-03 1.90E.03

3.62E-06 4.52E-06 5.62E-06 8.99E-06 7.89E-07 1.14E-06 1.43E-06 1.69E-06 2.46E-07 2.49E-07 5.40E-07 2.35E-06 7.70E-07 3.53E-06 2.43E-06

APPENDIX B The Onset of" Plastic Flow Particulate reinforced metal matrix composites consist of angular, platelet or block morphology, ceramic particles dispersed within a metal matrix. During temperature changes the behaviour of the composite can be modelled as the expansion or contraction of a rigid cavity, within an elastic/plastic matrix. The behaviour of such a system is soluble analytically for a number of simple geometries. The simplest approximation to the geometry of a particle reinforced M M C is the expansion of a thick walled spherical shell of internal radius a and external radius b where a and b are defined by the mean particle size and volume fraction of reinforcement Vp, i.e. lip = a3/b 3 (Fig. B1). Using the

Fig. BI. Schematic model of the deformation of a spherical shell of metal by the expansion or construction of a rigid spherical particle; particle radius = shell inner radius = a, shell outer radius = b.

P I C K A R D and DERBY:

DEFORMATION OF METAL MATRIX COMPOSITES

analysis of Hill [24], we would expect such a shell to yield plasticity throughout its thickness when the internal surface was displaced radially by U~ with

U~=~a[(1-v)(:)3-2(l-2v)ln(:)]

Table B1. Critical temperature change required to initiate full plastic flow on heating or cooling for a given temperature, C3 taken as 1.10 AT* (°C) Heating Cooling (IAT~'I) (IArtl)

(B1) T

where E is the matrix Y o u n g ' s modulus and v its Poisson's ratio. Uc the displacement of the cavity surface will be related to the difference in thermal expansion coefficient between particle and matrix (Act), and the temperature change (AT}. This thermal strain will result in a displacement U = ActATa (assuming the particle to be rigid). For the values of Vv used in this study ln(b/a)<<(b/a) 3 and so a simplified form of equation {BI) can be used to predict the onset of plastic flow when

0 50 100 150 200 250 300 350 400 450

try 1 - v For the platelet morphology of an M M C we would expect slight differences with equation (B2) and a more general equation would be Ci try ACtAT/> - -

(B3)

ev.

where C 1 is a geometrical constant close to one. Both try and E are expected to vary with temperature. The room temperature value of E for the A1 matrix is E* = 70 GPa. In the temperature range of interest the variation of E with temperature can be represented by the linear equation (25)

E(T) = E* (1 - C2T)

(B4)

where T is given in °Celsius and C 2 = 5.36 x 10-4/°C. The variation in yield stress with temperature for an 1100 series A1 alloy is given in Fig. B2. The data is available to cover the range from 0 to 380°C and can be approximately fitted to the linear relation ay = tr0(l -- T/To)

C 3 0 - o ( T 0 - - 7"2)

ACtAT* = Act IT - T_,[ =

201 185 167 149 130 lll 91 72 51 30

-- C2T2)

VoE* r0(1

(B6)

where T 1 is the initial temperature and T 2 is the new temperature (either an increase or a decrease) at which full plasticity occurs. The constant C 3 now includes any difference between the plasticity of the matrix and the matrix alloy in its unreinforced state, as well as the influence of particle shape. An interesting property of equation (B6) is its prediction of larger values of A T* on cooling than on heating, thus the

--375 350 321 289 251 205 146

values of AT* estimated from Fig. 8(a) and (b) correspond to AT* for the heating halves of the thermal cycle. Hence by substituting for T~ and T 2 the constant C3 can be estimated with Act ,~ 2 × 10 -5 °C -~ (see Table 1). From Fig. 8(a) C 3 = 1.14 and from Fig. 8(b) C 3 = 1.02. These are in very good agreement considering the relation between A T and AE is probably not truly linear (Appendix C). Table A1 shows the predicted values of AT* for heating and cooling cycles for a range of temperatures.

APPENDIX C Modification of the L~vy-Von Mises Flow Rule Models for MMCs As a starting point we take the model of Greenwood and Johnson [17] as developed from that of Anderson and Bishop [15], i.e.

AE=B ~Avtrx. V

(B5)

with a 0 = 38 M P a and T o = 521°C. Hence by including equations (B4) and (B5) into equation (B3) the following relation is found for the critical temperature change AT* required to initiate plastic flow

2551

(C1)

try

B t is a numerical constant calculated by averaging internal plastic strains throughout the material. In the main text [equation (4)] B~ = 5/6 per half cycle for the case of transformation stress [17]. The internal plastic strain (AV/V) can be estimated for low volume fraction of reinforcement as AV V - 3ACtVpAT

(C2)

where Act is the difference between the CTE of matrix and reinforcement and A T is the increment of temperature which occurs after internal plastic yield. Using this and the results of Appendix B to modify equation (C1) gives AE

3B~ Act Vptrx [2AT -- ATe" -- AT*]

(C3)

try

where AT~ is the critical temperature increment for plastic flow on the heating half-cycle and AT* that for cooling. F r o m Fig. B2 in Appendix B try is seen to have a strong dependence on temperature. Within the area of interest an approximate linear relation is used

40 D

3o

[]

~E

2O

D

try = tr0(1 - T/To).

D D

"~ 10-

Thus equation (C3) should be modified to allow for this temperature dependence

>,•

0

,

I00

.

,

200

.

,

.

300

,

400

.

500

Ternperoture (*C) Fig. B2. The variation in yield stress commercially pure AI(1100) as a function of temperature [30]. A M 38/12--L

(C4)

tro

LIJr,+~ +

fT2' AT ( T oa- TT)]

(c5)

2552

P I C K A R D and DERBY:

D E F O R M A T I O N O F M E T A L M A T R I X COMPOSITES

which on integration gives

and ax

AE = 3B t Act To V* - - [A Yl + A Y2]

(C6a)

AYE=

ln(\To_(T2_AT.)/I T0-3

ifTt
(C6c)

t70 or

with

AY~

=ln(To-(Tt+_~T:)~ \ To-T2 ,] ifT~+AT*up
or

AY I = 0

if T I + A T e ' >t T 2

AY2=0 (C6b)

if

T~ >1T2-AT*.

Equation (C6) reduces on expansion to equation (C3) if To is large. Equation (C6) predicts the mean strain increment in the matrix, thus averaged over the composite as a whole the strain per cycle will be related to the matrix volume fraction, i.e. Aec = A~(l - lip).