Analytical study on deformation behaviour of metal matrix composites

Journal~Ma~rialsP~cessingTechno~gy, 24(1990) 281-289

281

El~v~r

ANALYTICAL STUDYON DEFORMATION BEHAVIOUROF METALMATRIXCOMPOSITES N. KANETAKEand H. OHIRA Department of Materials Processing Engineering, Faculty of Engineering, Nagoya University, Nagoya464-01 (JAPAN)

SUMMARY The flow behaviour, stress-strain curve, of the metal matrix composite reinforced by short fibers or particles is calculated by connecting the Eshelby's inclusion model and the Ashby's secondary s l i p model. The theory is established for general combined stress states, in which the effects of a shape, a size, a volume fraction and a orientation of dispersing e l l i p s o i d a l p a r t i c l e are considered. A stress-strain curve up to large strain region and a y i e l d locus can be c a l c u l a t e d f o r some aluminium matrix composites dispersed SiC p a r t i c l e s or short fibers. Taking into consideration the three dimensional distribution of the e l l i p s o i d a l particles, the useful calculation for the practical composites is also possible. INTRODUCTION In the f i e l d of a metal matrix composite (MMC), r e c e n t l y , discontinuous fiber or p a r t i c l e dispersed composites are expected as a new structural mater i a l which may take the place of steel structures in an automobile or a machine industry.

Its useful performance is gradually recognized, and some t r i a l

pro-

ducts are made on an experimental basis. Nevertheless the MMC is not yet put to p r a c t i c a l use so much as expected. As against the development of composites with high performances, the technology for manufacturing complex shape products from those composite materials is hardly investigated up to date. The MMC dispersed short fibers or particles is possible to be deformed l i k e conventional metals. Therefore, in order to put the MMC to practical use for mass-products,

the deformation processing may be u s e f u l l y a p p l i c a b l e . The

intermediate products such as a bar or a plate are fabricated by some processes of casting or powder m e t a l l u r g y method. Then the h a l f MMC products must be formed subsequently to the f i n a l parts. In applying the conventional deformat i o n processing to the MMC, there are advantages in being able to use the conventional system as i t is for the MMC as well as in reducing the manufacturing cost of mass-products and in improving the mechanical property of the products. From the above mentioned point of view, the u p s e t t a b i l i t y and the f l o w stress of p a r t i c l e dispersed aluminium matrix composites have been investigated experimentally (refs. 1,2). IN the present work the flow stress under various

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282

combined stress states of the p a r t i c l e dispersed MMC are investigated theoretically. THEORETICAL CALCULATION Foundation of the theory As a strengthening mechanism of the particle reinforced composite, two main causes are considered. The one is increasing in the load carrying capacity by including hard particles, and the another is increasing in the work-hardening capacity of a matrix material by dispersing fine particles. I t is very useful, in general, to disperse long and narrow particles to a loading direction for the former mechanism, and to disperse homogeneously f i n e p a r t i c l e s in the matrix for the l a t t e r one. In the present work, both mechanisms are taken into account for calculating a stress-strain curve of the p a r t i c l e dispersed composites. For the former mechanism, the t h e o r e t i c a l c a l c u l a t i o n method proposed by Tandon and Weng (ref. 3) for the spherical p a r t i c l e dispersed composite is modified to the composite including various oriented e l l i p s o i d a l particles. The theory is based on the Eshelby's solution for an e l l i p s o i d a l inclusion (ref. 4) and Mori-Tanaka's concept of an average stress in the matrix (ref. 5). For the l a t t e r mechanism, on the other hand, the dislocation theory proposed by Ashby (ref. 6) for the work-hardening in a non-homogeneousmaterial is considered and combined with the above Eshelby's model. Load carrying capacity of the p a r t i c l e When the eigen-strain E* is generated in an inclusion buried in an i n f i n i t e e l a s t i c matrix, the average stress and strain of the inclusion were determined by Eshelby (ref. 4) as following. s = Ss~ o

= L(e-e*) = L(~*-E*)

(I)

S is the Eshelby's tensor, which can be determined from an elastic modulus and a shape of the particle, and L is an elastic moduli tensor.

~

OO ~

o0

Now consider a material in which the p a r t i c l e and the matrix have different e l a s t i c moduli l i k e a composite, say Ll and L0 r e s p e c t i v e l y , and l e t the material subject to external stress s o as shown in Fig. I. This problem was also analyzed by Eshelby. I f the

Fig. 1 Non-homogeneousp a r t i c l e in the infinete matrix

283

strain is E0 in the absence of the particle, the average stress and strain in the p a r t i c l e , oI and El , are perturbed and d i f f e r e n t from that of the matrix due to the different elastic moduli, that is ~=Oo+cpt and EI=Eo+EPt. As shown in Fig. 1 the perturbations of the stress and strain can be represented equival e n t l y by the stress and strain which generate in the p a r t i c l e subjected to the eigen-strain E* shown in Eq. (I). Then the stress in the p a r t i c l e is given by oI = gO+Opt = LI(Eo+EPt) = LO(Eo+EPt-E*)

(2)

At f i n i t e concentration of particles, say with a volume fraction f (O
(3)

oI = o2+~+oPt = LI(EO+~+Ept) = LO(EO+~+EPt-E*)

(4)

To satisfy the boundary condition, : -fo pt

(5)

Based on above equations, using the e l a s t i c moduli of p a r t i c l e and a matrix and the Eshelby's tensor, the s t r e s s - s t r a i n r e l a t i o n of both phases can be determined. However, when the plastic flow in the matrix becomes greater, the e l a s t i c moduli are not sufficient to calculate the deformation behaviour of the composite, and the plastic property of the matrix should be taken into account. Such a effect was f i r s t i n t r o duced by H i l l

(ref.

Particle

7) to a

similar model in his study of p o l y c r y s t a l p l a s t i c i t y . But Hill's

f o r m u l a t i o n is

very

complex to c a l c u l a t e because

U!

of some incremental c a l c u l a tions. A n u m e r i c a l l y simpler method was l a t e r suggested by B e r v e i l l e r and Zaoui (ref. 8). This

method e m p l o y s

the Matrlx

"secant" modulus, which is a gradient connecting a given point on a n o n l i n e a r stressstrain curve with an origin as shown in Fig.

2. Tandon and

Weng (ref. 3)

h a v e demonst-

rated

that the method is very

0

~2

Fig. 2 Schematic representation of stressstrain state of the constituents

284 suitable for the calculation of a p a r t i c l e reinforced p l a s t i c i t ~ Now the stress ( ~ ) and s t r a i n (e2) r e l a t i o n of the p l a s t i c a l l y deformed composite can be formulated as follows using the e l a s t i c and secant moduli. At f i r s t consider an i d e n t i c a l l y shaped comparison material with the property of the matrix, and l e t the composite and the comparison material both subject to the same stress a2. The strain in the comparison material e(O) is given by o 2 = LI0)~(0 ) where L(0 ) is the secant m o d u l i t e n s o r o f the comparison m a t e r i a l

(6) at the

a p p l i e d stress state. The superscript s and a parenthesis in a subscript r e p r e sent a secant modulus and a comparison m a t e r i a l r e s p e c t i v e l y . Because o f i n c l u d i n g p a r t i c l e s the average stress and s t r a i n in the matrix. G0 and e 0, are given by e 0 = 6(0)+~

(7)

G0 : aZ+B : LS(e(O)+~) (8) On the other hand, the average stress and s t r a i n in the p a r t i c l e , o I and el, are given by

(9)

e I : e0+ePt o I = Ll(e(0)+~+ePt) : L~(e(0)+g+ePt-e*)

(I0)

Using the Eshelby's tensor SO corresponding to L0,

the f o l l o w i n g r e l a t i o n

is

g i v e n from Eq. ( I ) . e pt = S~e*

(II)

And f o l l o w i n g equations can be a l s o given from above equations. : (LS-k~0~)e(0)+L~e_ - ~ opt : k~(eP~-e*) : L~(S~-I)e*

(12) (13)

= -f(S~-l)e*-(l-LS-ILI0))e(0 ) = -fePt+fe*-e(0)+L~-IL~0)e(0) where L s - I (I0),(II)

is

the

inverse

o f Ls,

(14)

Then e * is g i v e n as f o l l o w s

from Eqs.

and (14).

(LI-L~) [ ( f l + ( l - f )

S~)+L~ ]e* = -(kl-kS)(k~-Ik~0))e(0)_

Using above f o r m u l a t i o n s ,

the e i g e n - s t r a i n

(15)

e* can be c a l c u l a t e d

a p p l i e d s t r e s s 0 2, and the a v e r a g e s t r e s s and s t r a i n

from the

in both the m a t r i x and

p a r t i c l e s can be a l s o c a l c u l a t e d . Then the s t r a i n of the composite 62 is c a l c u l a t e d from the s t r a i n in both the m a t r i x a n d p a r t i c l e s . e 2 = ( l - f ) E 0 + f ~ 1 : L~-IL~0)E(0)+fe * Based on the above mentioned model, the s t r e s s - s t r a i n particle

(16) b e h a v i o u r o f the

r e i n f o r c e d composite can be c a l c u l a t e d using the e l a s t i c

the Eshelby's tensor of the p a r t i c l e

moduli and

and the s t r e s s - s t r a i n curve o f the matrix

material. Work-hardening capacity of the matrix The stress-strain curve of the matrix material to be known for calculating can be measured by a tensile test, and the curve can b e g e n e r a l l y represented

285

by a following equation. 0 0 = h(~O)n For

general

(17) combined

s t r e s s and s t r a i n

states,

the equation i s a l s o usable for

Mises's equiva-

%'4

(a)

~

i

(b)

(c)

Fig. 3 Schematic representation of

l e n t s t r e s s and s t r a i n .

the secondary s l i p model

Because f i n e p a r t i c l e s are dispersed in the com-

posite, the above stress-strain curve is not satisfactory as a flow property of the matrix material in the composite. During p l a s t i c f l o w in the matrix the moving of dislocations may be restricted by the particles or secondary dislocat i o n s may be generated around the p a r t i c l e s . The accommodating p l a s t i c f l o w shall give rise to a s u b s t a n t i a l l y increased dislocation density and hence to hardening of the matrix. Ashby (ref. 6) proposed a secondary s l i p model for explaining such a effect of a non-deforming p a r t i c l e in a p l a s t i c matrix as shown in Fig. 3. In the absence of the p a r t i c l e , at p l a s t i c a l l y deforming in the matrix a c i r c u l a r region shall deform e l l i p t i c a l l y

as Fig. 3(b). But the matrix around the p a r t i -

cle can not change i t s shape because of the non-deforming particle. In order to satisfy the geometrical effect, a secondary s l i p corresponding to canceling the displacement of the matrix should generate around the p a r t i c l e . As a result the dislocation density increases and the matrix is hardened more than in the absence of the particles. Following Ashby, the increase in the flow stress of the matrix for a composite containing equiaxed p a r t i c l e is given by 4o0 = kcE2~f/(l-f).b/d.~ 0 = k w ~ (18) where E2 is a young's modulus of the composite, d is a diameter of the particle, b is a Burger's vector and kc is a constant, 0.62 for aluminium alloys. Now the s t r e s s - s t r a i n property of the matrix material in the composite is represented as a following equation,

and i t ' s used for calculating the Eshel-

by's model. o0 = h ( ~ O ) n + k w ~

(19)

.Shape and orientation of the p a r t i c l e The p a r t i c l e is an e l l i p s o i d represented by a following equation. ~

x2 ÷

-

(dx/2) 2

-

y2

(dy/2) 2

Z2 + - -

(dy/2) 2

1

(20)

where dx and dy are a major and a minor diameter of the e l l i p s o i d respectively. The e l l i p s o i d a l particles are dispersed in three dimensions in the composite. and the coordinates fixed on the e l l i p s o i d can be r e l a t e d to those on the composite with two independent angular parameters.

9.86

When the e l l i p s o i d a l

particles

composite, the s t r e s s - s t r a i n

orient

in some d i f f e r e n t

directions

in the

curve o f the composite is obtained by a weighted

mean w i t h a r a t e o f each o r i e n t a t i o n s . CALCULATED RESULTS

250

Some c a l c u l a t i o n s were c a r ried

out

dispersed

for

SiC p a r t i c l e

2OO

aluminium matrix

composites produced by a powder extrusion

(ref.

I).

Here

t h e v a l u e s in l i t e r a t u r e

are

~150 100 50

employed f o r the Young's moduli,

the Poisson r a t i o s and the

....

0

I

0

Burger's v e c t o r of a SiC and a

0,05

strain

curve

o f the

matrix,

O. 20

E

spherical particles

cimens produced by e x t r u d i n g

250

pure a l u m i n i u m powder i s a p -

20%30%'

200

proximated t o the Eq. (17) and

"10%

'

•

.

5%

"~ 150

Fig. 4 shows t h e e f f e c t a volume

I

0,15

Fig. 4 E f f e c t o f volume f r a c t i o n o f

measured using spe-

used f o r c a l c u l a t i n g .

I

O, i0 Strain

pure aluminium. For the s t r e s s the curve

CONSIDER 2nd SLIP DISREGARD 2nd SLIP

fraction

(Vf)

s p h e r i c a l SiC p a r t i c l e s

of of

with a

i00 L 50

d~!dly=5

d i a m e t e r o f 3 ~m. In t h e f i gure

solid

lines

show t h e

r e s u l t s which are c a l c u l a t e d u s i n g Eq. (17) as a p r o p e r t y of

the

only

matrix,

the

crease

of

considering

mechanism f o r a load

in-

0

I

0

I

I

0,05

0.10 0.15 Strain E Fig. 5 Effect of volume fraction of e l l i p s o i d a l particles

0.20

250

carrying

c a p a c i t y , t h a t is the c a l c u l a t i o n , w h i l e broken l i n e s a r e results (19).

calculated Although

the

shown by b r o k e n dependent value

on t h e of

..... ~.f"

results

lines

are

constant

o f k c in t h e Eq. (18),

the effect

150

u s i n g Eq.

increasing

in

work-hardening c a p a c i t y o f the m a t r i x appears c l e a r l y .

Fig. 5

d~:PARTICLE SIZE

5olV. I

.

0

0

.

.

.

.

- -

0.05

VF:O

10%

..... 20% O. i0 0.15 O. 20 Strain

E

Fig. 6 E f f e c t o f diameter o f spherical particles

287 shows also the effect of the volume fraction an el l i psoi dal particle (dx/dy=5) dispersed composite, using Eq. (17) for the matrix. I t can be found that the flow stress of the composite containing e l l i psoi dal particles is much higher than spherical particles.

201

The effect of a diameter of the spherical particle is shown in Fig. 6. Considering Ashby's secondary s l i p model shown in Eq. (18), the e f f e c t of the diameter can be calculated, though it's impos-

8

200

i

dix/ i --2

~" 150

s i b l e in the Eshelby's model.

100 I / / /

Figs. 7 and 8 show the effect of an aspect ratio of an el l i psoidal p a r t i c l e (dx/dy) at i t s volume fraction of I0 %. Fig. 7 is the r e s u l t calculated for an u n i d i r e c t i o n a l

0

250

reinforced one. In the case of ment the e f f e c t of the aspect

,

: i00

~J

VF 10% dlx/dly:ASPECT RATIO

d~x 0

strain curves in tensile load-

0

I

I

0,05

0,10

el I ipsoidal

25O

d i r e c t i o n inclines more than

L oj

I00

c l e a r l y . The flow stress is

50

the lowest in the t e n s i l e

0

direction of not 90 but 60 The extruding process is useful to orient

short

unidirection.

fibers

in

an

But it's impos-

15°

3oo'

90°

"e 150

forcement with e l l i p s o i d a l p a r t i c l e s does not appear

8:0'°

200

to the reinforced direc-

tion, the e f f e c t of the r e i n -

0,20

Fig, 8 Effect of aspect r a t i o of e l l i p s o i d a l p a r t i c l e (random reinforcing)

p a r t i c l e s . When the t e n s i l e 30

I

0,15

Strain

ing to various directions for with

reinforcing)

150

Fig. 9 shows the stress-

an u n i d i r e c t i o n a l reinforced

0,20

, , ,,

,

u~ 50

not clearly.

0,15

2O0

the unidirectional reinforce-

composite

0 ,i0

p a r t i c l e (unidirectional

8 is a random d i r e c t i o n a l

while in random reinforcing is

0,05

Strain E Fig, 7 Effect of aspect r a t i o of e l l i p s o i d a l

reinforced composite, and Fig.

r a t i o appears remarkably,

ELLIPSOID

l ~ f ~ - ~ - - T M... VF=IO% u2y 50 F ~ -"'1 lu2, d,x/d,y:ASPECT RATIO / d,x 0k e , ,

~

"~,\ ~.i

ELLIPSOID

~

0

VF:IO%

d,x/dly:5

I

I

0,05

O, 10

Strain

I

0,15

E;

Fig, 9 Effect of loading direction (unidirectional reinforcing)

O, 20

288

s i b l e to o r i e n t p e r f e c t l y in

250

an unidirection, and the short f i b e r s are distributed around

200

o

~

e~

the d i r e c t i o n . Fig. lO is the

15o

r e s u l t of t e n s i l e stress-

i00

strain curves calculated under assuming such a case. They were c a l c u l a t e d for various

L

\ ~/z 50

s o l i d angles of d i s t r i b u t i n g around the t e n s i l e d i r e c t i o n , and in the solid angle e l l i p -

00

VF:IO,. d,x/d,y:5

~J/L / ' 0.05

O.,i0

, 0,15

21

O,

Strain E Fig. I0 Effect of d i s t r i b u t i n g states

soidal particles are dispersed

of e l l i p s o i d a l p a r t i c l e s

homogeneously. As predicted from the r e s u l t s of Fig.

9,

the d i s p e r s i n g o r i e n t a t i o n angle more than 30

decreases

200

o :1.

•

150

the e f f e c t of reinforcement i00

with short fibers. Fig.

II

shows e q u i v a l e n t

s t r e s s - s t r a i n curves under some combined stress states for an unidirectional reinforced composite. The d i r e c t i o n of a stress aI is corresponded

•

•

= 0:1:-1

ELLIPSOID

50

VF:IO%

dlx/dly=5 0

0

I

I

0,05

0,10 Strain

I

0,15 E

Fig. II Effect of stress states (unidirectional reinforcing)

to a major axis of an e l l i p soidal particle. At the stress state in which the load is subjected to the direction of a minor axis of the p a r t i c l e ,

100 ~

the flow stresses become lower. In order to assess the e f f e c t of various stress states sysJ

t e m a t i c a l l y , the f l o w stress at s t r a i n of 0.12 is shown on a p r i n c i p a l stress f i e l d

in

Fig. 12. The f i g u r e contains the r e s u l t s

for

composites

reinforced with e l l i p s o i d a l particles in unidirection (o l direction;

solid l i n e ),

two

directions ( ~ and 02; dotted

Uni-direction Two directions Three directions Random directions Fig. 12 Flow stress loci for various reinforced composites

0,20

289

line),

three directions

( ~,

o2 and o3; broken l i n e ) and random d i r e c t i o n s

(chain l i n e ) , The f l o w stress l o c i are e l l i p t i c a l in one or two d i r e c t i o n s . d i r e c t i o n s is c i r c u l a r ,

for the composites reinforced

The locus f o r the composite r e i n f o r c e d in t h r e e

and i t has higher flow stress than that reinforced in

random d i r e c t i o n s . CONCLUSION C o n s i d e r i n g two main mechanisms f o r strengthening, they are increasing in the load carrying capacity by hard p a r t i c l e s and increasing in the work-hardening capacity of a matrix by dispersing f i n e p a r t i c l e s ,

the deformation beha-

v i o u r of p a r t i c l e dispersed metal matrix composites were c a l c u l a t e d t h e o r e t i c a lly.

From some c a l c u l a t e d

r e s u l t s the e f f e c t s of a shape, a s i z e , a volume

f r a c t i o n and an o r i e n t a t i o n of the p a r t i c l e s and a stress state can be predicted f o r some SiC reinforced aluminium matrix composites. Taking into considerat i o n the three dimensional d i s t r i b u t i o n of the short fibers,

the useful c a l c u /

l a t i o n f o r the p r a c t i c a l composite produced by the extrusion was also possible. REFERENCES 1 2

3 4 5 6 7 8

N. Kanetake and N. Nakamura, Flow stress at large s t r a i n s of alumina p a r t i c l e dispersed aluminum powder composites, J of Japan Soc, Tech. P l a s t i c i t y , 31(351) (1990) 537-542. N. Kanetake, Deformation behavior and u p s e t t a b i l i t y of p a r t i c l e dispersed aluminum matrix composites, Proc. of 3rd Int. Conf. Tech. P l a s t i c i t y (Kyoto), published in 1990 by Japan Soc. Tech. P l a s t i c i t y , Tokyo, decided to publication. G. P. Tandon and G. J. Weng, A theory of p a r t i c l e - r e i n f o r c e d p l a s t i c i t y , J. Appl. Mech. (Trans. ASME), 55(1988) 126-135. J. D, Eshelby, The determination of the e l a s t i c f i e l d of an e l l i p s o i d a l i n c l u s i o n , and r e l a t e d problems, Proc, of Royal Soc., London, A241(1957) 376-396. T. Mori and K. Tanaka, Average stress in the matrix and average e l a s t i c energy of materials with m i s f i t t i n g i n c l u s i o n s , Acta M e t a l l . , 21(1973) 571574. M, F. Ashby, The deformation of p l a s t i c a l l y non-homogeneous a l l o y s , in: A. K e l l y and ~ B. Nicholson (Ed.), Strengthening Method in Crystals, E l s e v i e r , London, 1971, pp,137-192. ~ H i l l , Continuum micro-mechanics of e l a s t o p l a s t i c p o l y c r y s t a l s , J. of Mech. Phys. Sol,, 13(1965) 89-101. M. B e r v e i l l e r and A. Zaoui, An extension of the S e l f - c o n s i s t e n t scheme to P l a s t i c a l l y - f l o w i n g p o l y c r y s t a l s , J. of Mech. Phys, Sol., 26(1979) 325-344.