A simple model of deformation behavior of two phase composites

~

Acta metall, mater. Vol. 42, No. 4, pp. 1113-1123, 1994

Pergamon

0956-7151(93)E0042-2

Copyright © 1994 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/94 $6.00 + 0.00

A SIMPLE MODEL OF DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES

K.S.

RAVICHANDRAN UES, Inc., 4401 Dayton-Xenia Road, Dayton, OH 45432-1894, U.S.A. (Received 16 February 1993; in revised form 18 October 1993)

Abstract--The deformation behavior of two phase composites containing coarse rigid particles in ductile, plastically deforming matrices has been investigated using a simple, continuum mechanics model. The model has been derived on the basis of a unit cell, representative of the composite microstructure, which is idealized to a pattern of periodic cubic inclusions distributed uniformly in a continuous plastically deforming matrix. The stress-strain behavior is arrived at by dividing the unit cell into elements of matrix and inclusion and accounting for the constrained deformation of matrix while ensuring strain compatibility. The resulting closed form expressions have been used to predict the flow stress and stress-strain behavior of composites. The predictions have been compared with specific examples of deformation of composites for which numerical solutions, data from finite element calculations as well as from experiments are available. Typical composites considered include discontinuous particle reinforced composites and transversely loaded continuous fiber composites. A good agreement between the model predictions and data from finite element and numerical simulations as well as experimental measurements is found.

1. INTRODUCTION Composites offer attractive combinations of mechanical properties such as higher stiffness, strength, and superior resistance to creep and fracture relative to conventional structural materials due to the presence of hard and stiff second phases. Dispersion strengthening by sub micron size particles, which has been the key to the development of many structrual materials with high strength in the past, does not increase the stiffness or decrease the density, which are current requirements for light weight structural applications. The beneficial effects of reinforcement of metallic materials with coarse particles, whiskers and continuous fibers of high melting ceramic materials such as carbides, oxides and borides have been clearly demonstrated [1, 2]. Among these systems, particulate reinforced composites are of considerable interest in view of their isotropic properties and ease of manufacture. Optimal design of microstructures of such composites requires a detailed understanding of deformation mechanisms as a function of second phase volume fraction and matrix mechanical behavior. Deformation characteristics of composites with coarse equiaxed particles have been studied through numerical methods [3-9], finite element calculations [10-18] and micromechanics models which make use of Eshelby's solution of misfitting inclusion in a matrix [19, 20]. In general, in these approaches, the over all stress-strain behavior of the composite is constructed on the basis of average stresses and strains in the matrix and the inclusion. Finite element

methods (FEM) provide accurate stress-strain response of the composite if the insitu matrix deformation behavior is used in the calculations [16, 21]. Numerical schemes treat the composite as a homogeneous medium with unknown effective properties which are determined by infinitesimal incremental additions of second phase to the matrix (self consistent differential scheme (SCS) [3, 5, 9, 22]). A variant of this approach is the self consistent three phase model [6, 22] in which a composite sphere of inclusion and matrix, proportioned according to their respective volume fractions is embedded in an effective medium representing the composite. Micromechanical models such as the Mori-Tanaka [19, 20] can treat composites having either fine or coarse particles since it relies on the Eshelby's solution of misfitting inclusion to determine the average stresses and strains in the matrix and the inclusion. A brief review of the bases and assumptions of these models as well as the accuracy of their predictions relative to experimental data is available elsewhere [22]. In the present study, an attempt has been made to derive simple expressions to predict the deformation behavior of particulate composites by considering the basic unit cell representing the microstructure. During deformation, the plastic strain of composite is primarily contributed by matrix deformation while the second phases are rigid or elastically deformable. The deformation behavior of the unit cell is derived by using the basic relationships of deformation of parallel and serial arrangements of two phases loaded in isostrain and isostress configurations, respectively (Fig. 1). Previous studies [23, 24], have shown that the

1113

1114

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES

A

B

A B

(a) (a)

(b)

Fig. 1. Schematics of a bimaterial loaded in (a) parallel and (b) series.

stress-strain behavior of a continuous fiber composites having deformable fibers can be predicted from the rule of mixture approximation of stress-strain behavior of the constituents. Similarly, the stress-strain characteristics of transversely loaded bimaterial laminates can also be predicted [25, 26] from the behavior of the components. A combination of both approaches has been used in this study to predict the flow behavior of particulate composites. The primary objective is to see how accurately a simple model can represent the deformation behavior of composites relative to other complex methods. The predictions have been shown to provide a good estimate of flow stress and stress-strain behavior of discontinuously reinforced as well as transversely loaded continuous fiber composites. 2. UNIT CELL MODELS Expressions for the deformation behavior under uniaxial loading of two phase systems has been derived using an unit cell, representative of the microstructure. The composite microstructure is idealized as cubic inclusions of second phase distributed uniformly in a continuous matrix as illustrated in Fig. 2(a). Figure 2(b) illustrates the three dimensional (3D) geometry of the unit cell. Basically the method involves splitting the unit cell into parallel and series elements as illustrated in Fig. 3(a, b). The elements loaded uniaxially in parallel and in series are considered to undergo isostrain and isostress deformation conditions, respectively during elastic and plastic deformation. The two series elements, 3 and 4 of volume fractions V3 and II4 respectively in Fig. 3(b) add up to form an element 1 of volume fraction V~ in the parallel arrangement in Fig. 3(a). Then elements 1 and 2 of volume fractions Vl and V2 respectively, add up in a parallel arrangement for form the unit cell of Fig. 2(b). Figure 3(c) and (d) illustrate a similar two dimensional (2D) model of division of elements representing transversely loaded continuous fiber composites. If the stress-strain behavior of parallel and serial loading arrangements of both phases are known, the deformation behavior of the unit cell and hence the composite can be derived

,~

h

/

(b)

Fig. 2. Schematics of the idealized microstructure (a), and the unit cell (b), of the composite. using appropriate volume fractions of the individual elements. If the second phase or inclusion of volume fraction, Vp, is considered to be a cube of size a on the

mlllllll!!tlii

IIIIIIIIIIIII Illi n

(a)

(b)

Element

ÁÁÁÁÁÁÁÁÁÁÁÁÁ (c)

1 [~

Vl

2

v2

3 ~

v3

, i

v,

(d)

Fig. 3. Schematics of divisions of the unit cell into several elements, (a) and (b) representing the 3D model of particulate composites and (c) and (d) representing the 2D model of transversely oriented of fiber reinforced composites.

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES sides and if h is the dimension of the matrix material, as shown in Fig. 2(b), then Vp is related to the non-dimensional quantity, c = h/a as c

=F-' LVpJ1'" -

1 with Vp + Vm = 1

1

VI - - V2= 1 (1 + c) 2

1

(1 + c) 2

1

c

V3-(1+c )

V 4 : (1_4_ ~

and

.

(2)

The deformation behavior of the matrix subjected to an applied stress 0-, can be represented by 0- = EmC (7 =

K(EP)

n

for the elastic part

(3a)

for the plastic part

(3b)

with total strain in the plastic regime given by E = Ee + Ep. Era, E° and Ep are respectively the matrix modulus, elastic and plastic strain in the matrix. K and n are constants of power law fit to the plastic part of the stress-strain curve of the matrix material. If the composite element 1, made up of V3 and 1/4 is considered as a single material, then the elements V1 and V2 can be considered to share the applied stress, 0-c, analogous to continuous fiber reinforced composites, according to the rule of mixture as 0-c =

0"1 V I +

0"2 V 2 .

(4)

Since strain compatibility between V~ and V2 is required for both elastic and plastic deformation Ec =

E1 =

E2

(6)

with Va + Vb = 1. V, and Vb are respectively the volume frations of phases A and B; Ea and Eb are respectively the elastic moduli of phases A and B. Similarly the elastic modulus of two phases A and B, loaded in series, in a direction perpendicular to the interface [Fig. l(b)] is known as

E~E~ Ec EaV~+ E. Vo

(7)

Following the arrangement illustrated in Fig. 3, the modulus of composite can be calculated by using AM 42/4--F

Ec= (cEpEm + E2m)(1 -[-c)2- g 2m'~-gPgm (cEp + Em)(1 + C)2

(8)

in which Ep and Era are the elastic moduli of the particle and the matrix respectively. The above equation is compared with the experimental data on A1-SiC particulate composites along with the well known Hashin-Shtrikman lower and upper bound predictions [28] (for details, see Ref. [27]) in Fig. 4. A good agreement between the predicted and experimental data [29-32] can be seen. It is to be noted that only one scheme of division of unit cell (series arrangement first) has been considered in this study as a first step. Examination of the other possible scheme (parallel arrangement first) is beyond the scope of this investigation, but will be studied in future. The plastic strain in the composite for stresses above the yield stress can be derived in an analogous fashion and is discussed in the following. The total plastic strain of element 1 in Fig. 3(a), is the sum of the elastic strain of element 3 (elastic particle) and the plastic strain of element 4 in Fig. 3(b) under an applied stress of 0-1. Hence E1 =

E 3 V 3 -[- E 4 V 4 .

(9)

Substituting for appropriate volume fractions 1 E, =

c

(|

E3 ( I ~ C )

-I- E 4 C - - - ~- ~

(lO)

'

If the contribution from the elastic deformation of stiff particle, relative to the plastic strain of element 4, to the total strain is considered small, then C

E. = E, (1 + c~"

(5)

in which ec, El and E2 are the strains in composite, element 1 and element 2 respectively. The elastic strain of the composite can be directly calculated from the elastic modulus of the composite. The elastic modulus of the composite was derived in a separate study [27] using similar unit cell arrangement built up on isostress and isostrain elements. The procedure will be briefly described here. The elastic modulus of the isostrain (composite) element in Fig. l(a) is

Ec = Ea V, + Eb Vb

appropriate volume fractions of elements at different levels of divisions of unit cell. The elastic modulus of the composite can then be written as

(1)

in which Vm is the volume fraction of matrix material. The respective volume fractions Vl, Vz, V3 and V4 of the elements 1, 2, 3 and 4 respectively illustrated in Fig. 3(a) and (b), can be expressed in terms the parameter, c. These relations are

1115

(11)

From equation (3b), the plastic strain for element 4 is written as

500 ~ ¢~ ~v

400

"o 1.1.1 300

,

,

~ ...... D z~ O •

,

i

Eqn. (8) Hashin-Sht rikman [28] Ref[291 Ref. [30] Ref [31} Ref. [32}

i

t

,

.',/~ ....~."~ . . . " ~.." o.-°*.~.°'~" . . ~ * "

m

200 0

~

.-°°°o.°°°•"""•°".-'°..°°"• *•"" •

'

100 U.I 8

I 0.1

I 0.2

I 0.3

I 0.4

I 0.5

I 0.6

I 0.7

I 0.8

I 0.9

Volume Fraction of SiC particles, V p

Fig. 4. The predicted variation in elastic moduli of particulate composites in comparison with experimental data and Hashin-Shtrikman bounds.

1116

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES

It is to be noted that the plastic deformation of element 4 will be influenced by the constraint of rigid particles forming element 3, located above and below element 4 in the actual composite. Hence, its flow stress should be modified for the constraint. The influence of constraint in modifying the in situ flow stress of matrix has been the focus of several studies [4, 33-35]. As the ratio (h/a) of thickness of matrix material of element 4, h, to the size of the reinforcement forming element 3, a, decreases, the constraint experienced by the sandwiched matrix material increases since the effectiveness of rigid particles in preventing lateral plastic contraction would increase. This would increase the microscopic flow stress of the matrix in this region at all plastic strain levels. The modified flow stress can be written as (13)

O'ef t : O'f~)

in which O-fis the flow stress without constraint or in the monolithic form at any plastic strain and O~is a constraint factor dependent on volume fraction of rigid particles. One way to introduce the constraint factor for deformation in the plastic regime is to modify equation (12) to give a lower effective plastic strain for a given applied stress (in this case, O-~) as '4

=(o-l ~ TM.

(14)

•0K ]

In quantifying the constraint factor, ~b, the approximation for the yielding of a ductile interlayer sandwiched between rigid platens can be considered to describe approximately the deformation behavior of the matrix element 4 enclosed between the particles [34-37]. For a ductile interlayer deforming between rigid platens, the constraint factor, based on experimental data, is given by [38]

The stress applied to composite, a¢, is partitioned into stresses a I and O-2 in elements 1 and 2, respectively. Substituting for the volume fractions from equation (2) in equation (4) ac

-

al ( ( l + c ) 2 ~-a2 1

1~5 ) (1+

From equations (18) and (19) [ acc"(1 + c)2 O-2= ~(1 + c ) " + ( 1 -~-C)2Cn--c n1.

El

~-K/

~CC

"

(16)

The plastic strain in element 2 can be written as E2 = ( - ~ )

TM.

(17)

Since plastic strain compatibility should exist between elements 1 and 2 according to equation (5), from equations (16) and (17) ~,

./1 + c ' V = O-2~P~--) •

(18)

(20)

Since the stains in elements 1 and 2 should be equal to the overall average plastic strain of the composite, from equations (5), (17) and (20) the composite plastic strain, eP, is given by o'¢c"(1 + e) 2 E~P= K { ~ b ( l + c ) " + ( l + c ) Z c

] TM

"-c"}j

" (21)

The plastic strain in the matrix in monolithic condition, corresponding to an applied stress, aPm, is

The relative increase in flow stress of the composite with respect to that of the matrix, at a given plastic strain can be obtained by equating equations (21) and (22) and noting O-c= O-~. The resulting expression for normalized flow stress is O-Pmo-P=' ~P[~b (1 + c)"(lc,(l + + "~C)2Cn--c~ l 2c ) .

(23)

The stress-strain curve of the composite for any volume fraction of reinforcement can be determined from matrix properties by using equations

E~¢= O-¢/E¢ for the elastic part (o-¢< a °) (24a) E~--° [ O-¢c.(l+c)Z --Em-~- K{q~(1 + c ) " + ( 1 +c)Zc"-c"}

This equation is accurate in the range 1 ~< a/h <~50, permitting its use in composites with rigid inclusions up to lip = 0.98. Equation (15) offers a convenient way to introduce the sizes of the rigid particle and the matrix material between the particles, since these are related to each other through the volume fraction in equation (1) as a/h = 1/c. Substituting equations (14) and (15) in equation (11).

(19)

]l/n

for the plastic part (o-~> O-0) (24b) in which O-0 is the stress at which the matrix regions in the composite reach yielding. Unlike monolithic form, different regions in the composite reach yielding at different applied stress levels. Since the portion of material occupying element 4 in the composite element 1 (Fig. 3) is required to deform to a larger strain in order to maintain strain compatibility with element 2, this will reach yield stress first even while element 2 is in the stress range of elastic deformation. This could produce a nonlinear stress strain curve after the end of elastic deformation of the composite. However, significant macroscopic plasticity could be realized only after element 2 deforms plastically. Thus, the stress ~0 can be identified as the stress at which the element 2 in the unit cell reach the yield stress of the monolithic material, or the stress at which element 2 or the composite yields to the same prescribed stain as that of the matrix. Hence 0__ 0 O-c -- Ec£m

(25)

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES in which E°m is any prescribed yield strain of the matrix and a ° is the apparent yield stress of composite at this strain. To study the deformation characteristics of transversely loaded continuous fiber composites, similar derivations can also be made for a two dimensional model as illustrated in Fig. 3(c) and (d). The parameters, c, VI, I/2, V 3 and V4 related to fiber volume fraction, Vf, are then given by c=Fll

LV, j

VI-

1

1'2

- 1

(1 + c)

V2= 1 - - (1 + c)

1

C

F3 = ------~(l ) c+ and

F4 = ~ ) ' (cl+

(27)

Using the above relations, following the same procedure as that used for the three dimensional case, the elastic modulus of transversely loaded continuous fiber composite can be expressed as

Ec

=

EfEm(1 + c + c 2) + EZmc Ef(c + c 2)+Era(1 We)

(28)

The predicted variation in transverse modulus is compared with the experimental data of Chen and Lin [39] and Prewo and Kreider [40] for 6061 A1 alloy-boron fiber composites in Fig. 5. Also included in the figure are the data from finite element calculations [39] for square and hexagonal fiber arrangements in the transverse plane of the composite. A good agreement of the present calculations with the expreimental data can be seen. Analogous to the 3D case, the composite strain during plastic deformation is related to the applied stress and the volume fraction of fibers in terms of the matrix deformation constants as

I

The normalized flow stress is then given by o p [d~(l+c)"+(l+c)c~-c"] t 7m = c"(1 + e)

occ"(1+ c)

]1/.

EcP= K { q ~ ( l + e ) " + ( l + e ) c " - c " } J

.

(30)

The stress-strain curves can be calculated from E~ = ~rc/Ec for the elastic part (Oc < a °)

(31a)

0 [ a¢c'(1 + c) ]L,', EP=Em+ K { ~ b ( l + e ) " + ( l + c ) e " - - c " } for the plastic part (a c > a°).

(26) 1

1117

(31b)

It is to be noted that equations (24) and (31) for the deformation in the plastic region have the same exponents as that of the matrix, implying that the strain hardening characteristics of the matrix in the composite is necessarily the same as that of the monolithic form. Thus, the effect of the reinforemerit, according to the present model, is to increase the flow stress at a given plastic strain. This is expected, since coarse particles at low and medium volume fractions (~< 0.5) are not likely to influence the in situ strain hardening behavior of matrix in the composite. However, this is not true either when the particle sizes are small at dilute concentrations of particles ( ~<0.1) or at high volume fractions of coarse particles (>0.5). Interactions of particles with dislocations in the former case and dislocation pile-up at particles boundaries due to shorter glide spacing in the latter, would significantly increase the in situ work hardening rate of matrix material. 3. MODEL VERIFICATION WITH DATA 3.1. Comparison with numerical and finite element data Figures 6 and 7 illustrate comparisons of the predicted variation in normalized flow stress as a function of second phase volume fraction according to equation (23), with self consistent numerical calculations [3] and finite element simulations [12],

" (29) 3

,

i

,

,

~

~

,

,

,

400

~" Q. ~'~v ,,io

]28) Expt. Data; Chert & Lin [39] • Expt. Data; Prewo & Kreider [40] . . . . . . FEM Data; Square Array [39] - - - - FEM Data; Hexagonal Array [39]

350

- -

/

Eqn

[]

300 250

/ /

. -~[t,~,/ ..

-i

"~ zoo "O o

/ / 1 J /

~

150

~

lO0

~ Z

_m l,IJ

05

~. 1.8

...~ "'~

o "~

2.2

50'

02 1.4 0.t ] 0

o

0

,. 01

,. 02

, 03

, 0.4

, 05

,. 06

,

,

,

0.7

0.8

0.9

Volume Fraction of Fibers, Vf

Fig. 5. The predicted variation in elastic moduli of transversely oriented fiber composites in comparison with experimental data and FEM calculations.

0.1

0.2

0.3

0.4

0.5

Particle Volume Fraction, Vp

Fig. 6. A comparison of predictions of the present model with the data from self-consistent scheme (SCS) calculations of He [3] for particulate composites. The SCS data are presented in the same order as the predicted data, in terms of strain hardening exponent.

1118

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES 5

A

E

I~ o ~

- -

4.5

3

7

Eqn. (23); n =0

/

4

w ~

3.5

~

3

i

i

~E

i

,

i

,

Eqn. (30)

_-° .~

O

HU et. al (n =0.2)

..

o

Huet. al(n=O.1)

• H

= ~

2.2

U.

1.8

~

1.4

/i/

•

~

~

/,

/W // o=o2

/ /

/~.~

i

0.1

o

O M. "O

2.5

~

z

ca ~ O Z

1.5 1

1

0.Z

0.4

0.6

0.8

0

0.2

ParticleVolumeFraction, VP

0.4

0.6

0.8

FiberVolumeFraction,

Vf

Fig. 7. A comparison of predictions of the present model with the data from FEM calculations of Bao et al. [12] for particulate composites.

Fig. 9. A comparison of predictions of the present model with the data from FEM calculations of Hu et aL [16] for transversely loaded fiber composites.

respectively, for the case of particulate reinforced composites. Similarly, in Figs 8 and 9, the calculations using equation (30) are presented together with data obtained from self consistent method [3] and finite element method [16] respectively, for the case of transversely loaded continuous fiber composites. The comparisons are made for a range of strain hardening behavior varying from perfectly plastic (n = 0) to maximum linear hardening (n = 1). In general, the present calculations can be seen to be in good agreement with the numerical and finite element data. The agreement is very good for the case of particulate composite. However, the numerical data is available only for volume fractions up to 0.2 for comparison. The data for perfectly plastic deformation behavior is in close agreement with the finite element data for volume fractions up to 0.5 (Fig. 7). The deviations above 0.5 could be due to the differences in the assumed arrangement of second phases in the matrix between the present and the finite element study [12]. In the case of transverse deformation of fiber composite, a good agreement between the present

estimates and numerical data can be seen for n = 1.0, while considerable deviation to the extent of about 25% occurs at low strain hardening exponents. However, this can be said only for volume fractions up to which the data a r e available for comparison. In contrast, the present data agreed better with the finite element data for low strain hardening exponents (n ~< 0.2).

i

i

i

t

f

3.2. Comparison with experimental data

Systematic experimental data on two phase composites having rigid particle embedded in a metallic matrix capable of undergoing significant plastic strain are not widely available. Although a number of metal matrix composites such as AI-SiC particulate systems [29-32,41,42], two phase materials such as dual phase steels [14, 43] have been studied, complications arise due to the reaction between particulate and reinforcement, poor interfacial bonding, contiguity of particles and the difficulty of precise knowledge of in situ matrix mechanical behavior relative to that of monolithic matrix. However, some data free from above problems are available [44-46] for polymer based composites having rigid particle dispersions. Figure 10 compares the predicted normalized flow

/ 2.6

...... SCS Data from He {3]

5

/ /

~o

n =1.o o O~

E.

ro

.'*'

L 23

1

o.1

02

o.3

......

/L// /7

Eqn. (23) FEMdata;°

=0;.e, tt3]

/ ~

o=0.2

4 3.5

0.33

1.8

o

-

0.5

1.4

Z

4.5

0.4

FiberVolumeFraction, V~

o.s

~

3

IO 14. "O

2.5

~

2

~ O Z

1.5 1

02

Fig. 8. A comparison of predictions of the present model with the data from self-consistent scheme calculations of He [3] for transversely loaded fiber composites. The SCS data are presented in the same order as the predicted data, in terms of strain hardening exponent.

0.4

0,6

0.8

ParticleVolumeFraction, VP

Fig. 10. The predicted variations in normalized flow stress compared with experimental data of Young and Beaumont [44] and FEM data of Hom and McMeeking [13].

1119

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES 10

-

"~Oo

'

.

- -

0 2.6

D

EQn. . .(23) . . Expt.Oata;Ref.[45] Expt.Data;Re,.[,5]

/4/ /

/

.

E 9

L ~

n

/'L'~

0.1

--

2.2

E

~.8

.

.

.

.

n = 0.2

=0.2 ,~__~ 8675

~

.

Eqn. (23)

ii "O

~ Exp

4

=.

Z 1 .,~'T

~

o.2

i

i

i

I

I

I

0.6

0.4

E O

z

Z

1

Particle Volume Fraction, V

i

0

0.8

01

0.2

i

i

i

i

i

i

i

0.3

0.4

O.S

0.6

0.7

08

0.9

WC Volume Fraction, V

P

P

Fig. 11. The predicted variations in normalized flow stress compared with experimental data of Ishai and Cohen [45] generated at different strain rates.

Fig. 13. The variation in the predicted normalized flow stress in comparison with the experimental data of WC-Co systems from Ivensen et al. [47].

stress as a function of volume fraction with experimental data of Young and Beaumont [44]. A similar comparison is made with the data of Ishai and Cohen [45] from tests at different strain rates in Fig. 11 and also with the data of Turcsanyi et al. [46] for perfectly bonded composites in Fig. 12. Since the in situ strain hardening coefficient of the matrix in the presence of particle are not known, n values of 0, 0.1, 0.2 and 0.5 were used in the calculation [equation (23)]. A good correlation between the experimental data and the theoretically predicted trends for different strain hardening exponents of the matrix can be seen. The flow stress data as a function of volume fraction from cermets based on W C - C o are compared with the predicted behavior in Figs 13 and 14. In W C - C o systems, the angular WC grains dispersed in a continuous plastically deforming Co matrix contribute very little to deformation even at moderate temperatures ( < 1000°C). Hence, the deformation characteristics of these cermets, having nearly the same geometrical microstructural arrangement as that of the present model could be best illustrated. Figures 13 and 14 illustrate the data from Ivensen et al. [47] and Doi [36] respectively, in comparison with the predictions from equation (23). The strain hardening characteristics of the in situ Co binder

would be significantly affected by the smaller interparticle spacing relative to particle size, and hence the calculations with different strain hardening coefficients are shown in both figures. At large volume fractions the experimental data tend to be lower than the predicted behavior. There are two reasons for this behavior. First, the contiguity of WC grains, especially at low volume fractions of Co, could cause premature failure along W C - W C boundaries at a stress lower than that could be borne by constrained deformation of Co sandwiched between WC grains [36, 48]. Secondly, the assumption of rigid WC particle could be violated at such high stresses by the deformation of WC itself [49]. Although these factors are unknown, the calculated trends agree reasonably well with the data. The stress-strain curves of composites in uniaxial tension can be predicted by equations (24) and (31) for the particulate and transversely loaded continuous fiber composites respectively, from the matrix tensile behavior. However, not many systematic experimental data are available to evaluate the predictions. These are considerable data available [16,21,41,42] on A1-SiC particulate composites, which are influenced by the effects of residual stress

3

2022

,

° 0! 02

,

,

,

o,

"•:

I U.

E

n 6.2

t3 Expt.Data;Ref.{36] I Expt.Data;Ref. [36] -Eqn (23)

TOo 17 ¢~ ~)t/)

0.1 0v / ~ l ~ ~ n

9 ii D

1.8

1.4

Eqn. [231

m O Z

Z 1

~f~

S

1

0 0.2 0.4 0.6 0.8 Particle Volume Fraction, Vp

Fig. 12. The predicted variations in normalized flow stress compared with experimetnal data of Turcsanyi et al. [46].

I

I

0.1

0.2

0.3

i

I

I

I

I

I

0.4

0.5

0.6

0.7

0.8

0.9

WC Volume Fraction, V

P

Fig. 14. The variation in the predicted normalized flow stress in comparison with the experimental data at two plastic strain levels of W C ~ o systems from Doi [36].

1120

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES

and particle-matrix reaction after and during processing respectively. Evaluations [16,21] of finite element calculations with the data suggest that accurate knowledge of in situ deformation behavior of the matrix affected by microstructural changes due to processing, is necessary for a better agreement. However, the calculations of the present model can be compared with the finite element calculations reported elsewhere [10, 15, 50]. Figure 15 shows the comparison between the predictions of equation (24) with the stress-strain curve generated by finite element method [10] for A1-50% SiC composite, assuming a discrete arrangement of particles in a continuous matrix. Similarly, Fig. 16 illustrates the trend predicted by equation (31) along with the finite element calculations [15] for a transversely loaded, 6061 aluminum alloy reinforced with 0.5 volume fraction boron fibers. The experimental data [50] for a Ti-alloy-34 vol.% SiC fiber composite, loaded in transverse compression is compared in Fig. 17 with the prediction from equation (31). Since fiber reinforced composites have weak interfaces between the fiber and the matrix giving a tri-linear stress-strain response [50, 51] due to interface debonding during tensile deformation, these data were not considered. Since the compression data is free from the effects of debonding, these data were used for evaluation. In all these cases, the present model is seen to produce the stress-strain curves with good accuracy. 4. DISCUSSION The above evaluations demonstrate that the present model can adequately describe the unidirectional deformation behavior of two phase composites in the same way as rigorous, self-consistent and finite element calculations. In a previous study [52], the same unit cell based approach was shown to represent the steady state creep behavior of two phase solids accurately. The calculations [27] of elastic moduli of two phase systems also provided a better representation of experimental data for two phase systems

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Fig. 16. The predicted stress-strain curve for the transverse loading of a 6061-A1 alloy reinforced with boron fibers in comparison with FEM data from Karlak et al. [10]. relative to the bounds given by Hashin-Shtrikman calculations [28] based on the variational principle. It is instructive to compare and contrast the basis and assumptions on which the present model is constructed relative to those on which other methods of calculation of deformation properties of two phase solids are founded. First, in the elastic deformation regime, it can be seen that stresses are inhomogeneous in different parts of the composite. According to the division of elements employed in this study, higher stresses will be partitioned in element 1 relative to element 2. This can be understood physically, in the sense that if we envision elements 1 and 2 as two simple bars with different elastic moduli loaded by a remote force (applied stress X area of cross section of composite), strain compatibility would require that the bar having a higher elastic modulus bear a high proportion of the load. In the composite element 1, this results in a higher strain in portions of low modulus phase (matrix) to compensate for the low or no strain in the stiffer part of the element (reinforcement). The same mechanism is applicable when the composite deforms in a macroscopically plastic regime. Hence, the stresses in different parts of the 1200

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I

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Fig. 17. The predicted stress-strain curve for the transverse loading of a Ti-alloy reinforced with 34% SiC fibers in comparison with experimental data in compression from Majumdar and Newaz [50].

RAVICHANDRAN: DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES matrix surrounding the particle are required to be different for uniform compatible deformation of the unit cell both during elastic and plastic deformation of the composite. However, the transition from elastic to plastic regime occurs at different levels of composite applied stress in different parts of the matrix. Since the matrix portions (element 4) above and below the rigid reinforcement (element 3) are subjected stresses higher than the matrix in element 2, the former regions enter yielding and subsequent plastic deformation first, while element 2 remains in elastic region. In the transverse compression of fiber reinforced composites, these regions were experimentally seen to plastically deform first while the rest of the matrix undergoes elastic deformation [50, 51]. Such inhomogeneous deformation in regions located above and below the particle or fiber has also been illustrated in finite element simulations [11-13, 15]. This also is the likely reason why the calculated stress-strain curve is not accurate in the transition region (Figs 16 and 17). Further loading beyond the elastic-plastic transition, will eventually require element 2 also to deform plastically although still lagging behind element 4 in terms of absolute plastic strain. This process would continue until the highly strained regions in element 4 nucleate some form of fracture event, such as void nucleation, cleavage or interface debonding in the case of weakly bonded reinforcements. The criterion used in the present calculations assumes that gross yielding is achieved when the overall composite strain is equal to the monolithic matrix yield strain. In reality, the stress-strain curve should be modeled in three stages, namely elastic, elastic-plastic and fully plastic regimes. In evaluating the present model with finite element calculations in particular, it is clear that such inhomogeneous deformation is properly accounted for in finite element simulations since the elastic and plastic portions of monolithic stress strain curve are fed through a constitutive law and strain compatibility is ensured between different regions of the composite microstructure. However, the macroscopic composite behavior is eventually arrived at by averaging the stresses and strains. For instance, in unit cell finite element calculations [12, 13, 15], the macroscopic stress is determined as the net force in the loading direction divided by the area of cross section of the unit cell, allowing for uniform lateral contraction in the plastic regime. The macroscopic composite strain is determined by adding the displacements in the elements in the direction of applied force and dividing by the length of the unit cell. It is thus evident that the methods of macroscopic averaging of stresses and strains are nearly equal in the present model and finite element calculations. Thus the good correspondance between the simple model calculations and finite element simulations could be understood. However, finite element simulations provide detailed microscopic patterns of stresses and strains, aiding the location of first yielding or fracture precisely.

1121

In numerical calculations based on self-consistent methods [3, 5, 6, 9, 22], each of the two phase for which the deformation behavior is known, is successively embedded into a medium having the composite deformation properties as unknowns. The macroscopic stresses and strains are again arrived at by invoking strain compatibility between the phases. A shortcoming in this method is the assumption that at any stage during composite deformation, the stresses and strains in each phase are related to the composite properties by the simple rule of mixture. Although the stresses and strains in the embedded and the embedding phase are permitted to be different, within each phase, these are considered to be uniform irrespective of location of the phase, for example, matrix regions in the composite. However, the discussion presented earlier as well as finite element calculations [11-13, 15] and experimental observations [50, 51] reveal that in reality the stresses and strains are inhomogeneous in the continuous matrix phase. It is not clear whether this difference in matrix behavior reflects in the difference between the predictions of the present approximation and the self-consistent calculations. Due to a lack of systematic experimental data, only the data from polymer based composites were employed to assess the predictions from the model for the case of particulate composites. In metal matrix composites, the in situ matrix properties are sensitive to processing conditions and reactions with reinforcement. Hence, comparisons can be made only when matrix properties with identical processing conditions as that of the composite are known [41]. For example, aluminum alloy metal matrix composites show accelerated aging in the presence of reinforcements due to residual stresses and strains occurring after processing [53]. In addition, the particles in many cases are not equiaxed, leading to variations in mechanical properties to the extent influenced by the shapes of reinforcements [54]. For the WC-Co systems used in Figs 13 and 14, similar problems exist due to the dissolution of W and C and subsequent reprecipitation of compounds in Co [55]. However, a reasonable value of matrix flow stress chosen from the data for alloys of controlled composition was used in the model predictions. Additionally, the contributing effects to the in situ flow stress of Co, due to Hall-Petch type dislocation interactions with WC particles have also been suggested to be important [4, 36]. This effect could in principle can be incorporated into the model to accurately represent deformation behavior at large concentrations of WC particles. The present study is clearly an approximation of several microstructural features such as inclusion shape, size distribution, spacial arrangement of inclusions of the actual composite. Several studies [3, 12, 15, 18,41, 54, 56] show that the accuracy of predicted results depend on the these factors significantly. Since the primary objective of this study is to

1122

RAVICHANDRAN:

DEFORMATION BEHAVIOR OF TWO PHASE COMPOSITES

simplify the modelling approach to result in approximate equations for the deformation behavior, such factors have to be ignored. Details on the magnitude of these effects can be found in the references cited above. It is clear that although this simple model does not take into account explicitly the three dimensional stress fields and the likely interactions between particles, it provides an approximate yet reasonably accurate characterization of deformation behavior of two phase composites. It illustrates the progress of deformation from elastic to plastic regime while bringing out the necessity of stresses and strains being inhomogeneous to maintain strain compatibility between the different parts of the matrix. The constrained deformation of matrix at regions between the particles appear to be important in elevating the flow stress of composite. Following this approach, it should be possible to compute the stress-strain characteristics of the composites containing isolated ductile inclusions embedded in a continuous, brittle matrix material. This situation is just the opposite of the case modeled in the present study. It is also obvious that several possible arrangements of microstructure, for a given volume fraction of second phase, can be examined theoretically, to arrive at the best microstructural arrangement for maximum deformation resistance. Composites [57] based on eutectic microstructures often involve primary phases surrounded by eutectic mixtures of two phases. The flow behavior of microstructures of this nature, after a detailed quantification of the microstructure for volume fraction and morphology can also be estimated through an extension of the approximations outlined in this study. 5. CONCLUSIONS Approximate, closed form equations to predict the flow stress and the stress-strain behavior of composites containing rigid second phases, from the matrix flow characteristics, were derived through an unit cell representation of microstructure. Simple expressions provide accurate estimates of the elastic modulus and the flow stress at any plastic strain from a knowledge of constituent properties. A comparison of the predictions with data from finite element simulations and self consistent numerical estimations show that the present model describes the composite deformation behavior with good accuracy. The estimates of the increase in flow stress relative to that of the matrix, as a function of volume fraction were also shown to be in good agreement with experimental data on polymer based composites and WC-Co cermets. The stress-strain curves generated through this model also agree well with both the experimental and finite element data for the case of transversely loaded continuous fiber composites. The constrained deformation of matrix between rigid phases is seen as crucial in increasing the composite flow stress.

Acknowledgements--The author gratefully acknowledges

helpful discussions with Dr V. Seetharaman, UES, Inc., and Dr S. L. Semiatin, Wright Laboratory, Wright Patterson Air Force Base, OH 45433. REFERENCES

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