TRANSVERSE STRENGTH OF METAL MATRIX · PDF fileTRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES REINFORCED WITH STRONGLY BONDED ... bonded to the matrix, the composite will necessarily

Pergamon

Acta metall, mater. Vol. 42, No. 9, pp. 2983-2997, 1994 Copyright © 1994 Elsevier Science Ltd

0956-7151(94)E0108-S Printed in Great Britain. All rights reserved 0956-7151/94 $7.00 + 0.00

TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES REINFORCED WITH STRONGLY BONDED CONTINUOUS FIBERS IN REGULAR ARRANGEMENTS

D. B. Z A H L ~, S. S C H M A U D E R 1 and R. M. M c M E E K I N G 2

~Max-Planck-Institut fiir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestrasse 92, D-70174 Stuttgart, Germany and 2Mechanical Engineering Department, University of California,

Santa Barbara, CA 93106, U.S.A.

(Received 16 February 1994)

Abstract--The composite limit flow stress for transverse loading of metal matrix composites reinforced with a regular array of uniform continuous fibers is calculated using the finite element method. The effects of volume fraction and matrix work hardening are investigated for fibers of circular cross section distributed in both square and hexagonal arrangements. The hexagonal arrangement is seen to behave isotropically with respect to the limit stress, whereas the square arrangement of fibers results in a composite which is much stronger when loaded in the direction of nearest neighbors and weak when loaded at 45 ° to this direction. The interference of fibers with flow planes is seen to play an important role in the strengthening mechanism. The influence of matrix hardening as a strengthening mechanism in these composites increases with volume fraction due to increasing fiber interaction. The results for a power law hardening matrix are also applicable to the steady state creep for these composites. The influence of volume fraction on failure parameters in these composites is addressed. Large increases in the maximum values of hydrostatic tension, equivalent plastic strain, and tensile stress normal to the fiber-matrix interface are seen to accompany large increases in composite strength.

INTRODUCTION

Lightweight metals reinforced with ceramic fibers are potentially valuable in aerospace and transportat ion applications due to high specific strength and creep resistance. Composites reinforced with short fibers or particles have been the subject of considerable investigation, and such effects as inclusion volume fraction and shape [1-3], inclusion orientation [4, 5], particle cracking [5], particle clustering [6, 7], interfacial behavior [8, 9], Bauschinger effects [10], and thermal residual stresses [11,12] have been addressed. Cont inuous fibers provide the greatest strengthening, when loaded in the direction of the fibers. How- ever, when loaded in a transverse direction the reinforcement provides little strengthening. Jansson and Leckie [13] carried out calculations to model the behavior of cont inuous fiber composites under transverse cyclic loading. Brhm and coworkers [14, 15] and Brockenbrough and coworkers [16, 17] examined the influence of fiber cross section and arrangement on the thermal residual stresses and subsequent behavior of the composite under load, both mechanical and cyclic thermal. The cross-sectional shape and arrangement were seen to have little effect on the axial thermal strain response of the composite. How- ever, strong effects were noted on the thermal residual stress and strain distributions, as well as on the stress-strain behavior under transverse loading of the

composite, including the transverse elastic response. Square cross-sections were found to provide greater strength and stiffness than round cross-sections for a given fiber arrangement. A square arrangement of fibers loaded transversely in the direction of closest neighbors was seen to be stronger and stiffer than a hexagonal arrangement, which in turn was stronger and stiffer than the square arrangement loaded diagonally.

The hexagonal arrangement results seem to agree well with responses for a more random arrangement of fibers. The influence of cross sectional shape was found to have a lesser impact on the composite behavior than the arrangement of the fibers. Dietrich [18] found that composites with the hexagonal fiber arrangement can show appreciable anisotropy at large strains. Herath [19] and Gunawardena [20] have developed constitutive laws for strongly and weakly bonded composites, respectively, for implementation in finite element code. However, their results were obtained for one volume fraction only and for a specific set of material parameters.

The purpose of this paper is to further investigate the transverse behavior of continuous fiber composites and to systematically study the effects of fiber arrangement, volume fraction and matrix hardening. Only circular fibers will be studied here. The elastic response of two phase composites is reasonably well understood, so the focus of this

AMM 42/9---F 2983

2984 ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES

/

°°l 2 0 0 0 0

(a)

°°l

(b) Fig. 1. (a) Square arrangement of fibers with primary loading directions, and (b) hexagonal arrangement of fibers

with primary loading directions.

paper is limited to the fully developed plastic flow of these composites. The fibers are well bonded to the matrix so that no debonding or sliding is permitted at the interface. The finite element method is employed within the framework of continuum mechanics to carry out the calculations. Figure 1 presents the fiber arrangements considered in this work. A square arrangement of fibers is shown in Fig. l(a), with the loading directions at 0 ° and 45 ° indicated. Similarly, Fig. 1 (b) represents a hexagonal arrangement of fibers, with the loading directions at 0 ° and 30 ° shown. Note that in the hexagonal arrangement the 0 ° direction is identical to the 60 ° direction and the 30 ° direction is identical to the 90 ° direction.

MODEL FORMULATION

A plane strain model is used, since transverse strength is studied here and the fiber length is much greater than either the fiber diameter or fiber spacing. In fully plastic flow there is no plastic strain in the axial direction in a continuous fiber composite and therefore a plane strain model is appropriate. The

repeating cell used for the calculations is shown in Fig. 2. The boundary conditions are such that the lateral edges of the cell have zero average normal traction and zero shear traction. The loaded edges of the cell are given an average normal traction and zero shear traction. The cell is forced to remain a rectangle and cannot rotate. The fiber is rigid. Modelling the fibers as rigid does not influence the fully plastic behavior of the composite although elastic and transient response prior to full plasticity will be influenced by the fiber rigidity. In addition, the fibers are well bonded to the matrix so that no motion is permitted to occur on the fiber perimeter. Examples of finite element meshes of the different arrangements are shown in Fig. 2 for a volume fraction of 0.4. Figure 2(a) is a model for a square arrangement loaded in the 0 ° direction, Fig. 2(b) is for a square arrangement loaded in the 45 ° direction, and Fig. 2(c) is for the hexagonal arrangement, for both the 0 ° and 30 ° loading directions.

A continuum mechanics approach is used to model the composite behavior, thus eliminating the influence of size from the calculations. The uniaxial matrix stress-strain behavior is characterized by

a = ¢r0(~0) = EE E~<E0 (la)

ff = (7 0 E > 6 0

l l

(a) (b)

~(3o °) 4

(c)

Fig. 2. Finite element meshes for: (a) the square arrangement loaded in the 0 ° direction, (b) the square arrangement loaded in the 45 ° direction, and (c) the hexagonal arrangement loaded in either the 0 ° or 30 ° directions. All cells

shown represent a volume fraction of 0.4.

ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES 2985

for the perfectly plastic case, and by

a = e0 = EE E ~ % ( l b )

O" ~--...~ O" 0 E " ~ E 0

for the strain hardening cases. The parameter ~ is the axial stress, E is the axial strain, e0 is the yield stress in tension, E 0 = eo/E, E is Young's modulus, and N = l/n is the strain hardening exponent.

The J~ flow theory is employed with a yon Mises yield criterion to characterize the rate-independent matrix material. The solutions were calculated incre- mentally, with the stress increment related to the strain increment through

E ". v . (~ij = ~ Eij ~- ~ ~kk (~ij

3 SijSklEkl

where 6- 0. is the stress rate, i~ is the strain rate, 3 so= a!j--akk/3 is the deviatoric stress, G =~/Ssoso

is the tensile equivalent stress, v is Poisson's ratio, and E t is the current tangent modulus of the stress-strain curve. The last term in (2) is zero for an elastic increment.

Figure 3 presents the features of the overall stress-strain curves of primary concern in this work, following Bao et al. [1]. For the case of fibers perfectly bonded to the matrix, the composite will necessarily harden with the same strain hardening exponent, N, as the matrix, when strains are in the regime of fully developed flow. At sufficiently large strains the composite behavior is then described by

e = eN (3)

where ~ is the overall stress, E is the overall strain, and 6N is called the asymptotic reference stress. This is demonstrated in Fig. 3(a) for a non-hardening matrix, with ~0 being the limit flow stress. The asymptotic reference stress, 6N, can be determined by normalizing the composite stress by the stress in the matrix alone at the same overall strain, as indicated in Fig. 3(b). In a similar manner, a matrix which experiences a steady state creep response given by

~ = ~o( °" '] ~ (4) \ ~ 0 /

where ~ is the strain rate, ~0 is a reference strain rate, and n = 1/N is the creep exponent, will result in a composite whose steady state creep response is

-~o1%

Z 1 ---' / Matrix

Strain

(a)

Limit Flow stress, -~ O

" ~ N / ( ~ O

fi O

Composite, ~ /

. ~ Composite, ~ / o(e)

~ " Matrix, a(~)/c~ °

Strain

(b)

Asympto t i c Reference Stress, N

Fig. 3. (a) Limit flow stress, a0, for a non-hardening matrix, and (b) asymptotic reference stress, fiN, for work hardening

matrices.

described by 1

= i0 (5)

where #r~ has the value computed for rate independent power law hardening.

The ABAQUS finite element code [21] was employed using 8 noded two-dimensional plane strain biquadrilateral elements. An IBM RS/6000 Model 540 work station was used to carry out the calculations, which typically took 30 min to compute a stress-strain curve for a specific model with one set of material parameters.

RESULTS

Non-hardening matrices

The stress-strain curves for composites with fibers in the square arrangement and loaded in the 0 ° direction are given in Fig. 4 for a non-hardening matrix. No improvement in the limit flow stress is


seen for fiber volume fractions less than about 0.4, save that due to the plane strain condition. For these low volume fractions the composite is seen to be weaker than the matrix at strains slightly above the matrix yield strain, due to localized yielding around the fibers. At fiber volume fractions greater than 0.4 the composite strength increases with volume fraction, reaching a strength of 3 times the matrix yield stress at a volume fraction of 0.7. The fibers will come into contact, and the composite as modeled will thus have infinite strength, at a volume fraction of about 0.79.

Figure 5 is a comparison between the plane strain model for fiber composites and an axisymmetric model for particulate composites, both with non-hardening matrices. The plane strain model represents transverse loading of a material with continuous fibers while the axisymmetric case represents uniaxial loading of a material with spherical inclusions [1]. The results are compared for the same reinforcement cross-sectional area fraction rather than for the same volume fraction. That is, the area of reinforcements on a planar section is the same. In the plot, the appropriate matrix yield stress a0 for the axisymmetric case and (2/~/3)a0 for the plane strain case, is subtracted off, thus eliminating the strengthening due to the plane strain condition from the comparison. This renders the plot a rep- resentation of the increase in strength due to the reinforcements. The axisymmetric model for spherical reinforcements shows a slightly higher effective strength for a reinforcement diameter to cell size ratio, D/L, of less than about 3/4, where L is the center to center spacing between reinforcements. This ratio of 3/4 represents a volume fraction of about 0.44 for the continuous fiber case and about 0.28 for the case of spherical reinforcements. For higher values of D/L the continuous fiber model exhibits a greater increase in strength. It should be noted that the fiber case is stronger for all relative inclusion diameters due to the plane strain effect. However, when comparison is made for the same volume fractions, the axisymmetric model for spherical reinforcements results in a stronger composite for volume fractions greater than about 0.28. This is because an increase in volume fraction with spherical inclusions is characterized by a sharper increase in constraint, as indicated by D/L, than occurs for the same increase in volume fraction with fiber reinforcements. This is demonstrated in the extreme when the reinforcements touch and the composite becomes infinitely strong. This occurs at a lower volume fraction for spherical reinforcements than for fiber reinforcements.

Stress-strain curves for fibers in a square arrangement and loaded in the 45 ° direction are given in Fig. 6. The matrix is again non-hardening. It is clear that no strengthening above that due to the plane strain condition takes place, even for very large volume fractions. As before, the maximum volume fraction for this arrangement is about 0.79. As with

the 0 ° loading, the composite is seen to be slightly weaker than the matrix at strains in the vicinity of the matrix yield strain, due to localized yielding around the fibers.

Calculations for loading in both the 0 ° and 30 ° directions were carried out for non-hardening matrices with fibers in a hexagonal arrangement. A comparison of typical results in Fig. 7 shows the results to differ only in a transient stage between elastic and fully plastic behavior. The limit flow stress is independent of the loading direction for this arrangement as the fully developed flow patterns are identical. The 30 ° loading direction is seen to be stronger than the 0 ° loading direction only during the transient stage. However, this region is short lived, and the limit stress is achieved at strains of about twice the matrix yield strain.

Figure 8 presents stress-strain curves for the hexagonal arrangement, loaded in the 0 ° direction. Little strengthening, aside that due to the plane strain condition, is seen for volume fractions up to about 0.7. At a volume fraction of 0.8 this composite has a limit stress of about 3.6 times the matrix yield stress. The maximum volume fraction for this configuration is about 0.91 at which the fibers touch. Apart from the transient stage, the stress-strain curves for the hexagonal arrangement loaded in the 30 ° direction would be almost the same.

Figure 9 presents a summary and comparison of the limit flow stresses seen in the previous plots. The hexagonal arrangement provides slightly higher strengthening over the square arrangement at volume fractions less than about 0.4, whereas the 0 ° loading of the square arrangement provides greater strengthening for volume fractions greater than this. The square arrangement loaded in the 45 ° direction results in no strengthening apart from that due to the plane strain condition, regardless of volume fraction. A marked rise in strengthening is seen in the square arrangement loaded at 0 ° at a volume fraction of about 0.4. Similarly, the hexagonal arrangement experiences a marked rise in strengthening at a volume fraction of about 0.7. The improved strengthening of the hexagonal arrangement at low volume fractions is attributed to the restriction that shear deformations are only possible on 60 ° planes to bypass the fibers, rather than on the 45 ° planes on which the shear stress is maximum. This effect is more marked with increasing volume fraction.

The sharp rises seen in these curves can be attributed to the shear bands being impinged by the fibers. This effect is illustrated in Fig. 10. When a straight line can be drawn through the matrix in the maximum shear stress direction, there is no strengthening. Furthermore, the degree of strengthening correlates with the acuity of the plane just passing through the matrix tangential to the fibers. For the square arrangement with the 0 ° loading direction, the 45 ° shear planes become affected at a critical volume fraction of 0.3927, as shown in Fig. 10(a). Below this

ZAHL e t al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES

4

3.5

o 3

- 2.5

2 -o

~=~

1.5

O

Z 1

0.5

0

@

I ' ' ' ' l ' ' ' ' I ' ' ' ' I ' ' ' '

/=0.7

0.6

0.5

0.4

0.3 0.2 0.0

0 1 2 3 4 5 6

N o r m a l i z e d S t r a i n , ~-/ O

Fig. 4. Stress-strain curves for the square arrangement loaded in the 0 ° direction. The matrix is non-hardening.

2987

' I ' I ' ' I I '

o 1.5

o

' o 1

0.5

0

l Continuous Fibe

a O = O O

-05. , , , I , ~ , I , , , I , , , I , , ,

0 0.2 0.4 0.6 0.8

R e l a t i v e R e i n f o r c e m e n t S i z e , D / L

Fig. 5. Comparison of the plane strain results with results for an axisymmetric model with the same dimensions. The matrix is non-hardening.

2988 ZAHL e t al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES

1.5 . . . . I . . . . I . . . . t . . . . I . . . . I . . . . I . . . . I . . . .

1.25

I 0.6 0.5

0~ 0.75 0.4 (~

•

0.5

0.25

0

0 0.5 1 1.5 2 2.5 3 3.5 4

N o r m a l i z e d Stra in , ~- /e O

Fig. 6. Stress-strain curves for the square arrangement loaded in the 450 direction. The matrix is non-hardening.

1.5 . . . . I . . . . I . . . . t . . . . I . . . . I . . . .

D °

¢ / 3

t ~

0.5 1"4

O

Z

°°1/ 3oo

/ ;

f=0 .6

I I I I I I I I I I I I I I I , , , , I , , t , I , , , ,

0 0.5 1 1.5 2 2.5 3

N o r m a l i z e d Strain, ~ ' / e O

Fig. 7. Stress-strain curves for the hexagonal arrangement loaded in the 0 ° and 30 ° directions. The volume fraction is 0.6. The matrix is non-hardening.

Z A H L et al.: T R A N S V E R S E S T R E N G T H O F M E T A L M A T R I X C O M P O S I T E S

4 ~ ' ' ' ' I ' ' I ' ' ' ' I ' ' ' ' I , ' ' ' I ~ ' '

3.5 i °l o 3

2.5

1.5 ~ 0.7

O

Z 1

0.5

0 , , , , i , , ~ , i i , i , I , , , , I , ~ J , i ~ ~ ~ ,

0 1 2 3 4 5

Normalized Strain, ~/E o

Fig . 8. S t r e s s - s t r a i n c u r v e s f o r t he h e x a g o n a l a r r a n g e m e n t l o a d e d in t he 0 ° d i r ec t i on . T h e m a t r i x is n o n - h a r d e n i n g .

2989

,o

o 2.5

U 3

2 0

" 0 0

fi

Z

' ' ' I ' ' ' I ' ' ' 1 ' ' ' I ' ' ' I ' ' ~ l l " " ' " ' ~ l ' ' ~ ' I ' ' " " "'

@ f

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Volume Fraction, / Fig. 9. C o m p a r i s o n o f l imi t y ie ld s t ress , 6o, fo r b o t h s q u a r e a n d h e x a g o n a l a r r a n g e m e n t s w i t h a

n o n - h a r d e n i n g m a t r i x .


volume fraction, lines parallel to the maximum shear stress direction can pass entirely through the matrix. In this situation plastic flow can occur unimpeded by the fibers and so there is no strengthening. Above f * = 0.3927 the plastic flow has to accommodate the presence of fibers and so constraint develops accompanied by composite strength. On the other hand, when the square arrangement is loaded in the 45 ° direction, the 45 ° shear planes are intact until the fibers contact one another at a volume fraction of 0.7854, Fig. 10(b). In the hexagonal arrangement, a 45 ° line representing the maximum shear stress direction will always pass through a fiber no matter what the volume fraction. Thus there will be a degree of constraint in the plastic flow of the matrix and therefore some composite strengthening at all volume fractions. This feature is apparent in Fig. 9. However, above a critical volume fraction of 0.6802, no straight line can be drawn entirely through the matrix. In this situation, the pattern of plastic flow must be more complex than that prevailing below f * = 0.6802. This transition is accompanied by a significant increase in plastic constraint and consequently by more substantial composite strengthening, Fig. 9.

Some empirical results are interesting relative to Fig. 9. The results for both the square arrange- merit and the hexagonal arrangement can be approxi- mated as

go = a * (6)

/ , ~ 4 5 ° 45 °

l..

(a) (b)

(c)

Fig. 10. Critical volume fraction, for (a) the square arrangement, f*=0.3927, (b) the square arrangement loaded at 45 °, fmu = 0.7854, and (c) the hexagonal arrangement, f * = 0.6802. (fmx = 0.9069 for the hexagonal arrangement.)

Table 1. Constants for equations (6) and (7)

Square arrangement 2 - - % 14.2 0.345 0 ° loading x/~

Square arrangement 2 - - a~ 0.0 N / A 45 ° loading x/~

Hexagonal arrangement 2 - - % 0 + 0.26f) 27.2 0.634

for volume fractions less than J" and by

g0 = C1 ( f - ~ ) 2 + ~r g (7)

for volume fractions greater than f , where a *, Cl and .f are summarized in Table 1.

Equations (6) and (7) represent fits to the calculated limit stress as given in Fig. 9 for the volume fractions considered. It is possible that there are geometric interpretations of these results relating them to possible shear planes in the matrix and the size of the gap between fibers [22]. The volume fraction ~r has a value about 5 vol.% less than f * and presumably represents the situation in which the shear bands become sufficiently constrained to noticeably affect the plastic flow.

Hardening matrices

Stress-strain curves are shown in Fig. 11 for the square arrangement of fibers loaded in the 0 ° direction for composites with a strain hardening matrix and with a strain hardening exponent of N = 0.2. These results are given also in Fig. 12, however in the latter figure the stress is normalized by the stress which the matrix alone would experience at the same applied strain. At large enough strains, e.g. ( > 10%, these curves approach the asymptotic ~N/ao and therefore can be used to determine the asymptotic limit reference stress gN. In addition, these curves are similar to those depicted in Fig. 4. However the steady state stress in the hardening case is greater than the limit flow stress seen for the non-hardening matrices. This is attributed to an increase in effective fiber diameter due to localized yielding and strain hardening of the matrix around the fibers. Initial yielding takes place adjacent to the fibers. This matrix material then hardens to some extent and thus has a higher yield stress than the remaining matrix material. As a result, even when the matrix is yielding everywhere, the deviatoric stress will be higher around the fibers than in the rest of the matrix. This effect increases with increasing strain hardening exponent, N. The effect is made more evident in Fig. 13, where the asymptotic reference stress is plotted vs volume fraction for strain hardening exponents of 0 (non-hardening), 0.1, 0.2 and 0.5. The increase in strength due to fiber constraint during strain hardening increases with increasing volume fraction, but becomes significant only at volume fractions greater than the critical volume fraction, f * .

I ~ 8

o~ 6

2

ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES

12 , ' I ' ' ' I ' ' I ' ' ' I '

~ l N = 0 . 2

10 ?

0 J , ~ I , , , l a , , I ,

0 4 8 12

0.6 J

0.5

0.4

0.2 ~ ' ~ 0 . 0 - -

I I I I I I

16 20

N o r m a l i z e d S t r a i n , ~-/e O

Fig. 11. Stress-strain curves for the square arrangement of fibers loaded in the 0 ° direction. The matrix strain hardens with an exponent of N = 0.2.

2991

6

5

o~ 4

3

O

Z 2

1

@ I ' I ' ' ' I ' ' I

N = 0 . 2

I , 0 . 3 / ~ / ~

o 0 4 8 12 16

/--0.7

0.6

0.5

0.4

20

N o r m a l i z e d S t r a i n , ~ - / e O

Fig. 12. Stress-strain curves for the square arrangement of fibers loaded in the 0 ° direction. The matrix strain hardens with an exponent of N = 0.2. The stresses are normalized with respect to the stress the

matrix alone would experience at the same applied strain.

2992 ZAHL e t al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES

12

o

10

U

6

u

"~ 4 ,9.o o.,

g • < 2

N = 0.5

@ 0.2

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

V o l u m e F rac t i on , f

Fig. 13. Asymptotic reference stress, aN, for the square arrangement loaded in the 0 ° direction for matrix work hardening exponents N = 0, 0.1, 0.2 and 0.5.

3.5 ~ ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' '

N=0.5

3

~ 2.5

~1.5 0.1

0

I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

V o l u m e F rac t i on , f

Fig. 14. Asymptotic reference stress, ~N, for the square arrangement loaded in the 45 ° direction for matrix work hardening exponents N = 0, 0.1, 0.2 and 0.5.


10

9 io °

8

7

~a 6

5

l 0.2

0.1

0

~ -

f ' ,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

V o l u m e Fract ion , [

Fig. 15. Asymptotic reference stress, ~ , for the hexagonal arrangement for matrix work hardening exponents N =0, 0.1, 0.2 and 0.5.

2993

The asymptotic reference stress is plotted in Fig. 14 for the square arrangement of fibers loaded in the 45 ° direction for composites with strain hardening exponents of 0, 0.1, 0.2 and 0.5. In this case there is no critical volume fraction and the asymptotic reference stress rises gradually in all cases. The additional strength due to strain hardening, except when N = 0.5, is of the order of the strengthening due to the plane strain condition except for volume fractions which are high and probably impractical.

Figure 15 plots the asymptotic reference stress for composites in the hexagonal arrangement with strain hardening exponents o f N = 0, 0.1, 0.2 and 0.5. As with the case of a square arrangement loaded in the 0 ° direction, significant strengthening effects due to strain hardening occur only at volume fractions greater than the critical volume fraction, f * .

A simple expression relating ~N to #0 obtained by fitting to the numerical solutions takes the form

aN = 60 exp(C2 N f c3) (8)

where N is the strain hardening exponent, f is the volume fraction of fibers, and the constants C2 and C3 in (8) are dependent on the fiber arrangement and loading direction, as tabulated in Table 2.

These values for coefficients C2 and C3, used in equation (8), and in conjunction with equations (6) and (7), predict the composite asymptotic reference stresses, for the fiber arrangements and volume fractions considered here.

Failure considerations

Matrix failure in ductile matrix composites can be associated with void nucleation and growth initiated either in regions of high hydrostatic tension and plastic strain or at the fiber-matrix interface at a location of high tensile stress normal to the interface. Thus the maxima of these quantities found in the calculations are of interest. The results presented below will be of some value to those concerned with failure criteria for the transverse loading of fiber composites.

Hydrostatic tension and equivalent plastic strain are plotted in Fig. 16 for the square arrangement loaded in the 0 ° direction, for the square arrangement loaded in the 45 ° direction and for the hexagonal arrangement. The matrix is non-hardening for all cases and the results represent the stress and strain state after fully developed flow has been achieved. The results shown are the largest ratios found any- where in the matrix for a given volume fraction. The stress is normalized by the matrix yield stress whereas the strain is normalized by the composite strain L At volume fractions below the critical volume

Table 2. Constants for equation (8)

C2 C3 Square arrangement 4.53 1.21

0 ~ loading Square arrangement 2.88 1.50

45 ° loading Hexagonal arrangement 4.65 1.78

2994

O

a~ u

O °¢.¢ tt~

I

18

16

14

"12

10

8.0

6.0

4.0

ZAHL et al.:

, , , i , , , i , , , i , , , +

TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES

f* square J¢ hexagonal

+ ' ' ' I ' ' ' I ' I ' ' ' I ' ' ' I ' ' ' .

l Square-0 °

(I H - Sol id L ine

e pl - Dashed L ine eq

gonal

S

#

, . )'

' / Square-45 °

700

2.0 - - ' - / " " " " c ~ ' ~ ~ '" " / " I

0 . 0 ~ ~ . . . . . . . ° ° ' ° " ~ "

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V o l u m e Frac t ion , ]

Fig. 16. Maximum hydrostatic stress and maximum equivalent plastic strain. The matrix is non-hardening. The strain ~ is the composite strain.

I t o

600 tO

500 "~

tO

400

300 "~

200 m

100 "~

fract ion,f*, the magnitudes of maximum hydrostatic tension and equivalent plastic strain are relatively insensitive to volume fraction. Near the critical volume fraction, f * , the magnitude of both maximum hydrostatic tension and maximum equivalent plastic strain rises rapidly. Finally, sharp rises in these magnitudes are also seen as the volume fraction approaches the maximum volume fraction for the configuration at which the fibers touch each other. The high hydrostatic stress occurring when fibers nearly touch reflects the very considerable plastic constraint in those cases [22].

At volume fractions greater than about 0.35, the maximum hydrostatic stress in the square arrangement loaded at 0 ° is considerably higher than for the other two cases. The hexagonal arrangement gives rise to a maximum hydrostatic stress somewhat higher than for the square arrangement loaded at 45 ° . However, at volume fractions less than 0.35 the hexagonal arrangement results in a somewhat higher value than both the square arrangements. Similarly, at volume fractions above about 0.35, the maximum strain ratio is highest for the square arrangement loaded at 0 ° and lowest for the square arrangement loaded at 45 ° . As with the maximum hydrostatic tension, the hexagonal arrangement results in a slightly higher value of maximum strain ratio when the volume fraction is below about 0.35.

The approximate locations of the maxima are shown in Fig. 17. Symmetry and periodicity conditions determine the other locations in the composite where equivalent maxima are to be found. The capital letters A - F in Fig. 17(a) indicate the location of maximum hydrostatic stress for volume fractions from 0.2 to 0.7 for the square arrangement loaded in the 0 ° direction. It can be seen that for volume fractions less than f * the maximum occurs near the fiber-matrix interface at an angle of about 60 ° with the horizontal. The location shifts toward the top of the fibers in the 0 ° loading direction as the volume fraction is increased above f * . The location of maximum equivalent plastic strain in Fig. 17a is denoted by the lower case letters a-f. At a volume fraction of 0.2, the maximum occurs in the middle of the loading direction ligament between fibers. The location shifts to a site at the fiber-matrix interface making an angle of about 45 ° with the horizontal for a volume fraction of 0.3. With further increase in volume fraction this angle steadily decreases to about 20 ° w h e n f = 0.6. However, a t f = 0.7, the location is near the top of the fiber.

In Fig. 17(b) the location of maximum hydrostatic stress is indicated by the pound ( # ) sign, and the location of the maximum equivalent plastic strain is indicated by an asterisk (*) for the square arrangement loaded in the 45 ° direction for volume fractions


ranging from 0.2 to 0.75. In this model, the locations of these maxima do not change with volume fraction. The maximum hydrostatic stress is found near the center of the pockets of matrix surrounded by fibers, and the maximum equivalent plastic strain is found near the fiber-matrix interface making an angle of 45 ° with the horizontal.

The locations of maximum hydrostatic tension and maximum equivalent plastic strain in the hexagonal arrangement of fibers are given in Fig. 17(c). As in Fig. 17(a) the capital letters indicate the location of maximum hydrostatic stress, and the lower case letters indicate the location of maximum equivalent plastic strain. There is no result for f = 0.2. The location of maximum hydrostatic tension occurs near the fiber-matrix interface making an angle of about 35 ° with the horizontal for a volume fraction of 0.3. This location gradually shifts around the fiber to a location making an angle of about 55 ° at a volume fraction of 0.8. Although these are the maximum values computed, it is worth noting that hydrostatic tension of similar magnitude is also seen in the

ligament between the fibers in the vertical direction. The location of maximum equivalent plastic strain shifts steadily around the fiber from a position near the fiber-matrix interface making an angle of about 50 ° with the horizontal to a position making an angle of about 70 ° with the horizontal as the volume fraction is varied from 0.3 to 0.8.

The maximum values of the tensile stress normal to the fiber-matrix interface are plotted in Fig. 18 for each of the fiber arrangements and loading directions. As before, the matrix is non-hardening. The maximum stress is normalized by the matrix yield stress. Also plotted here is the location angle, ct, defined as the angle measured from the horizontal, which indi- cates the location of these maximum normal stresses. The curves seen here are very similar in shape to the curves of maximum hydrostatic tension seen in Fig. 16. The curves are characterized by a relatively uniform level at volume fractions below f * , a sharp rise in magnitude at f * , and a further sharp rise in magnitude as the maximum possible volume fraction is approached. At a given volume fraction, the square

a

t

C

t °

b

Key

Volume Fraction, [ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2-0.8

Maximum Hydrostatic Stress, a l l / % A B C D E F G #

Maximum Equivalent Plastic Strain, r~pleq/ ~ a b c d e f g *

Large Combination of aH/O o and gpleq/ E" ®

Fig. 17. Locations of maximum hydrostatic tension and maximum equivalent plastic strain for (a) the square arrangement loaded in the 0 ° direction, (b) the square arrangement loaded in the 45 ° direction,

and (c) the hexagonal arrangement.

2996

18 o

16

N 12

o Z 10

8 ofil x N 6

"~ 4

~ 2

0

0


f square f hexagonal

t , , , , , , , , , , , , , , ,,,, : , , , , , , , , , , , , , , , , ,

i } quare - 4 5 o

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V o l u m e Fract ion , f

Fig. 18. Maximum normal interracial stress and location. The matrix is non-hardening.

75

60

45

30

15

<

o

o ,-1

arrangement loaded at 45 ° has the lowest values for the maximum normal stress on the fiber-matrix interface. At volume fractions less than f * , the other two cases give rise to somewhat larger values of interfacial maximum normal stress. However, at volume fractions greater than f * , the value of maximum interfacial normal stress is much larger for the square arrangement loaded at 0 ° and the hexagonal arrangement than for the square arrangement loaded at 45 ° . At low and high volume fractions, the maximum interfacial normal stress for the square arrangement loaded at 0 ° and the hexagonal arrangement are comparable. However, at intermediate fiber volume fractions, ( f=0 .35-0 .5 ) , the square arrangement loaded at 0 ° gives rise to much higher values of interfacial normal stress than the hexagonal arrangement. At high fiber volume fractions when the fibers nearly touch, the high interfacial normal stresses reflect the considerable plastic constraint effective in those cases [22].

CLOSURE

Finite element calculations have been carried out to investigate the effect of fiber volume fraction, fiber arrangement, and matrix hardening on the transverse stress-strain behavior of continuous fiber metal matrix composites with strong fiber-matrix interfaces. The composite strength is seen to be sensitive to the fiber arrangement and loading direction, with a

square arrangement of fibers loaded in the 0 ° direction being the strongest and a square arrangement loaded in the 45 ° direction being the weakest. The requirement that shear takes place on non 45 ° shear planes in the hexagonal arrangement provides only modest increments of strengthening over the square arrangement for low volume fractions. Significant strengthening takes place only when the volume fraction of fibers is sufficient that the fibers impinge on the shear planes in the composite (45 ° shear planes for the square arrangement and 60 ° shear planes for the hexagonal arrangement). For the square arrangement loaded in the 45 ° direction, the 45 ° shear planes are present until the fibers contact one another. Thus most strengthening in this case is a result of the plane strain constraint. Hardening can provide additional strengthening, however this is also most significant at volume fractions greater than the critical volume fractions at which the fibers impinge the shear planes in the matrix.

The maximum values of parameters which lead to matrix failure, namely hydrostatic tension, equivalent plastic strain, and tensile normal stresses at the fiber-matrix interface, have been presented vs fiber volume fraction. Large increases in strength are accompanied, not surprisingly, by large increases in these failure parameters. However, the moderate strengthening seen in the hexagonal arrangement at volume fractions below f * does not significantly affect these parameters.


Acknowledgements--This work was supported by DARPA through the University Research Initiative at the University of California, Santa Barbara (ONR Prime Contract N000t4-92-J-1808). The work of D. B. Z. was also supported by a fellowship from the Max-Planek Geselischaft at the Max-Planck-Institut for Metallforschung, Stuttgart. The ABAQUS finite element code was made available by Hibbitt, Karlsson and Sorensen Inc., Pawtucket, R. I. through an academic license.

REFERENCES

1. G. Bao, J. W. Hutchinson and R. M. McMeeking, Acta metall, mater. 39, 1871 (1991).

2. V. Tvergaard, Acta metall, mater. 38, 185 (1990). 3. C. Horn and R. M. McMeeking, Int. J. Plast. 7, 255

(1991). 4. D. B. Zahl and R. M. McMeeking. To be published. 5. G. Bao and R. M. McMeeking, unpublished work

(1992). 6. T. Christman, A. Needleman and S. Suresh, Acta

metall, mater. 37, 3029 (1989). 7. T. L. Dragone and W. D. Nix, Acta metall, mater. 38,

1942 (1990). 8. D. B. Zahl and R. M. McMeeking, Mech. Comp. Mater.

Struct. Submitted. 9. T. L. Dragone, Effect of Interface properties on

the mechanical behavior of short fiber metal matrix composites, AIAA-91-0981 (1991).

10. J. Llorca, A. Needleman and S. Suresh, Scripta metall. mater. 24, 1203 (1990).

I 1. D. B. Zahl and R. M. McMeeking, Acta metalL mater. 39, 1117 (1991).

12. D. G. Taggart and J. L. Bassani, Mech. Mater. 12, 63 (1991).

13. S. Jansson and F. A. Leckie, J. Mech. Phys. Solids 40, 593 (1992)

14. H. J. B6hm, F. G. Rammerstorfer and E. Weissenbeck, Comp. Mater. Sci. 1, 177 (1991).

15. H. J. B6hm and F. G. Rammerstorfer, Mater. Sci. Engng A 135, 185 (1991).

16. J. R. Brockenbrough and S. Suresh, Seripta metall. mater. 24, 325 (1990).

17. J. R. Brockenbrough, S. Suresh and H. A. Wienecke, Acta metall, mater. 39, 735 (1991).

18. C. Dietrich, Ph.D. dissertation, Univ. of Stuttgart (1992).

19. K. Herath, Ph.D. dissertation, Univ. of California, Santa Barbara (1992).

20. S. Gunawardena, Ph.D. dissertation, Univ. of Califor- nia, Santa Barbara, (1992),

21. ABAQUS Users Manual, Version 4.7, c1988. Hibbitt, Karlsson and Sorensen Inc., Pawtucket, R.I.

22. D. C. Drucker, in High Strength Materials (edited by V. F. Zackay), Proc. 2nd Int. Symp. High Strength Materials, 1964, pp. 795-833. Wiley, New York (1965).

Documents

TRANSVERSE STRENGTH OF METAL MATRIX · PDF fileTRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES REINFORCED WITH STRONGLY BONDED ... bonded to the matrix, the composite will necessarily