Acta metall, mater. Vol. 42, No. 9, pp. 2983-2997, 1994
Pergamon
0956-7151(94)E0108-S
Copyright © 1994 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0956-7151/94 $7.00 + 0.00
TRANSVERSE STRENGTH OF METAL MATRIX
COMPOSITES REINFORCED WITH STRONGLY B O N D E D
CONTINUOUS FIBERS IN REGULAR A R R A N G E M E N T S
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 transportation
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.
Continuous fibers provide the greatest strengthening,
when loaded in the direction of the fibers. However, when loaded in a transverse direction the
reinforcement provides little strengthening. Jansson
and Leckie [13] carried out calculations to model the
behavior of continuous fiber composites under transverse cyclic loading. B r h m 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. However, 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
AMM
42/9---F
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 G u n a w a r d e n a [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
2983
2984
ZAHL et al.:
TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
°°l 2
/
00
00
(a)
°°l
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
(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.
a = ¢r0(~0) = EE
E~<E0
ff = (70
E > 60
l
l
(a)
(b)
(la)
~(3o°)
4
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
(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
for the perfectly plastic case, and by
a = e0
= EE
E~ %
O" ~--...~O"0
-~o1%
(lb)
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, E0 = 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 incrementally, with the stress increment related to the
strain increment through
(~ij = ~
E
".
2985
Eij ~- ~
v
Z
1 ---'
/
Matrix
Strain
(a)
Limit Flow stress, -~
O
.
~kk (~ij
Composite, ~ /
SijSklEkl
3
"~N/(~O
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
fi
O
.
~
~
Composite, ~ / o(e)
"
Matrix, a(~)/c~°
Strain
(b)
A s y m p t o 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
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
= 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
~ = ~o( °" ']~
\~0/
(4)
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
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
2986
ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
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 representation 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 et al.:
T R A N S V E R S E STRENGTH O F M E T A L M A T R I X COMPOSITES
4
I
3.5
o
'
'
'
l
'
'
'
'
I
'
@
3
-
'
'
'
'
I
'
'
2987
'
'
/=0.7
2.5
0.6
2
-o
0.5
~=~
1.5
0.4
O
Z
1
0.3
0.2
0.0
0.5
0
0
1
2
3
Normalized
4
Strain,
5
6
~-/
O
Fig. 4. Stress-strain curves for the square arrangement loaded in the 0 ° direction. The matrix is
non-hardening.
'
I
'
I
'
'
I
I
'
o 1.5
l
Continuous Fibe
o
'o
1
0.5
0
a O
-05.
,
0
,
,
I
0.2
,
~
,
I
,
,
,
0.4
Relative Reinforcement
I
,
,
0.6
,
I
= OO
,
,
,
0.8
Size, D/L
Fig. 5. Comparison o f the plane strain results with results for an axisymmetric model with the same
dimensions. The matrix is non-hardening.
2988
ZAHL et al.:
1.5
TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
. . . .
I
. . . .
I
. . . .
t
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
1.25
I
0~
0.6
0.5
0.4
0.75
(~
•
0.5
0.25
0
0
0.5
1
1.5
2
Normalized
2.5
Strain,
3
~-/e
3.5
4
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
. . . .
f=0.6
D °
°°1/3oo
¢/3
t~
1"4
O
/;
0.5
Z
I
0
I
I
I
I
0.5
I
I
I
I
I
I
I
I
I
1
Normalized
I
,
,
1.5
Strain,
,
,
I
,
2
~'/e
,
t
,
I
2.5
,
,
,
,
3
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.
ZAHL
4
~
et al.:
'
TRANSVERSE
'
'
'
I
'
STRENGTH
'
I
'
'
OF
'
'
METAL
I
'
'
MATRIX
'
'
I
,
'
'
'
I
~
i
~
'
'
°l
3.5 i
o
2989
COMPOSITES
3
2.5
1.5 ~
0.7
O
Z
1
0.5
0
,
,
,
,
i
0
,
,
~
,
i
1
i
,
i
,
I
2
,
,
,
,
I
3
,
~
J
,
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 h e h e x a g o n a l a r r a n g e m e n t
non-hardening.
'
'
'
I
'
'
'
I
'
'
'
1
'
'
'
I
'
'
l o a d e d in t h e 0 ° d i r e c t i o n . T h e m a t r i x is
'
I
'
~ll
""'"'~l'
'
~
'
I''""
@
,o
o
'
2.5
"'
f
U3
0
2
"0
0
fi
Z
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 limit yield stress, 6o, f o r b o t h s q u a r e a n d
non-hardening matrix.
hexagonal
arrangements
with a
2990
ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
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 arrangemerit and the hexagonal arrangement can be approximated as
go = a *
(6)
/,~45 °
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
0° loading
- 2- %
x/~
14.2
0.345
Square arrangement
45° loading
- -2 a~
0.0
N/A
Hexagonal arrangement
x/~
2
- - % 0 + 0.26f) 27.2
0.634
for volume fractions less than J" and by
g0 = C1 ( f - ~ ) 2 + ~rg
(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 * .
ZAHL et al.:
12
,
'
I
~l
8
o~
6
'
'
'
I
'
'
I
'
'
'
I
'
N=0.2
?
10
I~
2991
TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
0.6
J
0.5
0.4
2
0.2 ~ ' ~
0.0-0
J
,
~
0
I
,
,
,
4
l
a
,
,
I
8
I
,
I
~-/e
Strain,
I
I
I
16
12
Normalized
I
20
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.
I
I
'
'
'
I
'
'
I
N=0.2
@
6
'
/--0.7
5
o~
4
0.6
3
0.5
O
Z
2
0.4
1 I
,
0.3/~/~
o
0
4
8
Normalized
12
Strain,
16
20
~-/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 et al.:
TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
12
N = 0.5
o
10
@
U
6
0.2
u
"~
o.,
,9.o
0.1
4
g
•< 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Volume Fraction, 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
'
'
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I
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'
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
Volume Fraction, 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.
2993
ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
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 Fraction, [
Fig. 15. Asymptotic reference stress, ~ , for the hexagonal arrangement for matrix work hardening
exponents N =0, 0.1, 0.2 and 0.5.
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 anywhere 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
0~ loading
Square arrangement
45° loading
Hexagonal arrangement
4.53
1.21
2.88
1.50
4.65
1.78
2994
ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
f* square
+
18
O
,
O
°¢.¢
tt~
,
,
i
,
,
,
i
,
,
,
i
,
,
,+
'
'
'
I
'
'
'
I
'
I
'
'
'
I
'
'
'
I
'
'
l
Square-0°
(I H - S o l i d L i n e
14
700
'.
16
a~
u
J¢ hexagonal
gonal
e pl
"12
eq
Ito
600
tO
500
- Dashed Line
"~
tO
10
,
)'
.
'
400
/ Square-45 °
#
8.0
300
"~
200
m
100
"~
6.0
I
S
4.0
- -/ ' - / " " " " c ~ ' ~ ~
2.0
0.0
~
0
~
0.1
0.2
.......
0.3
0.4
'" " "I
°°'°
"~"
0.5
0.6
0.7
0.8
0.9
1
Volume Fraction, ]
Fig. 16. Maximum hydrostatic stress and maximum equivalent plastic strain. The matrix is non-hardening.
The strain ~ is the composite strain.
fraction,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
ZAHL et al.:
TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
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 indicates 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
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
t°
a
b
t
C
Key
Volume Fraction, [
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Maximum
Hydrostatic Stress, a l l / %
A
B
C
D
E
F
G
#
0.2-0.8
a
b
c
d
e
f
g
*
Maximum
Equivalent Plastic Strain, r~pleq/ ~
Large Combination of
aH/O o and gpleq/ E"
2995
®
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
ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
f hexagonal
f square
18
o
16
t,,,,,,,,,,,,,,
,,,, : , , , , , , , ,
,,,,,,,,,
75
N
o
Z
60
12
10
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45
o
8
ofil
x
N
6
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4
30
i
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o
,-1
15
quare - 4 5 o
2
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 Fraction, f
Fig. 18. Maximum normal interracial stress and location. The matrix is non-hardening.
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 . 3 5 - 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.
ZAHL et al.: TRANSVERSE STRENGTH OF METAL MATRIX COMPOSITES
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.
10.
I 1.
12.
13.
14.
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