Full text of DOI:10.1111/j.1151-2916.2002.tb00311.x

J. Am. Ceram. Soc., 85 [6] 1553–60 (2002)
journal
Failure of Crossply Ceramic-Matrix Composites
Michael P. O’Day and William A. Curtin
Division of Engineering, Brown University, Providence, Rhode Island 02912
The fast-fracture and stress-rupture of a crossply ceramic-
matrix composite with a matrix through-crack are examined
numerically to assess the importance of fiber architecture and
the associated stress concentrations at the 0/90 ply interface on
failure. Fiber bridging in the cracked 0 ply is modeled using a
line-spring bridging model that incorporates stochastic and
time-dependent fiber fracture. A finite-element model is used
to determine the stresses throughout the crossply in the
presence of the bridged crack. For both SiC/SiC and a typical
oxide/oxide, the fast-fracture simulations show that as global
failure is approached, a significant fraction of fibers near the
0/90 interface are broken, greatly reducing the stress concen-
tration. For fibers with low Weibull moduli (m < 10), the
tensile strength is thus nearly identical to that of a unidirec-
tional composite scaled by the appropriate fiber volume frac-
tion, while for fibers with larger Weibull moduli (m > 10),
there are modest (10؊17%) reductions in tensile strength.
Stress-rupture simulations show that initially high stress con-
centrations are relieved as fibers fail with evolving time near
the 0/90 interface and shed load away from the interface. For
a wide range of fiber properties, efficient load redistribution
occurs such that the crossply rupture lifetime is generally
within an order of magnitude of the unidirectional lifetime,
when the applied stress is normalized by the relevant fast-
fracture strength. Overall, stress concentrations at the 0/90
interface are largely relieved with increasing load or time due
to the nonlinear bridging response and preferential fiber
failure near the interface, resulting in crossplies that respond
very similarly to unidirectional composites.
through the 0 plies, leading to through-thickness matrix cracks that
are bridged by fibers in the 0 plies, as shown in Fig. 1(a). Bridging
only in the 0 plies leads to stress concentrations in the bridging
fibers near the 0/90 interface and failure of the composite is
ultimately caused by the failure of the bridging fibers. The stresses
in the bridging fibers of a crossply material have been determined
by Xia et al.6 for elastically homogeneous materials, while earlier
work focused on partially bridged cracks in unidirectional mate-
rials.7 All of these works used the classic line-spring model and the
bridging law of Marshall, Cox, and Evans,3 but with no fiber
failure. Stress concentrations alone are also not sufficient for
predicting failure; i.e., the tensile strength is not the unidirectional
strength divided by the maximum local stress concentration.
Damage, i.e., fracture of some fraction of the bridging fibers,
lessens the stress concentrations. The coupled phenomena of stress
concentrations and fiber damage, and their influence on damage
and strength in crossplies, that pervade the mechanics of compos-
ites with complex fiber architectures have not yet been studied.
The majority of the literature has simply neglected the fiber
architecture, the possible stress concentrations, and the local fiber
damage and proceeded to predict the tensile strength as if the
material were a unidirectional composite. In other words, if the
strength of a unidirectional composite of fiber volume fraction f is
uni
uts
␴
, then the tensile strength of a crossply or woven system of the
uni
same material has been estimated simply as (f1/f)␴ , where f1 is
uts
the fiber volume fraction in the direction of loading; typically f1 ϭ
f/2. This result has proven accurate in the prediction of strength in
a number of different CMC systems.8,9 One major reason for the
success of the simple model is that, at the failure stress, there is
typically a very high density of matrix cracks and, according to the
results of Xia et al.,6 the stress concentrations become small in
most cases. Not all composite systems have high crack densities
near failure, however. Some systems also have high fiber/matrix
interfacial shear stresses, which cause higher stress concentrations.
The important system of SiC/SiC can have both low crack
densities and high interfacial shear stresses. In fact, such condi-
tions tend to be optimal for design: low crack densities are usually
coincident with high proportional limits so that materials can
operate at reasonably high stresses with little or no damage, while
high interfacial shear stresses lead to higher composite strengths.
Under typical application situations of moderate stresses (well
below the ultimate tensile strength), CMCs with matrix cracks
must also survive at high temperatures for long times. In this case,
time-dependent fiber fracture occurs via slow crack growth of
existing flaws or other processes that can be highly stress-
dependent. The reduction in stress-rupture lifetime of crossplies
relative to unidirectional composites, due to architecture-induced
stress concentrations and accelerated fiber damage, has not yet
been studied.
I. Introduction
HE behavior of unidirectional ceramic-matrix composites
T
(CMCs) loaded in tension has been well established experi-
mentally and can be accurately predicted by existing models.1,2
However, because of their anisotropy, both in modulus and
strength, unidirectional composites are unsuitable for many appli-
cations. This has led to the predominant use of crossply and woven
composites. The analytical results for unidirectional composites
that relate constitutive fiber, matrix, and interface properties to the
stress–strain behavior and ultimate tensile strength (UTS) do not
apply directly to crossply systems. Analytic models for fiber
bridging, which play an important role in crossplies, are also not
generally applicable because the typical analyses (e.g., Marshall,
Cox, and Evans,3 Danchaivijit and Shetty,4 and McCartney5) are
only strictly valid for elastically homogeneous materials. Thus,
new methods of analysis are needed.
The first damage mode in most crossply CMCs is matrix
cracking. Matrix cracks typically start in the 90 plies and propagate
In this paper, we develop a coupled microscale/macroscale
numerical model to examine both fast-fracture and stress-rupture
in crossply CMCs. A finite-element (FE) model is used to
determine the macroscale stress distributions in the presence of a
matrix through-crack bridged by fibers in the 0 plies. Stochastic
quasi-static and/or time-dependent fracture of the bridging fibers is
then calculated based on the stresses obtained from the FE model,
and this microscale damage reduces the efficacy of the bridging
and leads to stress redistribution at the macroscale. Ultimately, the
accumulated fiber damage near the 0/90 interface becomes large
D. B. Marshall—contributing editor
Manuscript No. 187612. Received July 2, 2001; approved April 1, 2002.
This work was supported by NASA Glenn Research Center through Grant No.
NAG3-2100 and by the AFOSR through Grant No. F49620-99-1-0027, from the
Mechanics of Composite Materials program.
1553

1554
Journal of the American Ceramic Society—O’Day and Curtin
Vol. 85, No. 6
Fig. 1. (a) Representative section of periodic crossply composite, showing matrix crack and bridging in 0 plies; a unit cell is indicated by the dashed lines.
(b) FE discretization of the unit cell, with boundary conditions and undeformed geometry shown.
enough to cause unstable propagation of fiber fracture across the 0
ply, corresponding to global composite failure. We compare the
calculated fast-fracture strength to that estimated by the simple
models based on the unidirectional strength, and find generally
good agreement. This agreement stems from the fact that the
softening of the bridging due to fiber damage occurs preferentially
at the 0/90 boundary where the stress concentrations are high,
leading to stress redistribution and a reduction in stress concen-
trations. The stress-rupture lifetimes of crossplies are compared
with analytical and numerical determinations of unidirectional
lifetimes and, at the same normalized applied stress, there is
agreement within an order of magnitude for a wide range of fiber
parameters. The preferential and accelerated damage due to high
stresses at the 0/90 interface again acts to lessen the stress
concentrations and deter global failure. We conclude that, in all
cases, fiber damage largely relieves stress concentrations, resulting
in crossplies that behave similarly to unidirectional composites.
The remainder of this paper is organized as follows. In Section
II, the FE model, the fiber bridging model, and the fiber damage
evolution models are all presented, and the coupling between them
is described. In Section III, we present fast-fracture and stress-
rupture results for SiC/SiC and oxide/oxide crossply composites,
and compare their behavior to unidirectional composites. In
Section IV, we provide some further discussion and conclusions.
Ahn, and Takeda.2 Under a far-field uniaxial applied stress ␴
equilibrium leads to the relationship
ϱ
␴
2ls͑x,t͒
z៮
ϱ ϭ 1 Ϫ q͑T,t͒ 1 ϩ
T͑x,t͒
ͩ
ͪ
ͬ
ͫ
f
2ls͑x,t͒ T͑x,t͒
ϩ q͑T,t͒
ͩ ͪͩ ͪ
z៮
2
(1)
where f is the fiber volume fraction, T(x,t) is the local stress carried
in the unbroken fibers at time t, q(T,t) is the local fraction or
probability of fiber failure at the fiber stress T(x,t), ␶ is the
interfacial shear stress at the debonded/sliding fiber/matrix
interface, r is the fiber radius, ls(x,t) ϭ T(x,t)r/2␶ is the fiber slip
length determined from a shear-lag model, and z៮ is the average
matrix crack spacing. Although the unidirectional composite has
translational invariance in x, T(x,t) is written in anticipation of the
crossply system, where stress (and thus the fiber damage parameter
q(T,t)) may vary in x as well as in time t. Here, Eq. (1) is simplified
by taking z៮ 3 ϱ to model a single matrix crack leading to
␴
ϱ ϭ ͓1Ϫq͑T,t͔͒T͑x,t͒
(2)
f
(A) Fast-Fracture: In fast-fracture, the time-independent fi-
ber damage q(T) arises from existing flaws in the fiber. Specifi-
cally, a two-parameter Weibull model gives the probability of
failure in a length dz of fiber, over a stress increment ␴ to ␴ ϩ d␴
as
II. Model for Unidirectional and Crossply Composites
We consider both unidirectional and 0/90 crossply composites
containing a matrix crack that extends completely through all
matrix material perpendicular to the fiber axis. The crack is
assumed to pass around all fibers in the 0 ply, leaving them intact,
as shown in Fig. 1(a). Debonding along the fiber/matrix interface
is assumed to occur, with a residual interfacial sliding resistance ␶
acting across the debonded interface region. The composites are
assumed to contain a single matrix crack; the limitations of this
assumption will be discussed in Section IV. During loading, the
intact fibers bridging the crack in the 0 plies will exert closure
tractions on the crack surface.
mϪ1
m␴
Pf͑␴,d␴,dz͒ ϭ
dz d␴
(3)
m
L0␴
0
where ␴0 is the characteristic fiber strength at a gauge length of L0
and m is the Weibull modulus. A critical fiber strength ␴ and
c
critical gauge length ␦c ϭ r␴c/␶ can be identified for the problem,
with
m
1/͑mϩ1͒
␴ ␶L0
0
␴ ϭ
(4)
ͩ ͪ
c
(1) Fracture of Unidirectional Composites
r
In the unidirectional composite all fibers are parallel to the
direction of applied loading. A general expression for mechanical
equilibrium at the matrix crack plane has been derived by Curtin,
where we expect to find one flaw of strength ␴ in a length ␦c of
fiber. In the shear-lag fiber stress field around the matrix crack
c

June 2002
Failure of Crossply Ceramic-Matrix Composites
1555
(maximum at the crack plane, and decreasing linearly until the
far-field stress is attained), the probability of fiber failure is10
Substitution of Eq. (11) into Eq. (10) and integrating yields an
evolution equation for the flaw length. Using Eq. (12) the
evolution of flaw strength S(z,t) is
mϩ1
q͑T͒ ϭ 1 Ϫ exp͕͓ Ϫ 1/͑m ϩ 1͔͒T͑x͒ ͑1 ϩ m␣mϩ1͖͒
(5)
˜
˜
˜
t
␤
1/͑␤Ϫ2)
S͑z,t͒ ϭ ͓Si͑z͒␤Ϫ2 Ϫ ␴˜f͑z,tЈ͒ dtЈ͔
(13)
˜
˜
˜
˜
˜
͵
where ␣ ϭ fEf/Ec, with Ef and Ec the fiber and overall composite
Young’s moduli, respectively, and a tilde denotes a stress quantity
0
normalized by ␴c. Using Eq. (2) the applied stress ␴ can be
where the initial flaw strength is Si(z) and the following normal-
ϱ
related to the fiber stress T(x) by
izations have been introduced:
2
mϩ1
˜
␴˜ ϱ ϭ fT͑x͒ exp͕͓ Ϫ 1/͑m ϩ 1͔͒T͑x͒ ͑1 ϩ m␣mϩ1͖͒
(6)
t ϭ tC␴
(14a)
(14b)
˜
˜
c
␤
C ϭ
Ϫ 1 AY2K␤Ϫ2
Failure is the point at which no further increase in applied stress is
possible with increasing fiber stress T(x), or
ͩ ͪ
Ic
2
The probability of failure in length dz of fiber, over the stress
increment ␴ to ␴ ϩ d␴ is identical to Eq. (3), with the stress ␴
replaced by the initial flaw strength Si(z), thus
d␴˜ϱ
˜
dT
ϭ 0
(7)
mSmi Ϫ1
Pf(␴,d␴,dz) ϭ
dz dSi
(15)
so that the UTS of the unidirectional composite with a single
matrix crack is
m
L0␴
0
We simplify the fiber stress profile by neglecting any failure in the
far-field. According to the shear-lag fiber stress profile, the fiber
stress decreases linearly with distance away from the matrix crack
until the far-field fiber stress level is reached. The fiber slip length
ls(t) ϭ T(t)r/2␶ is the distance over which the fiber stress would
decrease to zero were it not interrupted by the far-field fiber stress.
The simplified fiber stress profile is
1/͑mϩ1͒
1
␴˜ uni ϭ f
eϪ1/͑mϩ1͒
(8)
ͫ ͬ
uts
1 ϩ m␣mϩ1
To apply the unidirectional result (8) to a crossply composite, a
scaling factor based on relative ply widths must be introduced. At
the matrix crack in a crossply composite, only the 0 plies carry
load. Thus, for a crossply composite having respective ply widths
of l0 and l90 the crossply tensile strength can be estimated as
z
␴ ͑z,t͒ ϭ T͑t͒ 1 Ϫ
f
͑for ͉z͉ Ͻ ls͑t͒͒
(16)
ͩ ͪ
ls͑t͒
which decreases linearly from the maximum value T(t) at the
matrix crack to zero at a distance ls away from the matrix crack.
The evolution of this simplified fiber stress profile is shown in Fig.
2. In this stress field, the probability of fiber failure anywhere
along the slip length at any time up to the current time is10
l0
cp
uni
uts
␴
uts ϭ ␴
(9)
ͩ ͪ
l0 ϩ l90
This is the analytic result for the UTS of a crossply composite
based strictly on the unidirectional theory. Effectively, this ne-
glects any contribution of stress concentrations in driving crossply
failure. In Section III(1), the predicted crossply tensile strength
from Eq. (9) will be compared with the FE results.
(B) Stress-Rupture: Under stress-rupture conditions, the
composite is subjected to a constant far-field stress while time
evolves, resulting in increasing fiber damage due to degradation of
the fibers. Eventually, the composite will have damaged enough
that the overall load level can no longer be maintained. The point
at which damage propagates unstably across the composite is the
stress-rupture lifetime at the given tensile stress. In this case, Eq.
(2) still holds, but the damage parameter q(T,t) becomes time-
dependent.
␤Ϫ2
˜
l ͑t͒
s
2
z
␤Ϫ2
˜
˜
˜
˜
q͑T,t ͒ ϭ 1 Ϫ exp Ϫ
T͑t ͒
1 Ϫ
͵
ͩ ͪ
ͭ ͫ
␦
c
˜
ls͑t͒
0
␤
m/͑␤Ϫ2)
˜
t
z
␤
˜ ˜
˜
dtЈ
ϩ
T͑tЈ͒ 1 Ϫ
dz
(17)
͵
ͩ
˜
ͪ
ͬ ͮ
ls͑tЈ͒
o
˜ ˜
where q(T,t) is now explicitly a function of time. The stress-
rupture lifetime of a unidirectional composite, at some constant
Here we assume that fiber degradation in time is governed by
slow crack growth of existing flaws in the fiber. A Paris law
describes the rate of crack growth as
da
ϭ AK␤
(10)
dt
where a is the current crack length, K is the crack tip stress
intensity factor, and ␤ and A are the (possibly temperature-
dependent) crack growth exponent and rate constant, respectively.
The stress intensity factor for tensile loading of a fiber is
K ϭ ␴ ͑z,t͒Y
ͱ
a
(11)
f
where Y is a geometric factor, and the fiber stress ␴f(z,t) is a
function of both position z along the length of the fiber relative to
the matrix crack at z ϭ 0 and time. The critical Mode I stress
intensity factor defines the strength S of a flaw of length a as
Fig. 2. Evolution of simplified shear-lag fiber stress profile with time
around a matrix crack at z ϭ 0.
ͱ
KIc ϭ SY a
(12)

1556
Journal of the American Ceramic Society—O’Day and Curtin
Vol. 85, No. 6
remote stress, can then be found by solving the coupled Eqs. (17)
and (2). An iterative scheme is used to obtain a self-consistent
solution at a given time. Time is then incremented by some small
amount, and a new solution is sought. The time at which no
converged solution exists corresponds to the stress-rupture lifetime
for a unidirectional composite.
where ␩ ϭ fEf/(1 Ϫ f)Em, and Em is the matrix modulus. We
include the effects of fiber failure, represented by the damage
parameter q as follows. We assume that in each small region of the
composite dx around point x there are a sufficient number of fibers
such that the local response is identical to that of a unidirectional
composite. The local fiber stress T(x) induces fiber damage q
which evolves according to either Eq. (5) or Eq. (17) for fast-
fracture and stress-rupture simulations, respectively. T(x) is ob-
tained as a self-consistent outcome of the FE calculation. Fiber
damage q due to the load T(x) then acts to weaken the bridging law
by reducing the fraction of fibers participating in the bridging as
(2) Finite-Element Model
The FE method is applied to simulate both fast-fracture and
stress-rupture. The following discussion relates to the FE model in
general; aspects unique to each simulation type will be mentioned
when appropriate. In formulating the macroscale mechanics prob-
lem, a continuum approach is adopted. The discrete fibers and
matrix are replaced by a homogeneous, isotropic material with
effective properties for each ply. The effective Young’s mod-
ulus for the 0 ply corresponds to that for uniaxial loading of a
single laminate along the fiber direction (longitudinal) and for
the 90 ply corresponds to the transverse Young’s modulus of a
single laminate. The effective properties were determined from
an Eshelby analysis,11 using the constituent properties given in
Table I.
Because of the periodic nature of the 0/90 laminate, a unit cell
geometry is appropriate. The unit cell and its FE discretization are
shown in Figs. 1(a) and (b), respectively. The left and right edges
of the unit cell correspond to the centers of the 0 and 90 plies,
respectively. The left and right edges are constrained in the x
direction and the z displacement of the top edge is prescribed, as
discussed below. The extent of the composite in the y direction is
assumed to be large enough such that plane strain conditions
prevail. The model height h is taken to be 3(l0 ϩ l90), which
ensures that far-field stress gradients in z are small. Approximately
1100 bilinear plane strain quadrilateral elements comprised the FE
mesh. The mesh is weighted to concentrate nodes near the matrix
crack plane and the ply interface. All relevant geometric parame-
ters are given in Table I.
The prescribed displacement of the top edge is unique to the
simulation being performed. In the fast-fracture simulation, the z
displacement of the top edge is monotonically increased until
composite failure occurs. In stress-rupture the goal is to maintain
a constant level of remote stress, which is accomplished as
follows. Over a very small time the composite is loaded to the
desired remote stress level. Because of the difference in 0 and 90
ply moduli, as well as the bridging stress variation along the matrix
crack, the remote stress is expected to vary in x along the top edge.
The average tensile stress over the entire top edge is to be held
constant at the desired level, but the FE model is loaded through
prescribed displacements. In general, as damage accumulates, a
greater incremental displacement is required to maintain the
constant remote stress level. In the FE procedure, the current
remote stress and remote stress history are used to predict the
necessary incremental displacement to satisfy the constant remote
stress condition. Fluctuations from the desired stress level are on
the order of 0.01%.
p͑u,q͒ ϭ ͑1 Ϫ q͒p͑u͒
(19)
where 1 Ϫ q is the local probability of fiber survival. The
dependence of the bridging law, p(u,q), on the local damage state
q is explicit, and position dependence x and time dependence, if
applicable, are implicit in u and q.
An iterative solution procedure is used to obtain a self-
consistent solution in displacements, fiber stresses, and fiber
damage. A typical FE iteration proceeds as follows. The incre-
mental displacement of the top edge is prescribed. For any iteration
(not yet converged) within this displacement increment, the
displacement field of the entire model was determined by the FE
solution from the previous iteration. The stress in intact fibers at
the ith node (position xi) bridging the matrix crack is calculated
directly from the nodal displacements as Ti ϭ p(ui,qi)/f(1 Ϫ qi),
where qi and ui were obtained in the previous iteration. Using Ti ,
the probability of failure at the ith node is found from either Eq. (5)
or Eq. (17), in accordance with the type of simulation being
performed. The line-spring stiffness contributions at the matrix
crack are then ki ϭ [dp(ui,qi)/dui]͉q , with p(u,q) given by Eqs. (19)
and (18). These are assembled intoi the global FE equations, which
are solved to provide the new displacement field. This procedure
is repeated until the displacement solution converges. When the
solution has converged, the next incremental displacement is
applied.
In fast-fracture simulations, evaluation of fiber damage by Eq.
(5) is clear. For stress-rupture Eq. (17) must be integrated
˜
numerically, as follows. At some time t and at every position x, the
bracketed term in the integrand of Eq. (17) must be integrated over
˜
the length of the fiber, up to the current slip length ls(t). Evaluation
of the first term inside the brackets is straightforward, but the
nature of the time-varying slip length ls(tЈ) in the second term
requires that the lower limit of the time integral be modified.
Within the present assumption of no far-field fiber stress, the slip
length and stress profile evolution along any fiber is shown in Fig.
2, where z ϭ 0 represents the matrix crack. When the position
integral in Eq. (17) is evaluated at some z*, the time integral is
˜
˜
physically meaningful only at times tЈ such that z* Յ ls(tЈ). From
Fig. 2, we see that the stress at z* is zero until the slip length
˜
increases such that z* Յ ls(t*). This means that for any time less
than t*, the point z* does not contribute to the time integral.
The effect of fiber bridging at the matrix crack plane is
introduced through a continuum nonlinear spring bridging law,
devised by Danchaivijit and Shetty4 (DS), which incorporates the
physically correct fiber stresses in the uncracked matrix material.
In the absence of any fiber damage and uniform remote loading,
the DS bridging law relates the closure traction p(u) to the crack
opening displacement 2u(x) according to
˜
Computationally, the lower limit of the time integral in Eq. (17)
˜
thus becomes t* with no loss in generality.
Several checks were used to verify the FE model. The FE
fast-fracture strengths of unidirectional composites were compared
with the analytical UTS, Eq. (8), and were in exact agreement. The
FE stress-rupture lifetime of a unidirectional composite was
compared with the analytical lifetime, obtained by solving the
coupled Eqs. (17) and (2), and again the FE results were in exact
agreement.
1/ 2
␩␴
16͑1 ϩ ␩͒2Ef f 2␶u͑x͒
ϱ
p͑u͒ ϭ
1 ϩ
ϩ 1
ͭͫ
ͬ ͮ
2
2
2͑1 ϩ ␩͒
␩ ␴ϱr
(18)
Table I. FE Input Material and Geometric Parameters
Ef (GPa)
vf
Em (GPa)
vm
f
␴
c
(GPa)
r (␮m)
␶ (MPa)
l0 (␮m)
SiC/SiC
Oxide/Oxide
269
372
0.16
0.2
310
124
0.16
0.2
0.342
0.248
2
1
7.5
6
97
25
156
122

June 2002
Failure of Crossply Ceramic-Matrix Composites
1557
III. Results
FE simulations were performed for a SiC/SiC composite and a
typical oxide/oxide. The overall trends observed were similar for
the two systems; thus we present fast-fracture results for the
SiC/SiC system and stress-rupture results for the oxide/oxide.
(1) Fast-Fracture Results
The normalized continuum bridging stress p(u,q)/␴uni and fiber
uts
damage parameter q are shown for the SiC/SiC composite in Figs.
3(a) and (b), respectively, at various levels of applied loading. At
loads less than 50% of the crossply UTS, a relatively large stress
concentration (approaching 2) exists but a negligible percentage of
fibers have fractured. With increasing load, fiber damage accumu-
lates more rapidly, particularly near the 0/90 interface, reducing
the stress concentration. At an applied stress of ϳ85% of the
crossply UTS, the region of the 0 ply at the 0/90 interface just
uni
reaches the limiting value of the unidirectional UTS ␴ . Since no
uts
uni
uts
region of the 0 ply can exceed ␴ , as the applied stress is
Fig. 4. Bridging law history at four points along the 0 ply (SiC/SiC, m ϭ
increased the near-interface region sheds load to fibers away from
the interface. Simultaneously, with increasing strain, fibers in the
near-interface region are damaging so rapidly that this region
actually supports a decreasing amount of stress. Figure 3(a) shows
that, at failure, the stress in the highly damaged near-interface
region is significantly below the unidirectional failure stress. The
evolving damage softens the near-interface region significantly
and decreases the stress concentration as the applied stress is
increased further. The net result is an almost complete elimination
of the stress concentration, and hence the tensile strength is well
approximated by the scaled unidirectional theory, as discussed
further below.
It is useful to examine the local bridging law at several locations
along the 0 ply. The DS bridging law, Eq. (18), describes p(u) as
a monotonic function of u, but the bridging is not monotonic when
fiber damage occurs. With damage, the bridging law has hardening
(dp(u)/du Ͼ 0) and softening (dp(u)/du Ͻ 0) regimes. Figure 4
shows the bridging history of four points along the 0 ply, up to
composite failure. As implied by the failure stress distribution of
Fig. 3(a), the most heavily damaged regions evolve on the
softening portion of the bridging law. Globally, the composite is
stable even though local regions of softening exist. Figure 4 also
shows that all points of the 0 ply basically follow the same
underlying bridging law; it is the degree of local stress concentra-
tion and fiber damage that determines how much bridging actually
occurs. The slightly different bridging described in Fig. 4 is solely
uni
uts
5); symbol denotes global crossply failure and stress is normalized by ␴
.
MCE-type law with p(u) ϭ Au1/2, where A is a constant indepen-
dent of remote stress, the curves in Figure 4 collapse to a single
curve and the bridging law of Eq. (19) is unique.
The FE determination of the UTS is compared with the simple
analytical prediction, Eq. (9), for both composite systems in Table
II. We have considered a balanced crossply (l90 ϭ l0) and l90
ϭ
2L0, which can roughly approximate current woven materials with
the longitudinal tows represented by the 0 ply and the transverse
tows and matrix-rich regions roughly represented by the wider 90
ply. At lower Weibull moduli (m ϭ 5) the scaled unidirectional
theory slightly overpredicts the UTS of the crossply SiC/SiC
system. For larger values of Weibull modulus (m ϭ 20), the
crossply UTS is significantly below the analytic theory. The
oxide/oxide system (not shown) shows slightly smaller stress
concentrations and greater ability to relieve them by damage.
Thus, the oxide/oxide UTS follows the trends found for SiC/SiC
but the differences with the scaled unidirectional prediction are
even smaller.
The variation of the Weibull modulus was performed at a fixed
value of ␴ (see Table I). As the Weibull modulus increases, the
c
stress range over which fiber damage occurs narrows. In the
limiting case of an infinite Weibull modulus, failure becomes
extremely brittle and occurs at the instant any point along the 0 ply
due to the ␴ dependence of the DS bridging law. Indeed for an
ϱ
Fig. 3. Fast-fracture behavior for balanced (l0 ϭ l90 ) SiC/SiC composite, m ϭ 5: (a) normalized bridging stress versus distance along 0 ply; (b) fiber damage
parameter versus distance along 0 ply matrix crack. Quantities are shown at various percentages of the applied stress to eventual crossply UTS.

1558
Journal of the American Ceramic Society—O’Day and Curtin
Vol. 85, No. 6
Table II. Comparison of FE and Analytical Fast-Fracture UTS (MPa)
SiC/SiC
Oxide/oxide
m ϭ 5
m ϭ 10
m ϭ 20
m ϭ 5
m ϭ 10
m ϭ 20
l90 ϭ l0
Eq. (9)
FEM
Eq. (9)
FEM
289
277
193
183
313
284
208
186
326
278
217
181
104
101
69
113
106
75
118
104
79
l90 ϭ 2l0
66
69
68
reaches the unidirectional tensile strength and hence stress con-
centrations play a major role. Damage tolerance in the fiber bundle
(lower m) is thus one key element for reduction of local stress
concentrations.
The bridging stress along the matrix crack at incipient failure,
uni
uts
normalized by ␴ , is shown in Fig. 5 for a range of Weibull
moduli. An increasing Weibull modulus results in less stress
redistribution to fibers away from the interface and thus larger
remaining stress concentrations. Therefore, with an increasing
Weibull modulus the scaled unidirectional theory will increasingly
overpredict the actual crossply UTS, as seen in Table II.
(2) Stress-Rupture Results
The main result of the stress-rupture analysis is Fig. 6, which
compares the stress-rupture lives of unidirectional and crossply
(oxide/oxide) composites at various remote stress levels, normal-
ized by the respective fast-fracture strengths. This normalization
permits a direct comparison of crossply and unidirectional com-
posites. Increasing ␤ values are shown to increase the difference
between unidirectional and crossply lives at a given load level.
Only when ␤ becomes large and the stress level is less than half of
the fast-fracture strength does the difference in lifetimes of
unidirectional and crossply composites approach an order of
magnitude.
The time evolution of the bridging stress depends on the ratio of
remote stress to fast-fracture strength of the crossply. At high
ratios (Ն90%) of the fast-fracture strength, significant fiber
fracture occurs just in loading up the composite. Thus the stress
concentration is partially relieved before any time-dependent
damage begins to accumulate, and time-dependent effects are
minor. Time-dependent effects are more prominent at lower ratios
of applied stress to UTS (ϳ50%), where initial stress concentra-
tions are larger due to less stress-driven damage. The bridging
stress and damage at various time increments are shown in Fig. 7,
with m ϭ 5 and ␤ ϭ 5, when the applied stress is 50% of the
crossply fast-fracture strength. Initially, with evolving time, the
Fig. 6. Oxide/oxide composite lifetimes at various normalized applied
stress levels ␴ϱ/␴
.
uts
near-interface region softens, reducing the stress concentration.
Damage increases rapidly in this area, resulting in load shedding
away from the interface, and continued softening near the inter-
face. Eventually, the near-interface region has no intact fibers (q ϭ
0), but since the applied stress is fairly low the composite is still
globally stable. Ultimately, this region of complete fiber failure
propagates into ϳ10% of the 0 ply just before failure. At this point,
the location of the maximum bridging stress actually is located in
the center of the 0 ply.
The role of fiber Weibull modulus in the stress-rupture behavior
of crossplies was also examined. In the fast-fracture simulations,
lower values of Weibull modulus were shown to contribute to
efficient load shedding away from the 0/90 interface. Figure 8
shows the bridging traction and damage at various times at 50% of
the fast-fracture strength, for m ϭ 20 and ␤ ϭ 5. As compared with
the m ϭ 5 system, the stress concentration is relieved to a lesser
extent. Damage near the 0/90 interface tends to accumulate faster
and the composite is unable to shed load away from the interface,
leaving the center of the zero ply with relatively little damage and
low stress. Immediately before failure, a stress concentration is
still present, and complete fiber failure has propagated into ϳ30%
of the 0 ply. With all other parameters unchanged, the composite
with m ϭ 20 fails an order of magnitude faster than the m ϭ 5
composite. Lifetimes of the m ϭ 5 and m ϭ 20 composites over the
full applied stress range are shown in Fig. 6. Similar to the
fast-fracture observations, a lower Weibull modulus again pro-
motes stress redistribution, resulting in crossplies that behave
similarly to unidirectional composites.
IV. Summary and Discussion
The fast-fracture and stress-rupture of crossply composites, and
the effects of local stress concentrations, have been modeled
numerically. In fast-fracture, interface stress concentrations do
induce local fiber damage but these, in turn, reduce the stress
concentrations at higher applied loads. The fiber Weibull modulus
Fig. 5. Normalized stress distribution of 0 ply at failure for various
Weibull moduli (SiC/SiC composite).

June 2002
Failure of Crossply Ceramic-Matrix Composites
1559
Fig. 7. Rupture behavior for a balanced (l0 ϭ l90) oxide/oxide composite with low m (ϭ5) and low ␤ (ϭ5) at remote stress of 50% of fast-fracture UTS:
(a) normalized bridging stress versus distance along 0 ply; (b) fiber damage parameter q vs distance along 0 ply; shown at various percentages of the
stress-rupture lifetime.
is shown to be the key parameter to obtain efficient load shedding
away from the highly damaged near-interface region. A low
Weibull modulus promotes stress redistribution and tensile failure
occurs at stresses only slightly less than those predicted by the
analytic theory, thus validating the general accuracy of the analytic
predictions. As the fiber Weibull modulus increases, the analytic
theory tends to overpredict the composite UTS. In addition, while
Eq. (8) predicts monotonically increasing UTS with increasing
Weibull modulus for unidirectional composites, the numerical
results suggest that, for the systems studied here, the crossply UTS
is relatively insensitive to changes in m, as shown in Table II. The
stress-rupture lifetime of crossplies is generally within an order of
magnitude of the corresponding unidirectional composite, for the
same normalized remote load level. Crossply lifetime is seen to
decrease with increasing Weibull modulus.
This work has been limited to a composite containing a single
matrix crack. For most composites loaded to failure, arrays of
parallel matrix cracks will form and saturate at some average
spacing.12 Thus a direct numerical comparison of the FE fast-
fracture strengths to actual crossply fast-fracture strengths is not
recommended. Also, if the average matrix crack spacing becomes
less than half of the slip length ls, then fiber damage can no longer
be calculated in the manner described above; the effect of fiber
breaks at neighboring matrix cracks will have to be considered
within the framework of Curtin, Ahn, and Takeda.2 The present FE
results will tend to overpredict the crossply UTS, but the trends in
comparing crossply and unidirectional behavior will be similar to
those found here. The single-matrix-crack assumption becomes
more appropriate in stress-rupture situations. Experiments show
that a typical unidirectional oxide/oxide composite loaded to less
Fig. 8. Rupture behavior for balanced (l0 ϭ l90) oxide/oxide composite with high m (ϭ20) and low ␤ (ϭ5) at remote stress of 50% of fast-fracture UTS:
(a) normalized bridging stress versus distance along 0 ply; (b) fiber damage parameter q vs distance along 0 ply; shown at various percentages of the
stress-rupture lifetime.

1560
Journal of the American Ceramic Society—O’Day and Curtin
Vol. 85, No. 6
than half of its UTS will contain less than 20% of the cracks
present at the fast-fracture point.13 Thus the average matrix crack
spacing for stress-rupture (at 50% of the fast-fracture strength) is
ϳ5 times larger than the spacing at the fast-fracture point and the
single-matrix-crack assumption is justifiable for the stress-rupture
simulations at such low stresses.
stress.15 The present work shows that the additional stress concen-
trations on the fibers caused by those matrix cracks, due to 0/90 ply
boundaries or free edges at the notch root or hole edge, should then
be greatly reduced by the stochastic fiber damage. Thus, the
notched composite strength can be accurately estimated using a
unidirectional strength value reduced by the net-section area
fraction.
The present results use the line-spring bridging model. Xia et
al.6 critically examined the line-spring model in applications to
problems of the type studied here. They concluded that in systems
with significant fiber/matrix interface slip, the line-spring model
overpredicts stress concentrations. A large-scale sliding (LSS)
model based on an FE model was proposed to correct this
deficiency. The stress concentrations in the LSS model were found
to be reduced with increasing stress, decreasing fiber/matrix
interfacial shear stress ␶, and decreasing matrix crack spacing z៮.
We have used the line-spring model for materials where the LSS
model is expected to be more appropriate. However, we have
shown that as failure is approached, the interface stress concen-
tration is largely eliminated; hence use of the line-spring model is
not a serious limitation. In general, use of the line-spring model
should predict a lower bound for the tensile strength. It is insightful
to reexamine the fundamental continuum approximation used in
bridging models, particularly the replacement of discrete fiber
stresses by a smooth distributed traction. This approximation
assumes that the stress concentration exists over many fibers along
the matrix crack. For the SiC/SiC system studied here, assuming a
rectangular fiber array, approximately seven fibers span the ply
half-width l0 . The stress concentration exists over ϳ40% of the
model width (see Fig. 3), or only three fibers. Thus, the continuum
approximation may not be extremely accurate for thin-ply mate-
rials; a model that incorporates discrete fiber bridging effects may
be more appropriate.
Acknowledgments
W. A. Curtin thanks Dr. H. Halverson and Dr. G. Morscher for useful discussions
pertaining to this work.
References
1W. A. Curtin, “Theory of Mechanical Properties of Cermaic-Matrix Composites,”
J. Am. Ceram. Soc., 74, 2837–45 (1991).
2W. A. Curtin, B. K. Ahn, and N. Takeda, “Modeling Brittle and Tough
Stress–Strain Behavior in Unidirectional Ceramic Matrix Composites,” Acta Mater.,
46, 3409–20 (1998).
3D. B. Marshall, B. N. Cox, and A. G. Evans, “The Mechanics of Matrix Cracking
in Brittle-Matrix Fiber Composites,” Acta Metall., 33, 2013–21 (1985).
4S. Danchaivijit and D. K. Shetty, “Matrix Cracking in Ceramic-Matrix Compos-
ites,” J. Am. Ceram. Soc., 76, 2497–504 (1993).
5L. N. McCartney, “Mechanics of Matrix Cracking in Brittle-Matrix Fibre-
Reinforced Composites,” Proc. R. Soc. London, Ser. A, 409, 329–50 (1987).
6Z. C. Xia, J. W. Hutchinson, A. G. Evans, and B. Budiansky, “On Large Scale
Sliding in Fiber-Reinforced Composites,” J. Mech. Phys. Solids, 42, 1139–58 (1994).
7D. B. Marshall and B. N. Cox, “Tensile Fracture of Brittle Matrix Composites:
Influence of Fiber Strength,” Acta Metall., 35, 2607–19 (1987).
8W. A. Curtin, “Stochastic Damage Evolution and Failure in Fiber-Reinforced
Composites,” Adv. Appl. Mech., 36, 163–253 (1999).
9W. A. Curtin, Comprehensive Composite Materials; Ch. 4.03. Edited by A. Kelly
and C. Zweben. Pergamon-Elsevier Science, Kidlington, U.K., 2000.
10H. G. Halverson and W. A. Curtin; unpublished work.
11T. W. Clyne and P. J. Withers, An Introduction to Metal Matrix Composites.
Cambridge University Press, Cambridge, U.K., 1996.
12P. Karandikar and T.-W. Chou, “Characterization and Modeling of Microcrack-
ing and Elastic Moduli Changes in Nicalon/CAS Composites,” Compos. Sci.
Technol., 46, 253–63 (1993).
Fiber damage represents one process by which composites can
reduce stress concentrations. Mackin et al.14 and others have
shown that localized multiple matrix cracking itself is an effective
mechanism in this role. Multiple matrix cracking that occurs
preferentially around a notch or hole due to the high elastic stress
concentrations is found to greatly relieve those large-scale stress
concentrations. Model calculations by Genin and Hutchinson
demonstrate how the loss of stiffness on cracking reduces the local
13D. S. Beyerle, S. M. Spearing, F. W. Zok, and A. G. Evans, “Damage,
Degradation, and Failure in a Unidirectional Ceramic-Matrix Composite,” J. Am.
Ceram. Soc., 75, 2719–25 (1992).
14T. J. Mackin, T. E. Purcell, M. Y. He, and A. G. Evans, “Notch Sensitivity and
Stress Redistribution in Three Ceramic-Matrix Composites,” J. Am. Ceram. Soc., 78,
1719–28 (1995).
15G. M. Genin and J. W. Hutchinson, “Composite Laminates in Plane Stress:
Constitutive Modeling and Stress Redistribution Due to Matrix Cracking,” J. Am.
Ceram. Soc., 80, 1245–55 (1997).
Ⅺ