Acta metall, mater. Vol. 43, No. 3, pp. 1119-1126, 1995
~
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AN ULTIMATE TENSILE STRENGTH DEPENDENCE ON
PROCESSING F O R C O N S O L I D A T E D M E T A L M A T R I X
COMPOSITES
J. M. DUVA I, W. A. CURTIN 2 and H. N. G. WADLEY3
~Department of Applied Mathematics, University of Virginia, Charlottesville, VA 2:2903,2Department of
Engineering Science and Mechanics & Department of Materials Science and Engineering, Virginia
Polytechnic Institute and State University, Blacksburg, VA 24061 and 3Department of Materials Science
and Engineering, University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 10 June 1994)
Abstract--The ultimate tensile strength (UTS) of metal and intermetallic matrix unidirectional composites
can be significantly lower than expected from the rule of mixtures prediction. One possible explanation
is that the fibers in the as-processed state are in a residual state of stress and in some cases are broken
because of the inhomogeneous nature of the densification during manufacture. Three main results emerge
from the effort to include the effect of this processing damage on the composite UTS. First is the
development of a simple but accurate analytical version of Curtin's model for predicting the stress~train
response and UTS of this class of composites. Second is the generalization of Curtin's model to include
both process induced fiber bending and fracture. Third is that the reduction in strength is a sensitive
function of the consolidation conditions; thus a link is established between the quality of the composite
and the conditions of its manufacture.
1. INTRODUCTION
Lightweight high-temperature metal (e.g. Ni, Ti) and
intermetallic (e. g. Ni3AI, NiA1, Ti3A1 or TiAI) matrix
composites reinforced with silicon carbide or aluminium oxide fibers are being developed for potential
applications in gas turbine engines and other
aerospace platforms [1, 2]. They are most economically produced in a two-step process. First, a "preform" is made by one of several methods including:
(a) spray deposition of (plasma melted) alloy droplets
onto linear fiber arrays, (b) infiltration of a fiber mat
with intermetallic alloy powder/polymeric resin slurries (tape casting), or (c) uniformly coating individual
fibers (or fiber tows) via a physical vapor deposition
process. Other preforms have also been used such as
foil-fiber-foil, but are considered too costly for practical use. In the second step, the preforms are cut to
shape, stacked to create the desired fiber architecture,
degassed and consolidated to near net shape using
either hot isostatic pressing or vacuum hot pressing
[3]. All three of the starting preforms result in composites with a uniform fiber spacing, a controlled
fiber volume fraction and potentially good quality.
Recent studies of the ultimate tensile strength
(UTS) of unidirectional intermetallic matrix composites have revealed a sometimes significantly lower
strength than expected from the rule of mixtures
prediction. Draper et al. showed that this apparently
anomalous result could not be attributed to fiber
strength degradation during processing because
chemically extracted fibers from as-processed,
untested samples had Weibull strength distributions
similar to the as-received fibers used for predicting the
strength [4]. Other studies support this; they have
shown that the extent of the chemical reaction (which
eventually does degrade the fiber strength) during a
consolidation thermal cycle (e.g. 8 h at 850°C) is
insufficient to cause fiber strength loss, at least for
SCS-6 fibers in a Ti-14AI-21Nb (wt%) matrix [5].
The loss of composite strength observed by Draper
et al. cannot be attributed to either a degradation of
composite's matrix strength, or to the residual stress
created by the mismatch in the coefficient of thermal
expansion (CTE) between fiber and matrix.
One possibility, which we explore here, stems from
the recent realization that the fibers in all metal and
intermetallic matrix composites are in a residual state
of stress (but one that is not created by the CTE
difference), and in some cases are even broken,
because of the inhomogeneous nature of the densification process for the preforms discussed above [6].
Nonuniform densification can occur, to a greater or
lesser extent, for all three of these preforms because
a nonuniform distribution of the matrix alloy can
exist in the lay-up resulting in fiber bending during
consolidation [6, 7]. An example of experimental evidence for this is presented in Fig. 1. It shows the
preform plane of an SCS-6/Ti-14A1-21Nb composite
after consolidation of a spray deposited preform [6].
The plane of the micrograph contains the longitudinal direction of the composite and is perpendicular to
1119
1120
DUVA
et al.:
CONSOLIDATION OF METAL MATRIX COMPOSITES
recently developed by Curtin [10, 11] for predicting
the stress strain response and the UTS of this class
of composites is generalized to include the effects of
both process-induced fiber bending and fiber fracture
so that the effect of these (process dependent) parameters can be included in strength predictions.
The key expression in our analysis is the function
q5 that gives the number of defects in a fiber segment
of length L that will fail under a uniform stress T. The
fibers of interest have a Weibull distribution of
strengths [5] and @ has the form
L T
ql(L,T) = ~00( ~ 0 ) m
Fig. l. A longitudinal section of an SCS6/Ti-14AI-21Nb
spray deposited preform after consolidation processing. The
straight edges of the fiber at the bottom indicate no bending.
The curved edges of the fiber at the top indicate some
bending out of the plane of the figure. The center fiber has
suffered severe bending and has fractured. The local bending
stresses responsible for this are caused by random asperity
contacts in the preform [6].
the pressing direction. The fibers, if perfectly aligned,
should appear as straight and parallel sided like the
one at the bottom of the figure. However, the fiber at
the top has a varying width consistent with bending
perpendicular to the plane of the micrograph, and
the center fiber has suffered a very large bending,
resulting in fracture during consolidation. (Note
white matrix material can be seen to have infiltrated
the cracks indicating failure during the densification
process.)
The fiber bending mechanism encounted in plasma
spray deposited preforms arises from the surface
asperities created by deposition (see Fig. 2) [6, 7].
When the preforms are stacked, voids are created into
which fibers deflect during subsequent densification.
These asperities are absent in tape cast and vapor
deposited preforms, but other nonuniformities in the
local matrix volume fraction occur over length scales
of 5-20 fiber diameters (e.g. because of uneven powder/resin mixing in tape cast preforms and crossing or
unevenly coated fibers in the physical vapor deposited
case). This results in the potential for similar bending
but by slightly different mechanisms [8].
Predictive models of fiber bending and fracture
during consolidation have been developed [7-9]. The
models analyze the inelastic response of unit cells like
those shown in Fig. 2 and combine these within a
statistical model of the matrix nonuniformity. This
enables the calculation of both the distribution of
fiber bending stresses and the number of fractured
fibers in a consolidated composite. The presence of
these stresses and process-induced fiber breaks are
not included in present treatments of the stress strain
response of metal and intermetallic composites and
could be a source of strength loss. Here the model
(1)
where L0, ~0 and m are Weibull parameters. Equation
(1) has the meaning that in a fiber segment of length
L0, there will be, on average, one defect that will fail
at a load cr0. Spatial variations of the stress in a fiber
due to bending and fracture found in a real composite
can be included in the analysis through the appropriate alteration of (1).
Three main results emerge from the work we
present. First is the development of a more simple,
accurate, but completely analytic version of the UTS
model first presented in [1 1]. Second is the generalization of the model to include the effect of fiber
bending and fracture on the UTS. The magnitude of
these stresses and the concommitant decrease in
strength from "ideal" behavior is one of several
possible measures of quality of the composite. Third,
we show this strength departure is a sensitive function
of the consolidation conditions and thus establish a
Macroscopic Strain Rate
Perpendicular to Fibers
PLASMA
SPRAY
PREFORM
q
2z (t)
2z
TAPE
CAST
PREFORM
~
J~
CONTINUO
MONOFILA
PREFORM
u
Fig. 2. A schematic representation of the bending mechanisms operating during the consolidation of the three preforms. The unit cells identified in the figure are analyzed to
obtain the distribution of bending stresses and fiber breaks
due to consolidation processing.
DUVA et al.:
CONSOLIDATION OF METAL MATRIX COMPOSITES
link between an attribute of quality and the conditions used for the composite's manufacture,
2. PROCESS M O D E L S
During the consolidation of composite preform
lay-ups, fiber bending and fracture occur by mechanisms like those shown schematically in Fig. 2. For
plasma spray deposited preforms, the bending has
been investigated by analyzing the behavior of a unit
cell [6]. This analysis includes the calculation of the
contact force F at an asperity required to cause its
inelastic deformation, the application of equal but
opposite forces to a fiber in a three point bending
configuration, and the calculation of the resulting
fiber deflection and bending stress. Inelastic deformation at asperity contacts during densification can
occur by plasticity (i.e. matrix yielding) and/or creep.
Elzey and Wadley [7] have shown that the force Fp
required to overcome the matrix material's resistance
to yielding (where the matrix material is considered
to be perfectly plastic) is
1121
The application of these forces to a three pointloaded beam of diameter df results in a fiber stress
that varies along the length and through the plane of
the beam. The maximum fiber tensile stress, aB, is
4F (t)l
°"= ~a}
(4)
where l is the length of the fiber bend segment
(governed by the asperity spacing) and the expressions for Fare given above by (2) or (3). The time
dependence for the force in (4) arises from (a) the
time variation of the applied consolidation load and
temperature (through the temperature dependence of
material parameters such as a 0, B and n), (b) the
evolution of fiber lengths supported between contacts
during densification, and (c) the time dependent
behavior of the creeping matrix.
At any moment during the consolidation, l and h
are random variables and a distribution of cell
lengths exist with differing values of a B. The distribution of a e can be calculated at any point during a
given consolidation process using a Monte Carlo
F o = 6naao(yp - h),
(2) scheme [7]. By comparing the calculated distribution
of fiber stresses with the Weibull distributed fiber
while for steady state creep, the force F¢ is
strength, it is possible to predict the probability of
fiber fracture and the resulting fiber fracture density
S'c "~ 1/n
(the number of breaks per meter of fiber). The extent
F~=(~cYc)
[2na(yc-h)]
(3)
of broken fibers and residual bending stresses depends on the material system and the consolidation
where ao is the uniaxial yield stress, B is the Dorn conditions.
constant, n is the stress exponent in the Norton steady
Figure 3 is a fiber fracture map for a
creep law, a is the asperity radius of curvature, h is
Ti-24AI-11Nb/SCS-6 composite consolidated from a
the original (undeformed) asperity height and yp, Yc
monotape preform. It shows contours of constant
are the deformed asperity heights after plastic and
numbers of process induced fiber breaks per meter of
creep deformation respectively. The dot over y~ indifiber as a function of the process densification rate
cates differentiation with respect to time.
and temperature [7]. Damage (and bending) is reduced by processing at high temperatures and at low
Decreasing damage
compaction rates because under these conditions
°8~-' I I I I 'I I
I'
I
low contact forces are required to activate sufficient
///////
] Ti-2' -'"0'SCS-6 I
asperity (creep) deformation to accommodate the
0.7
180
D=0.9
g
imposed deformation rate. The Elzey and Wadley
model can also be used to calculate the distribution
of bending stresses in a composite after consolida•NN 0.5
tion.
Figure 4(a)-(c) show the distribution of maxig
8
mum tensile bending stress (a~) that exist in this
composite system after consolidation under three
different sets of process conditions. It is clear that
8
processing in the 850-950°C temperature range can
~6
O3
result in significant amounts of bending and fiber
fracture. This bending stress is "frozen" into the
composite and will combine with applied loads to
ii
I
I
J
cause premature fiber failures, as described below.
850
900
950
1000
The Weibull parameters used in making these calcuTemperature (°C)
lations are a0 = 4.5 MPa, L o = 0.025 m and m = 13.
Fig. 3. A map of constant fiber break density (breaks/m) For latter calculations, a measured value of 0.10 GPa
contours plotted in the processing temperature-processing [12] for the interracial sliding resistance z and a fiber
densification rate plane. The number of processing breaks is
a strong function of processing temperature. The simulation radius r of 70/tm were used to compute a dimensionwas halted when the relative density D reached 0.9. Bullets less characteristic length 6c=365/~m introduced
mark the three processes described in Figs 4(a)-(c) [7].
below [defined in equation in (9)].
i
1122
DUVA et al.:
CONSOLIDATION OF METAL MATRIX COMPOSITES
0,5
950°C
0.1 mm/hr
~o = 0.037
tO
O
1.1.
t-
J
lb
(~'B (GPa)
0.5
I°
900oc
0_2 mm/hr
L~
0
10
O'B (GPa)
0.5
850°C
0.5 mm/hr
t,
O
~o = 0.657
D
O
kit¢_J
i'o
o
(~B (GPa)
Fig. 4. Figs 4(a)~c). Shows the distribution of bending
stresses due to three processes differentiated by the
processing densification rates (mm/h) and the processing
temperatures. The height of each column of the histograms
gives the length fraction of fiber subject to a bending stress
characterized by aB, the maximum axial stress due to
bending in the fiber. The dimensionless process-induced
break density P0, is defined in Section 3.
load carrying capacity of the matrix can be ignored.
That is, when the load carrying capacity of the matrix
has been exhausted, all additional load must be
carried by the fibers alone; we are assuming this
happens immediately. We note that in case the yield
strength is large and the matrix carries a significant
fraction of the load at the UTS, the following analysis
applies if an additional term (the matrix yield stress
times the volume fraction of the matrix) is included
[13]. Once the fibers begin to fail, it is assumed the
fiber/matrix interface debonds around each break
and the fiber slides, relative to the matrix, with a
sliding resistance, ~. The subsequent tensile properties
of the composite are controlled by the properties of
the fibers and by the fiber/matrix interface. In particular, the U T S of the composite depends on the
failure of the fiber bundle and on how load is
redistributed among the fibers in the presence of
broken fibers.
However, because of the interfacial sliding resistance z, broken fibers do carry some load away from
a fiber break point. For a fiber break occurring at
stress T, the stress in the broken fiber recovers linearly
from 0 at the break point to T over a slip length
1 = r T / 2 z on each side of the break. The load previously carried by broken fibers within + l of a break
must be redistributed to the unbroken fibers. (In this
context, unbroken means that section of any fiber
further than I away from a break.) We assume Global
Load Sharing (GLS) in which the additional load is
distributed equally among all unbroken fibers, that is,
there are no stress concentrations. The GLS assumption implies that the stress in a n y fiber at a n y point
further than l away from a break is precisely T; all
unbroken fibers carry the same load. The stress
distribution along an arbitrarily chosen fiber with
some breaks is shown schematically in Fig. 5. The
load T is related to the actual applied load per fiber
~ / f as determined below.
U n d e r the GLS assumption, the stress profile along
every fiber is similar to that shown in Fig. 5. If we
imagine a plane perpendicular to the fibers intersecting each fiber at one point, equilibrium requires that
the total applied load must equal the sum of the loads
carried by the fibers at the points of intersection.
Fibers with no breaks within l of the intersection
[
T
4i
~
rT
"c
,
I
60
3. THE UTS MODEL
Consider a uniaxial fiber-reinforced composite
containing a volume fraction, f, of aligned cylindrical
fibers of radius, r, and Young's modulus, El, embedded in metal or intermetallic matrix. We restrict our
attention to matrices whose cracking or yield strength
is well below the strength of the composite so that the
Fig. 5. A schematic representation of the stress distribution
in a typical fiber that has some breaks. The recovery length
is given by l = rT/2z where Tis the current maximum stress
in the fiber. Variations of the stress in the fiber due to the
presence of matrix cracks or other stress concentrators are
ignored.
DUVA et al.: CONSOLIDATION OF METAL MATRIX COMPOSITES
point carry a stress T, while fibers with breaks within
l of the intersection point carry a reduced load
proportional to the distance of the break from the
intersection point. However, since the stress profile
along every fiber is statistically identical and independent of the other fibers, the average stress carried by
the fibers at the intersection points must be the same
as the average stress along any single fiber. Thus, the
applied stress is
o" = Z
at(z) dz.
(5)
Here crr (z) is the stress profile in a fiber as shown in
Fig. 5, and L is a length much greater than the
average fiber fragment length.
Given the distribution of fiber fragment lengths
F ( x ) , where F ( x ) d x is that fraction of fiber fragments with lengths between x and x + dx, the integral
in (5) can be rewritten as
1123
These definitions imply that (I)(ac, 6c) = I. Note that
6c is twice the slip length at a c. We can then introduce
dimensionless parameters
T
T=--fi
cr
= p6c6 =
(7 c
(11)
O"c
and use the relation l = ac T/2 to reduce (8) to
5
-
1
= = (1 -
e
~').
(12)
f
P
The ultimate tensile strength of the composite corresponds to the maximum of the applied load 8 with
respect to T (which is proportional to the composite
strain). Note that f depends on the stress level T. In
the absence of any processing damage, the dimensionless break density in the composite is approximately equal to @ as given in (1).
In terms of dimensionless quantities
= ~m.
(13)
Using (12) and (13) together, the UTS is found to be
N
f L
f(x)
at(z) dz dx
do
(6)
6uvs
-f
do
where N is the number of fragments in length L. The
inner integral is simply T (x - l) if x /> 2l and Tx2/41
otherwise.
In previous work [10, l l] exact expressions were
derived for the fragment distribution F i n the absence
of initial damage, and adding damage due to processing in the form of breaks and/or a distribution of
residual bending stresses is a straightforward modification. To gain a physical insight of this, it is
useful to pursue a simplified model that can be
analyzed directly. To this end we approximate the
difficult to evaluate function F ( x ) by an exponential
distribution:
g ( x ) = pe o,-
(7)
where p = N/L. Substitution of (7) into (6) gives,
after some integration and algebra, the simple result
f =
(1 -
e
2pt).
(8)
Recall that the slip length l depends linearly on T.
Further, the composite strain is related to T through
E = T / E r as the composite strain is controlled by the
elastic deformation of the unbroken fibers. Hence (8)
is a constitutive relation for a fiber reinforced composite in the presence of a distribution of damage
characterized by p.
Equation (8) can be simplified further by introducing the characteristic length
=
-"E
(9)
and the characteristic stress
a~ =
( a,~zLo )l,'.,~ 1
- r
•
(lO)
--
auTs
~0r
~b ( m )
(14)
where ~b(m) can be accurately approximated by the
expression (1 - e 1/,,) (m/2)m/(m+ 1)
Equation (14) agrees with that obtained from an
exact analysis, validating the assumption that the
fragment lengths are exponentially distributed up to
the UTS. Beyond this load level, the damage evolution law of (13) is no longer accurate, and an exact
analysis must be used to predict fiber pull-out
lengths, the work of fracture and stress-strain behavior beyond the ultimate tensile strength.
4. UTS
CALCULATIONS
WITH
DAMAGE
PROCESS-INDUCED
4.1. Process-induced fracture
The effect of UTS (and the stress-strain behavior)
of combined process-induced fiber fractures and fiber
fractures due to an applied mechanical load can be
derived by the correct specification of the evolution
of the damage parameter ~. Process-induced breaks
could be included in one of two ways. First, if strong
defects in the defect population were subjected to
very large local stresses during processing so that
these defects, which would not fail under the applied
load, do fail, then it is appropriate to simply superpose "zero strength defects" (corresponding to these
fiber breaks with a dimensionless fracture density P0)
onto the Weibull distribution of defects to get
t~ = (P0 + :~m).
(15)
Alternatively, if the process-induced breaks are from
the weakest defects in the Weibull population of
defects, it is more appropriate to subtract these
defects from the Weibull distribution of defects so
that
(16)
DUVA et al.: CONSOLIDATION OF METAL MATRIX COMPOSITES
1124
08 07
~k'.'&X "X.
~
I
m=10
m =5
0.9 [
I
App roximale
II . . . .
Ex.~
o.8~I
I
0.6 [-
I
I
I
I
~
///,~...~
i
N
I
I
I
_
t3o = 0.657
°
",.~:X-.'-~.
-~ 0.5
,09
I
o.oo
0.3
0.2
0.4
0.i
0.0
~
~
~0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
~o 0.3
Strain/(~c/Ef)
0.2
o.-~o
0.0
~
I
I
1.0
2.0
3.0
Fig. 8. Shows the general features of the damage evolution
law given in (16) for the processing break densities associated with the processes described in Figs 4(a)-(c).
Damage Parameter Po
Fig. 6. Shows the reduction in the dimensionless tensile
strength of the composite as a function of the dimensionless
processing damage fi0 for Weibull moduli of 2, 5 and 10. The
processing breaks were superposed.
Because the bending forces are localized, the real
situation is likely to lie between these two extremes.
The first specification is conservative in the sense that
it overestimates the reduction in the UTS.
Figure 6 is based on (15) and shows the degradation of the UTS increases with the processing
damage parameter 70. It can be seen that the reduction in the UTS is less than 20% if the damage
parameter is less than unity (corresponding to about
270 breaks/m of fiber). Note the largest value oft~0 for
the three processes described by Figs 4(a)-(c) is about
0.66. Furthermore, the effects of process-induced
fractures are relatively weakly dependent on the fiber
Weibull modulus. As the process induced breaks
become more numerous, fewer breaks are induced by
applied load and the UTS becomes increasingly independent of the defect distribution in the fibers, that
is, of m.
0.8
0.7 I
VgTg] ~
Po = 0 ~
ExaeiModel
//"~--#o = 3
""
beMcd
0.6
ff 0.5
0.4
0.3
0.2
0.1
0.0
0.00
0.50
1.00
1.50
2.00
2.50
Strain/(c~/Er)
Fig. 7. Shows dimensionless composite stress strain curves,
based on (! 5), for three process-induced break densities. The
approximate model is valid only up to the UTS. The strain
to failure is a weak function of the process break density.
The processing breaks were superposed.
Figure 7 shows composite stress-strain curves for
a variety of/~0 values and m = 5, as computed with
(15), along with the results of the exact calculation.
The presence of processing breaks generally lowers
the tangent modulus as expected. However, the decrease in tensile strength is not accompanied by a
decrease in failure strain until the a m o u n t of damage
is high; the failure strain is nearly independent of the
processing damage even if the UTS is reduced by as
much as 30%. The figure also shows the breakdown
in the approximate constitutive relation embodied in
(8) beyond the UTS.
When the damage evolution is given by (16), the
stress-strain curve becomes identical to the case that
there is no processing damage once the stress/~"" has
been exceeded. Thus, if/~0 is sufficiently small, the
UTS is unaffected by processing breaks and only
the shape of the stress-strain curve changes. If the
a m o u n t of damage is large, failure will occur when
the stress-strain curve intersects the stress-strain
curve in the absence of damage, somewhere past its
peak. Figure 8 exhibits these features, and it is seen
that the prediction based on the simplified model in
this case may be poor because the post-UTS portion
of the curve is not accurate.
4.2. Process-induced bending
Process induced bending stresses like those shown
in Fig. 4 can be predicted by the consolidation
processing model of Elzey and Wadley [6] described
in Section 2. The bending causes local tensile and
compressive stresses to develop in the fiber, and the
distribution of stress in the composite upon the
application of a mechanical load is complicated.
Nevertheless, to a first approximation the bending
stresses can be linearly superposed on the tensile
stress due to mechanical loading if the effect of the
finite fiber curvatures on the overall stress distribution is ignored. Because of the highly nonlinear
nature of the fiber defect distribution, the reduction
of the probability of failure of a fiber element compressed by bending is not as great as the increase in
the probability of failure of a fiber element extended
DUVA et al.: CONSOLIDATION OF METAL MATRIX COMPOSITES
by bending, so the effects do not offset and bending
causes a net increase in the probability of failure at
a particular applied mechanical load.
The tensile stress in a fiber with process induced
bending is not uniform because of the through thickness variations across the fiber and because of the
variation in the amount of bending along the length
of the fiber. The variation caused by breaks is accounted for as described above. The nonuniformity
of the stress distribution due to bending can be
included in the U T S model through the appropriate
alteration of the damage evolution relation [that is, ¢
from (1)]. Instead of considering the fiber as a whole,
imagine breaking it into volume elements. In each
element the stress state, partly due to bending and
partly due to the applied mechanical load, is known.
The breaks in each element are added (integrated) to
get the total number of breaks in a fiber of length L
and radius r subjected to a mechanical load T and a
bending load zaB(l)/r, with z being the distance from
the neutral axis of the fiber. Thus (1) can be written
as
¢(r,L) =L~
,
x rx/75~-z2( T + zaJ(l)'/r ~"dl dz
ao
~
i
J
i
~
i
i
i
[ ~
0.8
P
o
~
0.7
0,6 0 ~ 3 6 5
0.5
0.4.-':
'"..
0.3
0.2
0.t
O.0
..... ±
•
__
I
t . .±
i
~-
0.0 0.2 0.4 0.6 o.a 1.0 1.2 1.4 1.6 1.8 2.0
Strain//(c~c/E/)
Fig. 9. Shows stress-strain curves for composites with
process-induced fiber bending and breaking associated with
the processes described in Figs 4(a)-(c). With bending
included, there is a large decrease in the UTS and the strain
to failure at values of P0 well below unit.
During processing residual stresses develop that are
insufficient to break (many) fibers, but reduce the
additional load required to cause large numbers of
breaks. This is particularly true for fibers with a large
Weibull modulus, so that the strength distribution is
narrow and very nonlinear.
5. S U M M A R Y
¢(T'L )=rcLoJ 1
_T+~ae(2)
i
(17)
/
where the integration over the cross section in the
direction normal to the bending has been executed.
N o t e that (17) is reduced to the simple form of (1) if
a B = 0. It is to be understood that the integrand is set
to zero if T + zaB/r < 0, as we are considering only
defects that fail in tension. A similar alteration of ¢
is discussed in detail in [13]. Nondimensionalizing
(17) with 2 = l/L and ~ = z/r gives
xv
1.0
0.9
1125
d~d~.
(18)
O"0
To use (18) in the exact algorithm for computing the
U T S [9], it is convenient to nondimensionalize each
length and stress with respect to 6~ and o-c as defined
in (9) and (10).
In defining the damage evolution law, the processinduced fractures are subtracted from the Weibull
distribution of defects as was done to obtain (16). The
stress-strain curves for composites with undamaged,
straight fibers and for composites with each of the
three processing damage states described in Fig. 4 are
presented in Fig. 9. It is evident, in contrast to the
results shown in Fig. 7 where only breaks were
considered, that there can be a significant reduction
in the U T S for values of t~0 well below unity, and that
the strain to failure is similarly reduced in the presence of bending damage. Thus, when present, bending stresses must be considered in making accurate
predictions of the U T S and could explain anomalous
measurements of fiber strengths reported in [3].
A simple, but accurate, analytic constitutive model
for metal or intermetallic matrix composites has been
proposed based on a simple way of looking at the
load carrying capacity of the fibers and the assumption that the fiber fragments are exponentially distributed. The model is useful for predicting the U T S
of the composite, but fails to accurately capture the
post UTS behavior of the material. A model that
quantifies process-induced fiber breaking and bending damage has been joined to the U T S model and
the significant effects of such damage on the U T S of
a typical composite have been computed. In doing so,
a direct link between the conditions under which a
metal/intermetallic matrix composite is fabricated
and its resulting properties has been established.
Acknowledgements--We are grateful to Professor Dana
Elzey for calculating the distributions of process stresses
used here, to James Groves for providing the micrograph,
and to the Advanced Research Projects Agency (William
Barker, Program Manager)/NASA (Robert Hayduk, Program Manager) for their support of this work under grant
NAGW 1692.
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