Composites Part B 28B (1997) 233-242
PII: S 1359-8368(96)00044-3
ELSEVIER
© 1997 Elsevier Science Limited
Printed in Great Britain. All rights reserved
1359-8368/97/$17.00
Densification of metal coated fibers by
elastic-plastic contact deformation
Haydn N. G. Wadley, Terry S. Davison and Joseph M. Kunze
Department of Materials Science and Engineering, University of Virginia,
Charlottesville, VA 22903, USA
(Received 7 September 1995; accepted 30 August 1996)
Near net shape titanium and nickel matrix composites can be produced by the hot isostatic/vacuum hot pressing of
alloy coated ceramic fibers. During the initial stage of consolidation, densification occurs by the inelastic deformation
of metal-metal contacts. When the temperature is low and the consolidation pressure is high, the dominant mechanism
of contact deformation is matrix plasticity. This densification process is investigated by analysing the elastic-plastic
contact deformation of aligned fibers. Applying a methodology developed for modelling the consolidation of alloy
powders and spray deposited composite monotapes, a contact yield criterion has been proposed and used to predict the
dependence of the relative density upon process conditions and matrix mechanical properties. The resulting
densification model contains unknown plastic flow (F) and contact area evolution (c) coefficients. A deformation
theory elastic-plastic finite element analysis of a representative coated fiber unit cell loaded in compression is used to
find these coefficients. The analysis shows the ceramic fiber significantly constrains plasticity in the alloy coating
resulting in fiber volume fraction dependent coefficients and a fiber volume fraction dependent density-pressure
relationship for coated fiber consolidation. © 1997 Elsevier Science Limited
(Keywords: metal matrix composites; contact mechanics; densification models)
INTRODUCTION
Metal matrix composites (MMCs) consisting of titanium or
nickel alloy matrices, reinforced with continuous alumina or
silicon carbide monofilament fibers have attracted interest
because of their relatively high specific stiffness, strength
and creep resistance 1. They can be manufactured by the hot
isostatic or vacuum hot pressing (HIP/VHP) of a preform
layup. In the past, many processing methods have been
pursued for the production of these preforms including the
powder cloth approach (and its continuous tape casting
analogue) 2, the foil-fiber-foil method 3,4 and plasma spray
deposition 5. Recently, interest in an alternative, physical
vapor deposition process for composite preform fabrication
has developed 1,6-10. In this approach, the matrix alloy is
first evaporated using either electron beam heating 1,6-9, or
sputtering 10 and is then condensed directly onto the fibers.
If the fibers are rotated during deposition, a (more or less
uniformly) metal coated ceramic fiber like that shown in
Figure 1 results.
A composite component can be manufactured from metal
coated precursors by either winding the coated fiber on a
mandrel, or packing them into a preshaped tool. These
preform laytips are then sealed inside a metal canister and
subjected to a high temperature consolidation process such as
hot isostatic or vacuum hot pressing 6, or perhaps roll bonding.
Contact deformation induced by the high applied pressure
eliminates the interfiber porosity while the high temperature
facilitates diffusion bonding of the constituents forming a
fully dense near-net shape composite component 10
The consolidation process can significantly affect the
performance of the component. It must result in complete
densification whilst ensuring minimal mechanical damage
to the fibers (e.g. by bending at fiber crossovers 1o,11, or
longitudinal matrix flow along the fibers 12) and chemical
reaction between the fiber and matrix alloy 13,14. Failure to
accomplish this can result in a loss of composite strength,
creep rupture life, fracture toughness and fatigue resistance
15,16. To avoid such damage during the consolidation of
spray deposited preforms, it has been very helpful to model
the consolidation process 17-20 and use process simulation
to design an optimal process schedule. Basically, models of
this type attempt to describe the micromechanisms of
densification, fiber damage, and fiber-matrix reaction in a
way that enables prediction of the effects of fabrication
pressure, temperature and time upon the relative density, the
extent of fiber microbending/fracture and the reaction layer
thickness 18
Following the approach of Wilkinson and Ashby 21, it has
usually proven sufficient to analyze the deformation of two
geometries; inter-powder particle or inter-monotape asperity contacts established at the start of compaction (so-called
Stage 1 densification) 21'22 and the collapse of isolated voids
as full density is approached (Stage II). During powder or
spray deposited monotape consolidation, the Stage I
geometry characterizes the situation encountered from the
233
Densification of metal coated fibers: H. N. G. Wadley et al.
Figure 1 A Sigma 1240(tungstencored) siliconcarbide fiberwith an electronbeam depositedTi-6AI-4V coating
beginning of consolidation up to a relative density of about
85%; the latter from about 95% to full density. For both
geometries, the models analyze the contributions from
plasticity (the time independent component of deformation),
power law creep, and diffusional flow and obtain the total
response by summing the individual contributions of the
three mechanisms acting on the two geometries.
When the consolidation temperature is low and the
contact stress is high (i.e. during initial loading or when high
consolidation pressures are used to either increase the
densification rate or reduce the opportunity for chemical
reactions between the fiber and matrix), plasticity is the
dominant mechanism of densification. Relationships
between relative density, applied pressure and the matrix
yield strength have been developed for Stage I consolidation
by analysing the inelastic flow at representative contacts
2~.23. For monotapes the resulting model contained two
important coefficients, F and c, identified by Bower et al. 24
for inelastic indentation and by Gampala et al. 23 for asperity
blunting. The flow coefficient, F, characterized the
constraint of the matrix to inelastic flow near the contact
while c related the contact area to the contact' s deformation.
Expressions for the flow coefficient have been obtained
for homogeneous (unreinforced) contacts using the slip
line field method (i.e. assuming rigid-perfectly plastic
behavior) z5-28. If the contact strain dependence of the
constraint is ignored, this approach gives the well known
value of 2.97 for F during indentation and blunting.
However, the large shape changes incurred during asperity
blunting results in a loss of the constraint provided by
surrounding elastic material. Recent finite element calculations show that F then decreases towards unity (i.e. the value
for an unrestrained parallel sided sample loaded in uniaxial
compression) as contact blunting (and densification)
progresses. Eventually, lateral constraints imposed by
adjacent contacts establish a new source of constraint and
the deformation resistance of the contact then rapidly rises
234
as the body approaches the incompressible limit at full
density.
Here, we extend this modelling approach to examine the
Stage I consolidation of metal coated fibers by deformation
theory plasticity. Expressions for F and c as a function of
either contact strain (or center-to-center displacement), fiber
volume fraction and matrix yield/work hardening properties
are obtained using a finite element method. It will be shown
that the presence of a rigid, very strong ceramic fiber has a
significant effect upon the contact yielding of the metal
coating. The expressions obtained for F and c are then
inserted into a simple Stage I densification model and used
to predict the relationship between relative density and
applied pressure, the matrix material's yield strength/work
hardening rate and the fiber volume fraction. Future work
will investigate the more complex case of creep densification and the significance of lateral constraint created by
multiple contacts.
DENSIFICATION MODEL
A coated fiber preform typically contains a variety of
packing geometries. When uniformly coated fibers are
carefully wound on a mandrel, a close packed hexagonal
array with an initial relative density of 0.906 can be
obtained. However, when fibers are placed in a die, the local
packing usually varies from place to place (i.e. random
packing) and includes regions of hexagonal, square and
triangular packing. For a random packing of aligned coated
fibers, the number of contacts made with a fiber (i.e. the
coordination number) is initially low ( < 4) and increases
with density towards a value near 6. Figure 2 shows the
cross section of a partially consolidated composite fabricated from silicon carbide fibers sputter coated with a
titanium alloy (similar to that shown in Figure 1) and
subjected to constrained uniaxial compression in a VHP.
Densification of metal coated fibers: H. N. G. Wadley et al.
Figure 2
"'Square" array of composite VHP consolidated silicon carbide (Sigma 1240) fibers sputter coated with Ti-6A1-4V
The area shown corresponds to a region of nearly square
packing though the die was initially randomly packed.
To illustrate the modeling approach, consider a square
array of coated fibers, subjected to a uniaxial pressure, P,
Figure 3. If the sides of the preform are not constrained,
lateral fiber expansion can occur and is analogous to the
situation encountered during the early stages of densification by VHP before the fibers have felt the affect of the die
side walls through lateral fiber-fiber contacts. In this case,
the average initial coordination number of a random
packing within the die would be low (i.e. around 4), and
the contacts formed perpendicularly to the loading direction
are subject to little lateral constraint (and can be ignored). It
is also similar to the case encountered during lubricated
roll bonding where, again, there is no physical barrier to
lateral fiber preform expansion (other than friction with the
rolls).
If a constant pressure, P, is applied to the preform, the
force acting on each of the contacts will be invariant
throughout the deformation process. For a square fiber
array, Figure 3, the contact force per unit length of fiber fc
balances the applied force. Thus,
fc = 2ero
(1)
where r0 is the radius of the matrix coated fiber. Each fiber is
seen to form two large contacts normal to the compaction
direction resulting in columns made up of contacting fibers
that support the applied pressure. Each of these normal contacts will be subjected to an identical contact force, fc, normal to the plane of contact.
As the applied pressure increases, the matrix initially
deforms elastically (see 29 for an analysis of the elastic
composite cylinder's contact problem). Plastic yielding
occurs once the contact stress, oC. equals or exceeds the
contact's flow stress. By analogy with asperity contact
deformation 23, we write a yield criterion for the contact in
the form
L
o~ = - - = F %
(2)
ac
where a~ is the contact length, F is a plastic flow coefficient and try is the matrix material's uniaxial yield
strength.
Classical studies of contact yielding for homogeneous
rigid-perfectly plastic materials were begun in the early
part of this century by Prandtl 25 and continued by Hill 26,
Tabor 27, and Johnson 3o. They have led to the well known
result, F = ac/% = 2.97, for a homogeneous rigid-perfectly
plastic body with a Mises constitutive law. More recently,
Gampala e t al. 22'23 using the finite element method and
235
Densification of metal coated fibers: H. N. G. Wadley et al.
Frictionless
Initial aeometrv
c~ntacts
v-t o
(b) During consolidation
r
ttt,
Figure 3
ttt
Square array of coated fibers with representative volume element
Akisanya et al. 28 using slip line field theory have
investigated the large strain blunting of hemispherical
asperities and homogeneous cylinders, respectively. They
determined that initially the flow coefficient was --3, but
rapidly dropped to - 2 with continued deformation. This
occurs because as the deformation progresses the region of
plasticity is constrained by a decreasing volume of elastic
material in these finite bodies. The coefficient for metal
coated fibers is also likely to exhibit analogous behavior and
will thus be a function of the relative density (or
equivalently the normal displacement imposed on the
contact).
As the deformation increases, the contact width, ac,
grows. Following the work of Bower et al. 24 and Gampala
et al. 23, we propose that for the two-dimensional contact of
aligned coated fibers, the contact area per unit length (i.e.
contact width), dependence upon the normal displacement,
236
m
h, can be written in the form;
1 a2
h -- 2c 2 ro
(3)
where a is half of the contact length (at), h is the normal
displacement, r0 is the initial coating radius and c is the
contact area coefficient 22,26. An area coefficient, c, equal
to unity arises if the contact length equals the length of
the chord created when a circle is truncated by h,
Figure 3(b). In early modeling studies 21 a uniform redistribution concept was used to obtain an expression for the
contact area dependence on densification. Applied to this
problem, it gives a value of c ~ 1.2. However, during the
contact blunting of an asperity, experiments and finite
element analyses both show that material displaced from
the contact often piles up at the periphery of the contact
and is not uniformly distributed 22,24,30,31.
Densification of metal coated fibers: H. N. G. Wadley et al.
As the contact area grows and the contact pressure drops
below that required for continued plastic yielding (as
determined by equation (2)), no further deformation can
occur at that applied pressure. Thus, if the contact width and
the flow coefficient are known as a function of the normal
displacement, rearrangement of equations (1), (3) and (4)
results in a simple expression for the pressure needed to
compress a preform by an amount, h;
-P=Oy F(h)c[2h] 1-2
circular quadrant by a rigid flat plate needs to be analyzed
(Figure 3(a)).
After a series of convergence tests with different types
and sizes of finite element meshes, the unit cell was divided
into a mesh consisting of 900 rectangular, second order
isoparametric elements with 2800 nodes. Second order
interface elements were used to model the line contact that
developed between the surface of the cylinder and the rigid
plate. Symmetry of the finite element mesh was maintained
by constraining the vertical and horizontal centerlines of the
coated fiber to remain straight. Thus, the horizontal
diametral line defining the upper surface of the cell was
constrained to remain horizontal as a uniform applied
displacement, h, was applied.
The ceramic fiber was modeled as an isotropic linear
elastic material. The outer metal layer was modeled as an
isotropic perfectly-plastic or bilinear strain-hardening
material (i.e. % = try + ke Gwhere e,~ is the plastic strain
and k the work hardening rate). Yielding of the outer layer
was assumed to be governed by the Mises condition,
(4)
Equation (4) defines the applied pressure, P, required to
cause a square array of metal coated fibers to undergo a
displacement, h, if the contact material has a yield strength,
Oy. It establishes a relationship between densification and
the pressure variable of the consolidation process. The
effect of processing temperature enters through the temperature dependence of ar.
Once the expression for h (equation (4)), is evaluated, the
relative density can be simply computed for the fiber array.
For example, for an unconstrained square array of metal
coated fibers, the relative density, D, is related to the normal
displacement contact, h, by:
1
)2)1/2
V/~((O" 1 -- 02) 2 "~ (0" 2 -- 0 3 ) 2 q- (03 -- O"1
= O" ~ O'y (6)
where ay is the matrix material's uniaxial yield stress and aj,
j = 1,3 are the principal stresses (in Voigt notation).
The governing stress-strain response of the outer layer
material in incremental form was taken to be,
71"
D= 4(1+70)(lW _ h)
(5)
dept = dX Og
Oaq
where w is the change in lateral width of a unit cell in the
array, Figure 3(b). It is shown below that w is only non-zero
for homogeneous cylinders. It is very small when a well
bonded ceramic fiber is present in a composite cylinder
with a fiber volume fraction of 0.25 and above.
where de pl is the plastic strain increment, dX, is a non-negative constant and g(a) is the plastic "associated flow"
potential. For the Mises flow potential used here, g(a) is
given in terms of the principal stresses by,
g(a) = f ( a )
FEM ANALYSIS
= ~(at
- o2) 2
(8)
where fla) is the Von Mises associated flow rule for an
isotropic material.
Detailed calculations have been conducted for representative metal-coated ceramic fibers with ceramic fiber
fractions of 0, 0.25 and 0.49. The metal coating has been
given the elastic and plastic properties of Ti-6A1-4V whilst
the (ceramic) fiber was assigned properties similar to those
of SiC (Table 1).
The commercial ABAQUS code was used to perform the
calculations via a PATRAN modelling interface.
If equations (4) and (5) are to be used to predict a densityprocessing relationship, the dependence of both the flow and
area coefficients on volume fraction of fiber and the extent
of deformation, h, needs to be determined. Here, a finite
element method is used to obtain F and c.
The FEM problem we consider is the plane strain
deformation of a composite cylinder consisting of a central
core that behaves elastically and an outer cylindrical shell
with either linear elastic-perfectly plastic or linear elasticlinear plastic hardening constitutive behavior. It is assumed
that the fiber-matrix interface is perfectly bonded. Thus,
continuity of displacement and tractions are assumed at the
interface. Initially, the semicircular surface of the matrix is
taken to be stress free and to make a frictionless diametrical
contact with the neighboring composite cylinders. Because
of the problem's symmetry, the blunting of only a plane
Table 1
(7)
FEM RESULTS AND DISCUSSION
For small normalized applied displacements (h/r0), the
coated fibers deformed elastically and the pressuredisplacement relation was approximately linear and
Material properties used in FEM analysis
Material
Young's Modulus (GPa) Poisson's Ratio
Uniaxial Yield Strength (Mpa)
Work Hardening Rate (MPa)
Ti-6AI-4V
SiC
110
390
840
~
775
--
0.324
0.240
237
Densification of metal coated fibers: H. N. G. Wadley et al.
MISES
STRESS
h / ro = 0.0154
CONTOURS
h / ro = 0.0462
h / ro = 0.0769
Homogeneous cylinder
.600 MPa ~.
/
(b)
840
840
6OO
400
3
Composite cylinder: 0.25 fiber fraction
400
a / ro = 0.338
840
Fiber
~-,ooMPa-~/\
Matrix /
~ L 2
(c)
3
355
]
382
Composite cylinder: 0.49 fiber fraction
6OO
840
I000 X'NX
840
Figure 4 Von Mises stress contours showing the evolution of the stress fields and contact area during deformation as a function of fiber volume fraction of
normalized consolidation displacement (h/ro)
agreed well with other studies of elastic composite cylinder
contact 29. As the applied displacement increased, the metal
coating began to flow plastically just above the center of the
contact. This occurred when the contact pressure was about
1.1 ay. As the applied displacement was increased, this
region of plasticity spread, eventually intersecting the
surface of the cylinders.
shows Mises stress
Figure4
238
contours for the three fiber volume fractions after normalized displacements, h/r0, of 0.0154, 0.0462, and 0.0769. For
the homogeneous case,
4(a), the plastic zone had
already reached the edge of the contact surface for the h/r0
= 0.0154; however, the center of the contact remained
elastic. As the displacement increased, the plastic zone
expanded vertically to the horizontal centerline of the fiber
Figure
Densification of metal coated fibers: H. N. G. Wadley e t
and more gradually spread laterally. Figure 4(a) shows that
even after a large normalized displacement of 0.0769, the
plastic zone of a homogeneous cylinder was still laterally
constrained by a shell of elastically deforming material.
Figure 4(a) also shows a rapid growth of contact width a s / f
r0 increased.
A small but significant lateral spreading of the cell also
accompanied the uniaxial compression of the homogeneous
cylinder. The finite element calculated width change
depended upon h and was well fitted by;
3.5
I
I
I
I
3.0
L•
g
2.5
"~
2.0
0
1.5
~
LL
~ ""0
s
"~
m
¢
¢
¢
==
m
_
-w
r0
r0
for 0 <- - - - 0.08
• Vt = 0.49
• Vf= 0.25
[] vr= o.oo
(9)
r0
The Mises stress contours for a 0.25 ceramic fiber
volume fraction cell are shown in Figure 4(b)for identical
normalized displacements to the homogeneous case
above. Plasticity was again seen to be initiated just above
the contact. As the applied displacement increased, the
plastic zone rapidly propagated upwards to the fibermatrix interface. After encountering the fiber, it extended
through the annular metal coating and eventually reached
the horizontal centerline of the fiber. Small elastically
deforming regions of the metal coating remained near the
ceramic/metal interface at the top of the cell and on the
periphery of the coating even after the largest normalized
displacement, (/fro = 0.0769). No detectable lateral spreading of the cell was observed, even for the largest normalized
displacements.
Figure 4(c) shows the Mises stress contours for the 0.49
ceramic fiber fraction case. The fiber had an even more
pronounced effect upon the plastic flow of the metal coating.
The plastic zone more rapidly reached the fiber-matrix
interface during deformation consistent with the higher
concentration of stress in the matrix at the fiber matrix
interface 29. This plastic zone then began to extend around
the annular metal ring. However, this was again inhibited by
the constraint of the well bonded fiber and resulted in
stresses within the plastic zone that were significantly
greater than the uniaxial yield strength of the matrix. A
larger volume of the coating was observed to remain elastic
for the higher fiber fraction case.
0.0
O.O0
I
I
I
1
0.02
0.04
0.06
0.08
Normalized
0.10
displacement h/r o
Figure 5
Finite elements results for the yield coefficient, F, as a function
of normalized displacement (elastic-perfectly plastic case)
and then decreased with normalized displacement as the
plastic zone reached the contact. For the perfectly plastic
composite samples, F ceased to significantly increase once
the plastic zone reached the outer boundary of the metal
shell (/fr0-0.015). Further deformation resulted in a slight
increase in F. Once the fully plastic behavior was
established, the flow coefficient increased with fiber
fraction.
The lower flow coefficient of - 2 observed here for the
onset of fully plastic flow versus - 3 from the slip line
analysis of Akisanya et al. 28 may be a consequence of the
two-dimensional (plane strain) cylindrical contact analyzed.
Akisanya et al. 2s assumed that the small size of the contact
relative to the cylinder diameter justified using the infinite
plate result for F which has a significantly higher constraint
because of the need for the slip lines to reach the more
remote free surface. The FEM results indicate that by the
point at which fully plastic flow is established in a
cylindrical contact (h/r0--0.01), the contact length is too
large (compared to the cylinder's radius) for this to be a
Flow coefficient, F
The flow coefficient, F(h), defines the magnitude of
the contact pressure required to cause inelastic contact
deformation. It is a function of the contact's normalized
displacement and can be found from the analysis of fiber
contact. First, the area of contact and the contact pressure
were found for a prescribed vertical displacement, h, applied
to the unit cell (Figure 3). Since the uniaxial flow stress (an
input to the model) was known, equation (2) could then be
used to obtain F(h).
The flow coefficient's dependence upon normalized
displacement for the three fiber volume fractions analyzed
is shown in Figure 5 for the elastic-perfectly plastic matrix
and in Figure 6 for the elastic work hardening matrix.
During the initial elastic contact, the flow coefficient of the
three samples rose rapidly towards a value of 2. For the
homogeneous cylinders, F reached a value of about 2.2
m
Elastic - Perfectly Plastic
1.0
0.5
w = 0.002 h + 4.38
al.
3.5
I
I
I
I
3.0
L~
2.5
"3
2.0
~E
0
0
0
1.5
o
1.0
U..
Work Hardening
• Vf = 0.49
• vt= 0.25
0.5
0.0
0.00
D Vf=O.O0
I
I
I
I
0.02
0.04
0.06
0.08
0,10
Normalized displacement h/r o
Figure 6
Finite elements results for the yield coefficient, F, as a function
of normalized displacement (linear work hardening case)
239
Densification of metal coated fibers: H. N. G. Wadley et al.
Table 2
Results for the flow coefficient dependence (F = m/hro + b)
Fiber volume fraction
0.0
0.25
0.49
Perfectly plastic matrix
Work hardening matrix
m
b
m
b
-7.7
1.5
1.5
2.3
2.2
2.5
-6.1
4.4
7.5
2.3
2.2
2.5
good approximation. The curvature of the cylinder in effect
decreases the volume of elastic material that constrains
plastic straining. The subsequent drop in F exhibited by the
unreinforced fiber arises from a further loss of elastic
constraint as the plastic zone spreads over the cylinder
surface, Figure 4(a), and is the two-dimensional analog of
the situation encountered in spherically symmetric (homogeneous) contacts 22. For the reinforced composite cylinders, this constraint loss mechanism is found to be
compensated by the development of constraint created by
the hard to deform ceramic fiber and the imposed continuity
of displacement at the fiber-matrix interface. This "locksup" material near the interface and creates a resistance to
flow that increases with normalized displacement and fiber
fraction. Figure 6 shows that matrix work hardening also
increases the contact's flow resistance, particularly when the
fiber fraction is high. This occurs because the elastic fiber in
a composite cylinder limits the volume of plastically
deforming material and requires larger strains to accommodate the imposed displacement. These results indicate
F(h) to be a strong function of the fiber fraction and the
matrix work hardening rate.
It is convenient for densification modeling to obtain a
simple functional form for the dependence of F on h/ro.
Figures 5 and 6 show that the F versus h relation becomes
linear after a 0.3% deformation (corresponding to less than a
0.25% increase in the relative density of a square cylindrical
array). Thus, F can be approximated by a linear function
with a slope, m, and an intercept, b, that depend upon the
fiber fraction and matrix strain hardening rate (the straight
lines in Figures 5 and 6), Table 2. Note that the intercepts
are unaffected by the work hardening rate of the matrix; this
causes only an increase in the slope of the F versus h/ro
relation.
where the contacts exhibit c > 1 and matrix "piles-up" at
the periphery of the contact 23. Values of c < 1 for fibers
arise because of the relative ease of lateral spreading in
regions remote from the contact. This is retarded for
spherically symmetric contacts by elastic hoop stresses 2 2
The presence of an elastically deforming ceramic fiber
within a composite cylinder also inhibits the lateral
expansion and accounts for the approach of c towards
unity as the fiber fraction increases. Finally, we note that a
uniform redistribution calculation (like that used in earlier
models of particle c o n t a c t 21,32,33) results in a value of c =
1.12 for a square fiber array; a significant overestimate of
the growth rate of the contact. Other researchers e2 have
found the opposite to be true for the blunting of
homogeneous spherical contacts. The difference lies in the
fact that our analysis is for a two-dimensional contact as
I,,,,,,
0.5
The area coefficient, c, was calculated for each applied
displacement, h, by determining the contact length (from
finite element results) and plotting a/ro against ~ / r o,
Figure 7. The slope of this relationship gives the area
coefficient, c. The area coefficient also appeared to be
independent of the matrix work hardening relationship as
seen in the values of c shown in Table 3 for the six cases
studied.
The values for c were found to be always less than unity
consistent with a contact length smaller than the chord
length of a circle truncated by an amount, h. This
phenomenon has been referred to as "sinking-in" of the
contact 31 and contrasts with results for spherical contacts
240
I
I
I
(a) Elastic - Perfectly Plastic
0.4
~
0.3
o
0.2
E
0.1
0
Fiber
fraction:
009.04
. ~
~
0 . 0 ~ ~'r
0.0
v,=
0.49
v,--o.as
.0.~5
,,.~
Vf = 0.00
• O,847
z
I
0.1
I
0.2
r
,
I
0.3
I
0.4
i
i
0.5
q-2h / r
0.5
~.
Area coefficient, e
I
(b) Work Hardening
.c:~
.~b
0.4
0.3
O
o
•' ~
o
0.2
er
/~"
"~,
~ 0.1
g
0
Z
Slope: fraction:
.;,~,"
/.,,~
j a r
o.o ~
0.0
-
I
0.1
L
0.2
i
0.3
=o.9,~
v,=o.49
0 0.886
~;/=0.25
• 0.825
V;f=0.00
i
0.4
0.5
~Tr
Figure 7
Data and curve fits for calculating the area coefficient, c, (a)
elastic-perfectly plastic and (b) linear work hardening
Densification of metal coated fibers: H. N. G. Wadley et el.
Table 3
0.85
Values for the Area Coefficient, c = a / 2 v / ~ r o
Volume fraction
of fiber
Perfectly plastic
matrix
Work hardening
matrix
0.0
0.25
0.49
0.84
0.90
0.96
0.82
0.89
0.95
i
0.84
.#
0.83
¢- 0.82
"10
~> 0.81
opposed to the three-dimensional case studied by others.
The uniform redistribution model overestimates the area of
the contact compared to the finite element solution and
therefore overestimates the applied pressure needed to
achieve a desired density.
°
• c = 0.90, Perfectly plastic
-~ 0.80
¢¢
0.79
u
0.78
0.0
finite element model
3.25
c = 0.90, Work hardening
finite element model
c = 1.12, Perfectly plastic
uniform redistribution model
I
0.5
I
1.0
Normalized pressure
DENSITY PREDICTIONS
1.5
m/Oy
Figure 9
Using equations (4) and (5) and the relationships developed
above for the flow and area coefficients, one can determine
the pressure needed to achieve a prescribed vertical
displacement and thus density. Because F(h) and c depend
on fiber fraction, we can use the resulting model to assess
the effects of varying the volume fraction of fiber upon the
densification process. This is shown for the fiber volume
fractions used in the finite element calculations in Figure 8.
Additional results for intermediate fiber fractions were
obtained by linearly interpolating the dependence of F(h)
and c upon the fiber fraction. The maximum normalized
displacement imposed for the finite element analysis, h/ro =
0.08, corresponded to a final density of 0.83 for the
unreinforced case and 0.85 for the reinforced case because
of the larger lateral spreading of the homogeneous fiber.
These results show that the density of the homogeneous
cylinder rises more rapidly with pressure than for the
reinforced cases; it is a consequence of deformation
softening associated with the loss of constraint, Figure 5.
The additional constraint provided by a fully bonded elastic
fiber/metal matrix interface eliminates this "softening" and
results in a fiber fraction dependent densification response.
The effect of the matrices' work hardening rate upon
0.85
0.84
[~
o
'
~
i
c
0.83
B
¢" 0.82
0,)
>~ 0.81
- Perfectly Plastic
•--
-~
n"
Calculated values:
////
////
0.80
.
• v,= 0.00
~
o v~= O.lS
=v, o28
0.79
rl VI= 0.40
tn Vf = 0.49
!
0.78
0.0
I
0.5
Normalized pressure
Figure 8
of fiber
L
1.0
1.5
P/ay
Plastic densification model results for different volume fractions
Plastic densification model results illustrating the effect of work
hardening and the significance of the area coefficient (volume fraction of
fiber, 0.25)
densification is relatively significant for a composite
cylinder, Figure 9, because of the larger plastic strain
within the more localized contact deformation field of a
composite cylinder. Figure 9 also shows the more
significant consequences of using a uniform redistribution
estimate for c ( = 1.12) as opposed to the finite element
calculated case. The increase in the pressure needed to
achieve a prescribed density is a direct consequence of the
increased contact area (and thus reduced contact pressure).
Taken together, the results of Figures 8 and 9 show a strong
influence of fiber fraction upon the densification of metal
coated fibers and indicate that it will be important to account
for this phenomenon when attempting to predict the
observed densification response of these materials 6,Jo
SUMMARY
A finite element analysis has been used to examine the
elastic-plastic contact deformation of metal coated fibers.
The evolution of the contact area and stress fields were
obtained as a function of normalized displacement, and
from this the contact's flow and area coefficients were
calculated as a function of the volume fraction of fiber and
the matrix work hardening rate. The presence of well
bonded elastic fibers in a composite cylinder was found to
significantly inhibit the inelastic flow of the matrix,
resulting in an increased flow coefficient for a given density
or normalized displacement. The presence of matrix strain
hardening also increased the flow coefficient of the
composite cylinder contacts. The area coefficient was less
than unity and independent of the normalized contact length
and the work hardening relationship. While the area
coefficient increased with fiber fraction, it never exceeded
unity. The low values of c compared to spherical contact
results arose from a reduced resistance to lateral spreading of
the unit cell analyzed. These results have been used to predict
the densification-pressure relationships for a square array of
metal coated fibers. A strong fiber fraction effect is predicted.
241
Densification of metal coated fibers: H. N. G. Wadley et al.
ACKNOWLEDGEMENTS
13
T h i s w o r k h a s b e e n s u p p o r t e d b y the A d v a n c e d R e s e a r c h
P r o j e c t s A g e n c y (W. Barker, P r o g r a m M a n a g e r ) a n d the
N a t i o n a l A e r o n a u t i c s a n d S p a c e A d m i n i s t r a t i o n (R.
14
Hayduk, Program Monitor) through grant NAGW-1692
a n d t h r o u g h the A R P A J O N R f u n d e d U R I p r o g r a m at U C S B .
15
16
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