PII:
Acta mater. Vol. 46, No. 1, pp. 193±205, 1998
# 1997 Acta Metallurgica Inc.
Published by Elsevier Science Ltd. All rights reserved
Printed in Great Britain
1359-6454/98 $19.00 + 0.00
S1359-6454(97)00215-2
INELASTIC CONTACT DEFORMATION OF METAL
COATED FIBERS
D. M. ELZEY, R. GAMPALA* and H. N. G. WADLEY{
Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903
(Received 16 April 1997; accepted 23 June 1997)
AbstractÐMetal matrix composites (MMCs) can be synthesized by aligning metal coated ceramic ®bers
in a shaped container and applying pressure at elevated temperatures. The high stresses created at the
contacts between neighboring ®bers cause inelastic matrix ¯ow that ®lls inter®ber voids, reduces inter®ber separations and results in composite densi®cation. The rate of densi®cation depends on the contact's resistance to ¯ow. Current contact mechanics models are unable to adequately predict this
resistance because they do not account for the e€ect of the (elastic) ®ber. Closed-form solutions for
contact stress±displacement rate and contact area±strain relationships are used to describe metal coated
®ber blunting as a function of ®ber volume fraction, matrix material non-linearity (i.e. creep stress
exponent) and ®ber packing geometry. The solutions contain two unknown coecients (c and F) which
are evaluated using the ®nite element method. A simple model for the consolidation of coated SiC
®bers is developed in terms of the coecients, c and F. The model indicates that materials with a low
creep stress exponent are more dicult to densify as the ®ber volume fraction increases whereas, perfectly plastic materials exhibit a relatively weak dependence on the ®ber volume fraction. # 1997 Acta
Metallurgica Inc.
1. INTRODUCTION
Physical vapor deposition (PVD) processes are
beginning to be used to produce metal coated ceramic ®bers which can be used to create continuous
®ber-reinforced metal matrix composites (MMCs)
[1±4]. By applying reactive metals such as titanium
via a vapor phase process allows deposition of the
matrix onto a ceramic reinforcing ®ber (e.g. SiC,
Al2O3) held at a relatively low temperature
(06008C), thus avoiding chemical attack of the ®ber
or its coating. By rotating the ®ber during deposition, a relatively uniform thickness coating can be
applied [3]. Once a suitable thickness of alloy coating has been deposited onto the ®ber, a fully dense
composite component can be synthesized by aligning a bundle of the ®bers and subjecting them to a
consolidation process such as hot isostatic or vacuum hot pressing (HIP or VHP) [5].
During the consolidation step, pressure is normally applied at elevated temperatures (T 00.6 Tm,
where Tm is the absolute melting temperature) causing compressive stresses to develop at contacts. This
results in creep ¯ow into the voids between ®bers
and densi®cation of the composite [6]. Creep ¯ow is
enhanced by the retention of a ®ne grain size in the
metal phase. An optimal consolidation process schedule should exploit this and result in complete den*Now at Concurrent Technologies Corporation,
Johnstown, PA, U.S.A.
{To whom all correspondence should be addressed.
193
si®cation while minimizing ®ber/matrix reactions
and bending or fracture of ®bers [7]. The retention
of matrix voids in incompletely densi®ed components, excessive interfacial reaction or large ®ber
bending can all lead to potentially serious degradation of the tensile, creep and cyclic strengths of
fabricated components [8, 9]. The identi®cation of
an optimal processing path is dicult by experimentation alone because of the competing phenomena
(grain growth, matrix creep, ®ber/matrix reaction
and ®ber bending) activated by a consolidation process. This has stimulated an interest in the development of process models [10]. For example, models
have recently been developed for predicting the
evolution of density, interfacial reaction zone thickness and ®ber microbending/damage during the
consolidation of titanium matrix composite (TMC)
monotapes produced by plasma spray deposition
[11±13]. After experimental testing, these models
have been used to successfully design optimal processes [14] and even new feedback control algorithms [15, 16] that expand the combinations of
metal alloy and ®ber that can be successfully composited. This paper deals with the development of
an analogous model for early stage consolidation of
metal coated ®bers.
During the consolidation of metal alloy coated
®bers by either HIP or VHP, the externally applied
pressure is transmitted internally throughout the
body via points where the coated ®bers contact
(Fig. 1). As material is displaced at the contacts,
the centers of the coated ®bers approach, resulting
194
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
Fig. 1. The use of PVD processes to deposit alloy coatings
onto ceramic ®bers followed by consolidation o€ers a potentially low cost means for manufacturing continuously
reinforced metal matrix composites.
in overall densi®cation. Initially, the voids are much
larger than the average contact width, but as densi®cation proceeds, the voids eventually become small
relative to the contact width. This leads to a transition from densi®cation best described by contact
mechanics to one more accurately modeled as the
collapse of isolated voids in an inelastic continuum.
This transition of mechanism is similar to the geometry transition from Stage I to Stage II encountered in the modeling of powder or monotape
densi®cation [17]. The contact deformation of powders and monotapes during Stage I has been modeled by relating the applied pressure and overall
change in density to the behavior of a single (representative) contact. Contact forces and the resulting
densifying displacements are then related using
results from contact mechanics [e.g. 17].
The ®rst Stage I densi®cation models for powders
and monotapes drew upon an extensive contact
mechanics literature (see for example Johnson [18]).
Much of this had focussed on the analysis of indentation, in which a rigid body (an indenter) penetrated the plane surface of a softer substrate
(typically treated as an in®nite half-space). These
analyses had been conducted to aid the interpretation of hardness testing experiments. The contact
deformation occurring at interparticle contacts
during consolidation is better described as a blunting process. Although blunting is the more accurate
physical description of interparticle contact deformation, the widely available results of indentation
analyses continue often to be used to model consolidation [e.g. 17, 19±21].
In the in®nitesimal plastic strain limit, the di€erences between a blunting and indentation analysis
are insigni®cant. However, recent work [22] has
identi®ed important di€erences between indentation
and blunting for the large deformations characteristic of the end of Stage I densi®cation. In particular, it has been observed that during blunting, the
average contact stress required to cause continued
plastic deformation is strongly in¯uenced by the
size of the deforming body. This is in sharp contrast
to indentation, where a semi-in®nite volume of
material is typically available for deformation. In
the ®nite size blunting body, the zone of inelastically deformed material gradually expands away
from the contact (at the expense of the elastically
deformed volume), the elastic constraint gradually
disappears, and a signi®cant softening occurs. This
explains why the contact stress required to cause
further plastic deformation changes little with depth
during indentation, but is found to be very dependent on the displacement during blunting [23].
Because of this, indentation analyses cannot provide
an accurate prediction of contact blunting behavior
beyond relatively small deformations.
A further e€ect hampering the successful application of indentation solutions to contact blunting
during densi®cation is the external constraint
imposed on a contact by the close proximity of
other contacts. The average number of contacts per
particle (referred to as the particle coordination
number) is around seven for a random dense packing of monosized spheres [24], but is typically three
for aligned, randomly packed cylinders [5]. The coordination number increases with density, reaching
a maximum of about 12 for spheres (forming tetrakaidecahedra) and six for cylinders (forming hexagonal
close
packed
structures).
Initially,
deformations are localized at the contact and are
una€ected by the presence of other contacts, but as
densi®cation proceeds, the growing deformation
zones quickly begin to interact. While the in¯uence
of interacting deformation zones on a metal's resistance to contact deformation has been investigated
for blunting cylindrical and spherical contacts,
indentation models have necessarily neglected this
e€ect.
There appears to be some disagreement as to the
origin of the e€ect of interacting deformation ®elds
on the blunting resistance: the slip-line ®eld analyses
of Ogbonna and Fleck [25] and of Akisanya and
Cocks [26] led them to propose that the interaction
of plastic ®elds leads to softening. The authors in
[25] likened contact to the deformation of a ¯at
plate of thickness, 2H, compressed uniaxially
between platens of width, 2a; their analysis showed
that as the a/H ratio increased from 0 to 1 (allowing the opposing deformation ®elds to approach
and interact), the required pressure decreased from
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
3sy (where sy is the uniaxial yield stress) to about
sy. However, it has been shown [23] that comparable softening occurs even during the inelastic
blunting of isolated contacts, where no interaction
is possible. This softening was attributed to the loss
of elastic constraint as the plastic zone expanded
(especially as the plastic zone reached the contact
face or free surfaces of the blunting sphere).
Softening by this mechanism is not predicted by
indentation models (nor observed experimentally
during indentation testing) for semi-in®nite bodies
because the plastic zone remains small relative to
the elastic ®eld, and therefore experiences a relatively constant elastic constraint. However, the softening e€ect does become evident during indentation
when the substrate volume becomes ®nite [27].
Thus, the slip-line calculations due to Ogbonna and
Fleck [25] are consistent with the view that contact
interactions may lead to softening in the same way
that the interaction of a plastic zone with a free surface does, namely by causing a loss of elastic constraint.
While contact interaction may initially lead to
loss of elastic constraint and thus to softening, it
also causes plastic ¯ow to become redirected and
intersection of adjacent ¯ow ®elds. The increased
resistance to blunting of closely spaced contacts
after large deformation observed by Gampala et al.
was a result of this redirection of plastic ¯ow; a
process in which material is essentially extruded
into adjacent voids. The presence of a rigid ®ber in
the metal coated ®ber problem encountered here is
likely to signi®cantly perturb matrix ¯ow during
contact blunting. While recent developments in contact mechanics have focussed on extension of earlier
linear elastic or elastic±perfectly plastic analyses to
nonlinear material behavior, e.g. strain hardening
plasticity [28], power-law creep [29, 30, 22] and compressibility [31], with the exception of the work of
Davison et al. [32, 33], few have yet addressed the
e€ects of a ®ber upon the deformation ®elds near a
contact.
In general, the contact of ®ber-reinforced cylinders has received much less attention than homogeneous cylinders. Narodetskii [34] developed an
approximate, closed-form solution to the problem
of a heterogeneous cylinder deforming elastically
under the in¯uence of diametrically opposed
(radial) loads. More recently, Davison et al. have
improved on this result, obtaining an exact solution
[32, 33]. It was noted that the presence of the ®ber
leads to a concentration of stress (most notably the
hoop stress component) at the ®ber/matrix interface, causing plastic yielding to occur at a lower
contact stress than in the homogeneous case.
In addition to the elastic analysis, Davison [32]
also brie¯y considered the elastic±plastic contact deformation behavior of alloy coated ®bers for ®ber
volume fractions of 25 and 50% using FEM and
compared these results with those for the homo-
195
geneous case. It was found that although the onset
of yielding was lowered by the presence of the ®ber,
the stress to cause expansion of the plastic zone
increased with ®ber fraction. For example, at
a/r = 0.33, sc/sy=1.95, 2.5 and 3.2 for ®ber fractions of 0, 0.25 and 0.5, respectively. Furthermore,
in contrast to the homogeneous case, no softening
was observed for the ®ber reinforced cases. Finally,
Schuler et al. [35] also recently applied a ®nite element model to analyze the consolidation of metal
coated ®bers and compared their predictions with
experimental results for Ti±6Al±4V-coated SiC
®bers. Thin yttrium marker layers, incorporated
into the matrix coating as concentric rings during
vapor deposition, were used to track matrix ¯ow
for comparison with FEM results.
The elastic and rate-independent plastic behavior
of contacting alloy coated ®bers is important for
low and intermediate temperature consolidation,
such as during cold isostatic pressing (CIP).
However, consolidation is usually performed at elevated temperature, where the densi®cation is
accomplished by power-law creep of the alloy coating. While the creep strain rate±stress relation is a
sensitive function of microstructure, stress and temperature, the simplest uniaxial stress (s)±strain rate
(EÇ ) relationship for a power-law creeping material is
expressed as
n
s
e_ ˆ e_ 0
;
…1†
s0
where s0 and E0 are the reference stress and strain
rate, and n is the creep stress exponent. The contact
deformation behavior of metal coated ®bers is anticipated to depend strongly on the material nonlinearity, n. For example, Gampala et al. [22] found
that when n is high, the deformation within a blunting hemisphere tends to become highly localized
near the contact, and relatively large contact stresses are needed to maintain a given blunting displacement rate because of the signi®cant constraint
imposed on the inelastic deformation zone by the
surrounding elastic (or more slowly creeping) material. For the same reason, materials having a high
stress exponent experience the least in¯uence of
adjacent contacts. Analyses which neglect the constraining e€ect of neighboring contacts are thus
most accurate when applied to this class of materials. Materials having a low stress exponent on
the other hand, exhibit a strong sensitivity to lateral
constraint, as evidenced by a rapidly rising contact
¯ow stress. It has therefore been observed that
although intially, high-n materials are most resistant
to consolidation, low-n materials may rapidly
``harden'' due to the strong in¯uence of neighboring
contacts. Any approach used to model the contact
of metal coated ®bers should thus account for both
the e€ects of material nonlinearity and ®ber packing geometry.
196
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
Fig. 2. Schematic illustration showing the consolidation of a square array of metal coated ®bers with a
unit cell (a single quadrant of a coated ®ber and its contact) identi®ed for analysis.
2. MODELING APPROACH
During the compressive contact of metal coated
®bers, the presence of an elastic ®ber will signi®cantly a€ect the local distributions of stress and
strain rate within the matrix contact, and therefore
the contact's stress±displacement rate relationship.
Here, we apply a closed form solution for the contact stress±displacement rate relationship for the
contact of two representative metal coated ®bers.
The analysis assumes that the ®ber remains elastic
at the consolidation temperature and that the (metallic) matrix deforms by power-law creep. A ®nite
element approach solves for unknown coecients
relating the ¯ow stress to the contact pressure and
the blunting displacement to contact area. These
are a function of ®ber fraction, matrix nonlinearity
and type of packing. The coecients can then be
used to derive expressions for densi®cation, also as
a function of ®ber fraction, matrix nonlinearity and
packing coordination.
A random dense packing of aligned, monosized
alloy coated ®bers typically has a starting relative
density of between 75 and 80%. This corresponds
to an average number of interparticle contacts per
particle (particle coordination number, Z) of
between 3 and 4. In a random packing, the number
of contacts per particle varies during densi®cation
from about 4 to a maximum of 6. The resistance to
deformation at any particular contact depends on
the number of contacts per particle, Z, due to interaction between adjacent inelastic zones. In order to
explore the in¯uence of contact interaction, two
limiting cases are considered: a square array corresponding to low density (78%) with a particle coordination number, Z = 4, and a hexagonal close
packed array corresponding to high density (91%)
with Z = 6. The approach is ®rst developed, and
results presented, for the square array, followed by
results for the h.c.p. structure.
Figure 2 shows a square array of parallel, alloy
coated ®bers undergoing compression in the y-direction. Each coated ®ber is constrained laterally so
that no material is able to move in the x-direction
beyond the limit of a square cell enclosing a typical
coated ®ber. The ®bers' length is assumed to be
much greater than their diameter, and thus no
strains are taken to occur in the z-direction, i.e. in
the direction of the ®bers. The macroscopic strains,
Ex and Ez, are thus zero. Since each coated ®ber is
identical and experiences the same loading, any
single coated ®ber may be selected for analysis. Due
to symmetry, this may be further reduced to a unit
cell consisting of a quarter circle (Fig. 2).
The spatially distributed strains obtained by ®nite
element analysis can be collapsed to a form suitable
for consolidation modeling by using an approach
developed for the power law creep blunting of
spherical contacts [22]. There, it was shown that the
contact stress, sc, bears the same power-law relation
to the e€ective strain during blunting (de®ned ashÇ /
aEÇ 0) as does the uniaxial stress (s) to the strain (E)
in the alloy matrix, with the addition of a parameter whose value depends on the shape of the
blunting contact, the material non-linearity and
blunting strain. The contact stress±blunting rate solution for a given blunter geometry (i.e. conical,
spherical, etc.) was then shown to have the form
!
a h_ 1=n
sc
ˆ F n;
:
…2†
s0
r a_e0
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
Here, s0 and E0 are the reference stress and strain
rate de®ned by equation (1), a/r is the ratio of the
contact radius to the radius of the undeformed
hemisphere and F is a ¯ow coecient; an as yet
undetermined, dimensionless function of the creep
stress exponent, n, and the blunting strain, a/r. For
F = 1, equation (2) describes the uniaxial powerlaw creep of a cylinder of height, a, subjected to an
axial compressive stress, sc, i.e. Norton's creep
equation is recovered when F = 1. During contact
blunting, F is almost always greater than 1, re¯ecting the increased resistance to deformation due to
constraint imposed by material surrounding the
zone of contact deformation. The value of the ¯ow
coecient depends on the evolving stress and strain
rate ®elds within the deforming contact, and is evaluated using ®nite element analysis.
In previous studies of spherical asperity blunting,
closed-form solutions for F (n,a/r) were obtained by
®tting polynomial expressions to ®nite element
results obtained by analysis of a representative unit
cell using di€erent values of the stress exponent, n
(see Table 2, [22]). This is essentially the same
approach that had been used to analyze the indentation of power-law creeping solids [29, 30], with the
exception that the self-similar character of the
indentation response allowed F to be expressed
independently of a/r.
As the blunting displacement, h, increases, the
contact semi-width, a, increases at a rate which
depends on the degree to which deformation concentrates near the contact. For materials having a
large stress exponent, n, strains are highly concentrated in the region of the contact where stresses
are greatest, and therefore the contact area grows
most quickly with h. It has been shown [22], that
for a power-law creeping spherical contact, the
blunting displacement±contact radius relationship
can be expressed as
2
1
a
hˆ
;
…3†
2c…n†2 r
where c (n) is the contact area coecient. For
spherical asperity blunting, c increased with increasing stress exponent, n, re¯ecting the observation
above that for a given blunting displacement, the
size of the contact is greater for materials with a
higher n-value [22]. Equation (3) has also been
shown to be a valid description of the evolution of
contact size during ball (Brinell) indentation
[29, 30]. The area coecient, c, was also determined
in these studies from ®nite element analysis of a
blunting contact for a range of stress exponents
[22].
During consolidation, a relation between the contact semi-width (a) and the blunting displacement
(h) (i.e. equation (3) is needed when balancing
applied forces with internal (interparticle) forces (so
satisfying equilibrium) since it allows the contact
197
force to be related to the average contact stress
(sc=L/2a). Rearrangement of (3) gives c = a/
(Z2rh), which physically, de®nes c as the ratio of
the actual contact radius to the nominal contact
radius, where ``nominal'' refers to the radius of a
section through a hemispherical indenter that is
parallel to the plane of the contact, but located a
distance h above the point of initial contact (see
Fig. 2). The condition, c >1, indicates that
deformed material is piling up at the periphery of
the contact. Piling-up is favored by materials having
a high stress exponent since deformation tends to
become concentrated near the contact. When c <1,
the actual contact width is less than the nominal
width, a condition referred to as ``sinking in'' [e.g.
22].
Contact relationships of the form given by
equations (2) and (3) for the power-law creep blunting of hemispherical asperities, have been shown to
be valid for indentation and blunting of linear elastic materials (where n = 1), perfectly plastic and
strain hardening plastic solids (n 4 1) and powerlaw creeping materials [28, 29, 22]. We hypothesize
that they are also applicable to the power-law
creeping contact of alloy coated ®bers, with the additional requirement that the ¯ow and area coecients will be functions of the ®ber volume fraction,
i.e. F(n,nF, a/r) and c (n, nF). In that case,
!1=n
sc
a h_
ˆ F n; F ;
;
…4†
s0
r a_e0
hˆ
2
1
a
;
2c…n; F †2 r
…5†
and the area and ¯ow coecients, c and F, can be
determined from ®nite element analysis of an
appropriate unit cell, e.g. that in Fig. 2 for a square
packing.
3. NUMERICAL COMPUTATION
3.1. Constitutive model
The creep deformation of a representative coated
®ber contact (see Fig. 2) was analyzed using the
®nite element method. The ceramic ®ber was
allowed to deform elastically, while the matrix was
treated as elastic-power-law creeping. Continuity of
displacements was imposed at the ®ber/matrix interface. Elastic deformations were calculated assuming
isotropic, linear elastic behavior with elastic property data for the matrix and ®ber as given in
Table 1. The matrix creep strain rate components,
Eij, were calculated from the multiaxial power-law
creep relation
nÿ1
3
se
Sij
e_ ij ˆ e_ 0
;
…6†
2
s0
s0
where Sij (=sij-skkdij/3) are the deviatoric stress
198
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
Table 1. Elastic properties of composite constituents
Constituent
Matrix
Fiber
Young's modulus
(GPa)
110
450
Poisson's ratio
0.324
0.3
components, se(=3SijSij/2) is the von Mises equivalent stress and s0 and E0 are the reference stress
and strain rate, respectively. Since the area and ¯ow
coecients, F (n, nF, a/r) and c (n, nF), are independent of the reference stress and strain rate, these
were assigned (arbitrary) values of 10.0 MPa and
0.01 sÿ1, respectively. Calculations were performed
for ®ber volume fractions, nF=0, 0.25, and 0.5 and
stress exponents, n = 1, 2, 5, and 10.
3.2. Square array
FEM calculations were performed using the
ABAQUSTM code under plane strain conditions.
Due to symmetry, the deformation at the contact of
two alloy coated ®bers in a square array was analyzed by considering a single quadrant of coated
®ber (Fig. 2). The ®nite element mesh used consisted of plane strain, second order (8-noded rectangular), reduced integration elements. The mesh
was developed using a mesh re®nement scheme that
enforced multi-point constraints; elements farthest
from the contact were relatively coarse and become
increasingly re®ned as the region of contact was
approached. For the homogeneous case, 521 elements were used; for a ®ber volume fraction,
nF=0.25, 480 elements were used to represent the
matrix with 41 elements for the ®ber; for nF=0.5,
384 matrix elements and 137 ®ber elements were
used. The total number of nodes in all cases was
2726. Second order, plane interface elements were
used at contact interfaces (treated as frictionless).
Elements along the vertical and horizontal symmetry planes of the unit cell were subjected to
displacement boundary conditions in which elements are allowed to slide along, but not penetrate, the boundary. Two contact interfaces were
de®ned as rigid ¯at surfaces perpendicular to the
vertical and horizontal symmetry boundaries.
Contact interface elements were used to treat the
conformation of the cylinder's surface with the contacts during blunting. The contact boundaries,
which are also symmetry planes, were again treated
as frictionless.
Contact blunting was e€ected by imposing a concentrated load, L, acting vertically, at the center of
the alloy coated ®ber (i.e. at the intersection of the
vertical and horizontal symmetry boundaries).
Displacements of the ®ber center, h, and the contact
semi-width, a, were then recorded at speci®c time
intervals. The average contact stress, sc, and the
blunting displacement rate, hÇ , were calculated from
this data, allowing F and c to be calculated using
equations (4) and (5).
3.3. Hexagonal array
The symmetry of the h.c.p. array subjected to
hydrostatic compression allows the analysis to be
carried out for 1/12th of a cylinder. We obtained
results for the unit cell shown in Fig. 3, i.e. for
1/6th of a typical alloy coated ®ber. This was derived by re¯ecting the solution for 1/12th cylinder
about a radial line located at 308 from the vertical.
The larger unit cell allows the proximity between
adjacent contacts to be seen more clearly. The ®nite
element mesh consisted of plane strain, second
order (8-noded rectangular), reduced integration elements (352 elements representing the alloy matrix
and 15 elements representing the ®ber). Elements
along the vertical and inclined symmetry planes of
the unit cell were subjected to displacement boundary conditions in which elements are allowed to
slide along but not penetrate the boundary. There
are two contact interfaces, de®ned as rigid, ¯at sur-
Fig. 3. Unit cell used to analyze the contact blunting behavior during isostatic compression of a hexagonal array of metal coated ®bers. Symmetry conditions allow consideration of one-half of a typical
contact (i.e. 1/12th of a cylinder).
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
199
Fig. 4. Increasing the ®ber volume fraction leads to greater
concentration of inelastic strain near the contact
(a/r = 0.33).
Fig. 5. The in¯uence of stress exponent on the distribution
of inelastic strain during metal coated ®ber contact blunting (nF=0.25): in addition to concentrations near the contact, strains also accumulate at the ®ber matrix interface.
faces perpendicular to the vertical and slanted symmetry boundaries. Contact interface elements were
again used to treat the conformation of the cylin-
der's surface with the contacts. The contact boundaries were treated as frictionless. As for the square
array unit cell, contact blunting was e€ected by
200
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
imposing a concentrated load, L, acting vertically,
at the center of the alloy coated ®ber. The average
contact stress, sc, and the blunting displacement
rate, hÇ , were calculated as described previously,
allowing F and c to be found from equations (4)
and (5).
4. RESULTS AND DISCUSSION
4.1. Square array
4.1.1. Spatial distribution of inelastic strain. The
in¯uence of the ®ber reinforcement on the distribution of (equivalent) inelastic strain can be seen in
Fig. 4(a)±(c). Results are shown for ®ber volume
fractions of 0, 0.25, and 0.5. The stress exponent is
n = 2 for all three cases, and the normalized contact size, a/r = 0.33. The presence of the ®ber leads
to increased strain concentration and moves the
peak strain from the contact's outer edge to a
region located directly above the contact centerline.
A second key region of strain concentration occurs
at the ®ber/matrix interface. The strain localizations
increase with increasing ®ber volume fraction. The
signi®cant in¯uence of the ®ber in localizing strain
is most clearly seen by comparing Fig. 4(a) for a
homogeneous cylinder with Fig. 4(c) for the nF=0.5
case; except near the contact, the alloy coating is
practically undeformed when a ®ber was present.
An example of the in¯uence of the stress exponent on the strain distribution in a coated ®ber
(nF=0.25) is shown in Fig. 5 (a/r = 0.33).
Increasing n from 1 to 5 to 10 is seen to result in
only a slight increase in the equivalent inelastic
strain. Peak strain concentrations are seen, in order
of magnitude, on the contact centerline, at the contact's outer edge, and at the ®ber/matrix interface.
4.1.2. Flow coecient, F. Figure 6 summarizes the
dependence of the ¯ow coecient, F, (as de®ned
by equation (4)) on a/r. Figure 6 (a)±(c) illustrate
F (a/r) for n = 1, 2 and 10 for the three volume
fractions, nF=0, 0.25 and 0.5. For the low-n cases
(n = 1 and 2), the ¯ow coecient increases monotonically with a/r for all ®ber volume fractions, indicating that an increasing contact stress is required
to achieve the same rate of blunting, hÇ . We note
that for the n = 1 (linear viscous) case, F becomes
very large, especially for the high ®ber volume
fraction (nF=0.5), for which F exceeds 12 at
a/r = 0.65. Other things being equal, low-n
materials with high ®ber fractions will therefore be
signi®cantly more dicult to consolidate. The very
mild di€erence between the homogeneous and the
nF=0.25 cases in Fig. 6 (a)±(c) is remarkable and
Fig. 6. Evolution of the ¯ow coecient, F, with increasing
contact strain, a/r, showing the in¯uence of stress exponent and ®ber volume fraction on the resistance to creep
contact blunting. Results for the in¯uence of n on F are
summarized for nF=0.25 in (d).
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
indicates the possibility of applying models for the
contact of homogeneous cylinders (wires) to predict
consolidation behavior of alloy coated ®bers when
volume fractions of ®ber are at or below 25%. It
also suggests that local thin regions of metal coating may be very dicult to creep consolidate for
high ®ber fraction systems.
The consistent increase in the ¯ow coecient
with increasing ®ber content [Fig. 6(a)±(c)], is not
surprising; the strengthening e€ect is attributable to
the decreased volume of coating material available
to ®ll the given void volume, the increased ®ber/
matrix interfacial area (recall it is modeled as perfectly bonded) and the high elastic sti€ness of the
®ber itself. Thus, while the onset of ¯ow may
indeed occur at a lower contact stress when a ®ber
is present (as suggested by Davison et al. [33]), the
pressure required to signi®cantly consolidate alloy
coated ®bers will never be less than that for unreinforced alloy wires, and for low n-values could be
several times higher.
As n increases to 10 [Fig. 6 (c)], the dependence
of F on a/r becomes more complicated; at ®rst F
increases with continued blunting, followed by softening, then another cycle of mild hardening/softening, followed ®nally by abrupt hardening. The peak
¯ow coecient (prior to the ®rst softening) is seen
to be about 1.95 for the homogeneous case, 2.1 for
the 25 vol.% ®ber case and 2.5 for a 50 vol.% reinforced case. Examination of strain contour data
indicates that the initial hardening (i.e. increase in
contact stress needed to maintain a constant rate of
blunting) is caused by the rapid drop in strain of
material surrounding the zones of highest strain
rate. This is a direct consequence of the steep
power-law relation between stress and strain rate
for high-n materials. Because the stress, and hence
strain rate, concentrations occur initially within the
coating's interior, as these zones expand, they eventually reach an outer surface of the coating, and
constraint is quickly lost (recall the contact surface
is frictionless and the free surface is traction-free).
Additionally, the volume of constraining material
diminishes as the high rate deforming zone expands.
Thus, a peak in the ¯ow coecient is observed, followed by softening as constraint is lost. We note
that eventually, the matrix material will ®ll up the
available void volume in the laterally constrained
cell as the relative density approaches 1. Since the
matrix material is incompressible, the ¯ow coecient must then increase without bound.
From a practical viewpoint, if the results for
n = 10 are compared with those for n = 1 and 2
using the same scale for the ordinate [e.g. Fig. 6
(d)], the n = 10 softening/hardening behavior on
the interval 0.2Ra/r R0.6 appears very minor.
High-n materials, for which the deformations are
most highly concentrated in the region of the contact, therefore exhibit an almost deformation-independent contact ¯ow stress. F can then be
approximated to be roughly independent of a/r.
This arises because the highly deformed zone
always remains small relative to the total volume of
matrix, and is therefore most like the situation
encountered in indentation (for which F isusually
independent of indentation depth). For high-n materials that approach the perfectly plastic limit, to a
fair approximation F depends only on the volume
fraction of ®ber, and is given by
F…F † ˆ 2:05 ‡ 1:4F :
nF
1
0.0
0.25
0.5
0.0
0.25
0.5
0.0
0.25
0.5
0.0
0.25
0.5
2
5
10
…7†
Figure 6(d) shows the in¯uence of the stress exponent on F for a ®xed ®ber volume fraction of 0.25.
It is seen [as in Fig. 6 (a) and (b)] that no softening
occurs for the low-n materials (i.e. n <5). This
occurs because when n is low, the stress and strain
rate ®elds are much more uniformly distributed,
and so the e€ect of diminishing material constraint
is reduced. However, the lateral constraint, due to
the presence of neighboring ®ber contacts, is present
for all cases, and therefore F again tends to in®nity
as a/r 4 1. While the ¯ow coecient is greatest for
high-n matrices for a/r <0.45, the rapid hardening
exhibited by the low-n matrices causes them to o€er
the greatest resistance to ¯ow at high a/r, and
causes the ¯ow coecients to cross in Fig. 6(d).
This phenomenon was also observed during the
creep blunting of spherical contacts [22].
In general, the ¯ow coecient depends on a/r, nF
and n. Table 2 summarizes the a/r-dependence of
the ¯ow coecient for various n and nF cases analyzed. Inserting any one of these expressions for F
into equation (4) provides a closed-form solution
Table 2. Numerical ®ts for the ¯ow coecient, F (square array)
n
201
F (a/r)
0.54 ÿ 1.14(a/r) + 8.16(a/r)2
0.56 + 0.71(a/r) + 7.04(a/r)2
1.84 ÿ 3.49(a/r) + 30.0(a/r)2
1.07 + 0.50(a/r) + 2.51(a/r)2
0.75 + 1.15(a/r) + 3.07(a/r)2
2.23 + 1.51(a/r) + 4.18(a/r)2
0.74 + 10.68(a/r) ÿ 32.78(a/r)2+32.65(a/r)3
0.74 + 10.68(a/r) ÿ 32.78(a/r)2+32.65(a/r)3ÿ32.78(a/r)4+32.65(a/r)5
0.55 + 24.32(a/r) ÿ 108.8(a/r)2+238.0(a/r)3ÿ249.9(a/r)4+100.7(a/r)5
ÿ2.58 + 81.69(a/r) ÿ 506.7(a/r)2+1427(a/r)3ÿ1874(a/r)4+936.6(a/r)5
ÿ4.19 + 111.2(a/r) ÿ 680.8(a/r)2+1895(a/r)3ÿ2459(a/r)4+1208(a/r)5
ÿ4.56 + 115.0(a/r) ÿ 654.9(a/r)2+1706(a/r)3ÿ2078(a/r)4+960.5(a/r)5
202
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
Table 3. Numerical solution for c (square array)
n
Fig. 7. The contact area coecient, c, as a function of
creep stress exponent: increasing n or ®ber volume fraction
has the e€ect of concentrating strain near the contact,
causing an increase in c; the stress exponent has little e€ect
on c when ®ber is present.
for the contact stress±blunting displacement rate of
contacting alloy coated ®bers.
4.1.3. Area coecient, c. Finite element results for
the evolution of contact size during blunting were
used with equation (5) to compute the area coecient, c. It was con®rmed that to a good approximation, c is independent of a/r. The results revealed
only a mild in¯uence of ®ber volume fraction on
the evolution of contact size. As might be expected,
increasing the volume fraction of ®ber results in larger contact size for a given blunting displacement.
This is consistent with the observation that increasing the ®ber volume fraction tends to concentrate
deformation more highly near the contact. As noted
above for the ¯ow coecient, F, the area coecient
is also only slightly a€ected by an increase in ®ber
volume fraction from 0 to 25%. Thus, models for
the contact deformation of homogeneous cylinders
may adequately predict the area coecient behavior
of alloy coated ®bers for ®ber volume fractions up
to or slightly greater than 25%.
Varying the stress exponent for a ®xed ®ber fraction was found to have little in¯uence on the evolution of contact size. At any given blunting
displacement (ordinate), the contact size increases
only slightly with n. As shown in Fig. 7, where the
area coecient is plotted as a function of 1/n, c is a
much stronger function of n for the homogeneous
case. This ®gure shows clearly that the introduction
of the ®ber suppresses the n-dependence of the area
coecient. It is a consequence of the ®ber's tendency to concentrate deformation near the contact
(note the di€erence between the nF=0.25 and 0.5
cases in Fig. 7), regardless of the stress exponent.
From a practical viewpoint, Fig. 7 indicates that for
low ®ber fractions, the dependence of c on n can
also be neglected in the presence of the ®ber reinforcement, and the results for a 25 vol.% ®ber
nF
1
2
5
10
0.0
0.25
0.5
1.41
1.22
1.40
1.20
1.24
1.40
1.23
1.24
1.40
1.31
1.24
1.40
provide a good approximation to the homogeneous
case (or vice versa).
Table 3 summarizes results obtained for the area
coecient as a function of stress exponent and ®ber
volume fraction. These results, in combination with
equation (5), provide a reasonably accurate, closedform solution for the contact size±blunting displacement relation for contact deformation of alloy
coated ®bers by power-law creep.
4.2. Hexagonal close packed array
4.2.1. Flow coecient, F. Finite element results
were obtained for stress exponents of n = 1, 2, 5
and 10 and for a ®xed ®ber volume fraction of
0.25. It was found that strain concentrations at the
®ber/matrix interface are much less evident for an
h.c.p. structure subjected to isostatic pressing, in
agreement with the observations of Schuler et al.
[35]. Results for the ¯ow coecient for the h.c.p.
packing are shown in Fig. 8 (a)±(d) and compared
with results for the uniaxial compaction of a square
array (nF=0.25). It can be seen that the results for
the square array extend to higher a/r-values than
for the h.c.p. structure. This is because at any given
value of a/r, the relative densities of the square and
h.c.p. arrays are di€erent; the h.c.p. structure has a
starting density of 0.92 (a/r = 0) and approaches
full density as a/r 4 0.55, while the square array
has a starting density of 0.78 (a/r = 0) and full density corresponds to a/r = 1. It can be seen that the
¯ow coecient is lower initially for the h.c.p. packing (for all stress exponents), indicating the in¯uence of interaction between contacts in close
proximity. As noted previously, this softening e€ect
due to interaction of contact deformation zones has
been observed by Ogbonna and Fleck [25] using
slip-line theory to investigate perfectly plastic contact deformation. It is viewed here as a manifestation of elastic (or more slowly deforming
material) constraint loss. However, as blunting proceeds, and a/r increases, the ¯ow coecient rises
rapidly for the h.c.p. case. This is a consequence of
increased lateral constraint imposed by neighboring
contacts, and requirement for material to ``extrude''
into the vacant interstices of the hexagonal packing.
The in¯uence of the stress exponent on F for the
h.c.p. case is similar to that for the square array; at
low a/r, F increases with increasing n due to
increased localization of deformation when n is
high (leading to more e€ectively constrained ¯ow).
At high a/r on the other hand, F is greater for low-
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
203
Table 4. Numerical solutions for the ¯ow coecient, F (h.c.p.
array)
n
F (a/r)
Range
2
1
2
5
10
0.25 ÿ 4.46(a/r) + 94.5(a/r) ÿ437.7(a/
r)3+788.8(a/r)4
0.03< a/r <0.385
0.45 + 1.53(a/r) + 22.2(a/r)2ÿ107.6(a/
3
4
r) +192.2(a/r)
0.05< a/r <0.397
1.13 ÿ 6.15(a/r) + 71.1(a/r)2ÿ257.0(a/
3
4
r) +315.6(a/r)
0.05< a/r <0.425
ÿ0.48 + 29.5(a/r) ÿ 178.5(a/
2
3
4
r) +446.4(a/r) ÿ385.2(a/r)
0.086< a/r <0.417
n materials because the more homogeneous deformation enhances the e€ect of lateral constraint.
Figure 8 shows that for low n-values, ¯ow is
more dicult for an h.c.p. array. However, with
increasing n, the h.c.p. packing's contact ¯ow coecient becomes less than that of a square array . For
high-n materials, it may therefore be concluded that
more closely packed structures will bene®t from interaction softening and the applied pressure per contact will be lower during consolidation. However, it
must be kept in mind that the overall pressure
required for consolidation depends not only on the
force per contact (as determined by F), but also on
the number of contacts per particle, which is higher
for the h.c.p. case. Approximate expressions for F
(a/r) are given in Table 4 for creep stress exponents
of 1, 2, 5 and 10, and a ®ber volume fraction of
nF=0.25.
4.2.2. Area coecient, c. The FEM results for the
area coecient indicate that the contact size grows
more rapidly with blunting displacement for the
h.c.p. case as compared with the square array. This
is re¯ected in Fig. 7, which includes c (n) for the
h.c.p. case (nF=0.25). Clearly, the area coecient
depends strongly on the coordination number,
increasing from 1.23 for the uniaxial compaction of
a square array (Z =4) to 1.62 for the h.c.p. case
(Z = 6).
5. APPLICATION TO CONSOLIDATION
MODELING
The contact blunting model above can be used to
predict the time-dependent, overall density, D (t), of
a random close packing of aligned metal coated
®bers subjected to an external hydrostatic pressure,
p. The contact coecients, F and c, provide a
simple, accurate description of the behavior of a
single blunting contact, as described by equations
(4) and (5). Inserting appropriate values for F and c
with the help of Tables 2±4, the rate at which the
Fig. 8. Comparison of the ¯ow coecient, F, for square
and h.c.p. packing for stress exponents of 1, 2, 5 and 10:
initially, the h.c.p. array densi®es more easily (contact interaction softening), but rapidly hardens as full density is
approached (corresponding to a/r = 0.55 for a hexagonal
array).
204
ELZEY et al.:
INELASTIC CONTACT DEFORMATION
centers of two metal coated ®bers approach for a
given force normal to the contact, as well as the
evolution of contact area, are given by these relationships. With the behavior of a single representative contact known, it is then only necessary to
relate the stress and displacement for a single contact to the applied pressure and overall relative density. The ®nal result is a pressure±density relation
for a bundle of coated ®bers of the form
p
D
a a
ˆ
F n; F ;
n;
s0 2prl
r
r
p
… D p r
D 1=n
;
…8†
DdD
…1 ÿ D†
Z0 D ‡ Z 0
a
e_ 0 D
D0
where D = (1 ÿ D0/D), l is the length of the coated
®bers, and Z0 and Z' are constants specifying the
average number of interparticle contacts per ®ber as
a function of density. Equation (8) can then be integrated to ®nd D(t) given the tabulated values for F
and c obtained from this analysis.
6. SUMMARY
Models for predicting the densi®cation behavior
of metal alloy coated ceramic ®bers rely on accurate
predictive relationships for the deformation response of representative interparticle contacts. The
present study has developed closed-form solutions
for the contact stress±displacement rate and displacement±contact size relationships for a single alloy
coated ®ber contact. Since interaction with neighboring contacts strongly in¯uences contact behavior, two packing geometries/loadings have been
considered: constrained uniaxial compression of a
square array and isostatic compression of a hexagonal close packed array. These two cases approximate the contact interaction at low and high
relative densities, respectively. Closed-form solutions for the ®ber-reinforced contact problem are
proposed, based on the extension of existing models
for inelastic blunting of homogeneous contacts.
Finite element analyses were used to evaluate the
unknown (¯ow and area) coecients, F (n, nF, a/r)
and c (n, nF), in these expressions.
In addition to the closed-form solutions presented
in the paper, the analysis provides signi®cant physical insight into the mechanisms of ¯ow during the
contact blunting of metal coated ®bers. The inclusion of a (elastic) ceramic ®ber enhances the resistance to contact blunting: the strengthening e€ect
is mild for ®ber volume fractions up to 25%, so
that existing models for the contact deformation of
homogeneous alloy wires may be used to approximate the consolidation behavior of metal coated
®bers within this range of ®ber content. The ®ber
strengthening e€ect increases nonlinearly for
volume fractions beyond 0.25 and must be
accounted for if accurate predictions of contact
pressures are to be obtained. The addition of ®ber
also enhances the growth rate of the contact as two
metal coated ®bers approach during contact blunting. This is a result of the tendency of the ®ber
(and its interface with the matrix) to concentrate
stress (and therefore deformation) in the vicinity of
the contact. Finally, it was found that initially, contact blunting occurs more easily (lower ¯ow coecient) when metal coated ®bers are packed to
higher relative densities, so that contacts interact
more extensively with their neighbors.
AcknowledgementsÐThe interpretation of this work has
bene®tted from helpful discussions with Professor N.
Fleck. The authors would like to express their appreciation to the Defense Advanced Research Projects Agency
(Dr Anna Tsao, Program Manager) for support through a
contract with Integrated Systems, Inc., Santa Clara, CA.
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