Pergamon
0956-7151(95)00279-O
Acra mater. Vol. 44, No. 4, pp. 1479-1495, 1996
Elsevier Science Ltd
Copyright 0 1996 Acta Metallurgica Inc.
Printed in Great Britain. All rights reserved
1359-6454/96 $15.00 + 0.00
POWER-LAW CREEP BLUNTING OF CONTACTS AND ITS
IMPLICATIONS FOR CONSOLIDATION MODELING
R. GAMPALA,
D. M. ELZEY and H. N. G. WADLEY
School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 27 February
1995; in revised form 26 June 1995)
Abstract-Densification
occurs primarily by power-law creep of interparticle contacts during the initial
stages of the elevated temperature consolidation of metallic powders and metal matrix composite (MMC)
preforms. Past approaches to its modeling have relied upon results from indentation analyses. Here,
contact deformation is viewed as a blunting process and closed-form solutions for the contact stressdisplacement rate and the contact areastrain relationships are proposed for laterally constrained contact
blunting by power-law creep. These solutions contain unknown coefficients which are evaluated using the
finite element method. The contact stress-displacement rate relationship during blunting is found to be
controlled by the degree of constraint imposed on the flow of material near the contact: initially, the loss
of internal (material) constraint dominates, leading to a strain softening; this is followed by hardening
in which the influence of externally imposed lateral constraint (due to the presence of neighboring contacts)
becomes dominant. Strain softening decreases in importance and lateral constraint hardening occurs
earlier as the creep stress exponent decreases. The blunting results are used to revise densification and fiber
microbending/fracture models. It is shown that, depending on the creep stress exponent, past indentationbased models can substantially over- or underestimate the true rate of densification.
1. INTRODUCTION
Many of today’s high performance
structural
materials (e.g. high strength tool steels [l], titanium and
nickel-base
alloys [2, 31, refractory
metals [4], ceramics [5] and metal/intermetallic
matrix composites
[6,7]) are manufactured
to near-net shape in a twostep process. First, monolithic powders [8], (tape cast)
powder/ceramic
fiber preforms [9] or spray-deposited
monotapes
containing
unidirectional
fibrous
reinforcements
[lo] are produced
(Fig. 1). These are
then encapsulated,
evacuated and consolidated
to full
density using processes such as hot isostatic or vacuum hot pressing (HIP/VHP) [ 1 I]. During HIP/VHP,
a combination
of high pressure and temperature
is
used to consolidate
the initially porous preforms by
inelastic (plasticity/creep)
flow. In the case of fiber
reinforced composites,
this must be done while also
avoiding damage to the fibers [12]. Numerous efforts
have been made to understand
and model these
processes
[e.g. 13-161 and to use this insight to
optimize the consolidation
process [17].
The development
of these models began by recognizing that at the start of the densification process the
externally applied stresses are internally supported at
powder particle (or surface asperity) contacts. During
consolidation,
high stresses rapidly develop at these
contacts and the resulting contact deformation
displaces material from the contact into adjacent (interparticle) voids. When the relative density is low
(typically less than about 90% of the theoretical
density), classical results from contact mechanics
have been used to estimate the contact stress required
for inelastic flow [13, 181 and simple “uniform redistribution”
models used for estimating
the contact
area [19]. Since contact deformation
occurs by a
combination
of elastic, plastic and creep mechanisms,
contact mechanics solutions for each mechanism are
used in the models. These fundamental
results are
then used to develop predictive models describing the
relationships
between
densification/fiber
damage,
process conditions (temperature,
pressure and time),
the preform’s internal (contact) geometry and material properties.
The relative contribution
of each densification
mechanism
depends upon temperature.
When the
process temperature
is low (T < 0.4T,, where T,,, is
the melting temperature),
inelastic deformation
at the
contacts of metals and their alloys occurs predominantly by plasticity and the most important material
property
is the (temperature
dependent)
yield
strength. At higher temperatures,
dislocation and/or
diffusional creep are dominant and in this case the
power-law creep exponent and the activation energy
are the key material properties [20]. In earlier work,
this contact deformation
has conventionally
been
treated as an indentation
process, and contact mechanics results from analyses of perfectly plastic or
creep indentation
have been used to develop densification and fiber damage models.
Wilkinson and Ashby [21] were the first to propose
a creep contact model for analyzing the densification
1479
1480
GAMPALA
et al.:
CREEP BLUNTING OF CONTACTS
a) Monolithic
metal/alloy powders
b) Tape cast
cotiposite slurries
C) Spray-deposited
composite monotapes
Fig. 1. Micromechanics-based models for predicting the
overall densification response of powders and composite
monotapes (e.g. spray-deposited or tapecast preforms) rely
on accurate analyses of the deformation behavior at representative interparticle contacts.
of powders by hot isostatic press9g. Their approach
was based on a treatment
of creep indentation
by
Marsh [22] and Johnson
[23], who observed that
beneath a blunt indenter,
material was displaced
more or less radially from the point of first contact,
and so could be analyzed as the expansion of a cavity.
Fischmeister
and Arzt [24] subsequently
derived a
model for interparticle contact deformation
by likening the contact to a flat punch indenting a power-law
creeping half-space. Helle et al. [25] applied this to the
prediction
of hot isostatic pressing diagrams
for
powder consolidation.
These diagrams
have subsequently been widely used to analyze and optimize
conditions
for the densification
of powders under
hydrostatic pressure [5,26,27]. More recently, Kuhn
and McMeeking
[28] have extended Helle et al.‘s
result for powders to predict densification
under
arbitrary stress states and therefore their model also
relies on indentation
analysis to predict the contact
deformation
behavior. Likewise, indentation
results
have been used in analyzing
the deformation
of
asperity contacts (between spray-deposited
composite
monotapes)
for modeling MMC densification
kinetics and fiber microbending/fracture
[15, 161.
The use of indentation
theory for predicting the
response of contacts in these problems is an overly
simplistic approximation
of the actual physical process encountered
during contact deformation.
For
instance, the deformation
field remains self-similar
during indentation
[i.e. the stress and strain (rate)
fields change little with continued
indenter
penetration once fully inelastic deformation
has been
established]. This leads to simple indentation
depthindependent
relationships
between the average contact stress (acting over the face of the indenter) and
the rate of penetration.
However, the deformation
during contact blunting occurs in a finite-size body,
and increasingly
significant interactions
occur with
the body’s boundaries as deformation
proceeds, Fig.
2. When the contact strain is very small, the size effect
is likely to be negligible and the results of indentation
may be sufficient for modeling.
However,
during
further densification
by time-independent
plasticity,
large contact strains are developed
and the strain
field-asperity
surface interaction has been shown to
lead to a contact strain-dependent
distribution
of
stress and strain in the contact and a resulting
displacement-dependent
average contact stress [29].
The contact flow stress in this situation has been
shown to drop rapidly from an initial value of 2.97~~
(where oY is the uniaxial yield stress) towards unity
because of the loss of “material constraint”.
Thus,
indentation-based
models for densification only estimate the actual consolidation
behavior and may be in
error, particularly as the relative density increases (i.e.
when the contact strains are large).
A second limitation of the indentation
approach is
its inability to account for the strong influence of
neighboring contacts. Again, when the contact strain
is very small, surrounding contacts do not perturb the
flow criterion,
but as densification
proceeds,
additional
lateral (neighboring)
contacts
are established, material flow at the primary contact extends
through the asperity or particle and becomes more
constrained,
requiring higher contact stresses to sustain deformation.
Indentation-based
models do not
incorporate
the effect of lateral constraint
upon a
contact’s deformation.
Recent models for the blunting of hemispherical
contacts by plasticity have accounted for the imposed lateral constraint
(see for
example Gampala et al. [29], who used finite element
techniques
or Akisanya and Cocks [30] who used
slip-line field theory and finite element analysis).
GAMPALA et al.:
CREEP BLUNTING
Plastic indentation
of a half-space
OF CONTACTS
1481
Plastic blunting
of a contact
Fig. 2. Indentation theory has often been used to predict the blunting of contacts, even though indentation
and blunting are different. The most significant differences arise from the finite size of the deforming body
during blunting.
These analyses show an initial decrease in flow stress
(due to a loss of material constraint)
with contact
strain followed by a strong increase in flow stress due
to the imposed lateral constraint. No similar analysis
for the high temperature
(i.e. creep) response
of
deforming contacts is available.
A third limitation of the indentation approach is its
inability to predict contact area evolution. Uniform
redistribution
techniques have been used to estimate
the contact area evolution with density [24]. However, these ignore the “piling-up” of a matrix near the
contact and are expected to give inaccurate predictions of contact area evolution. Since the product of
the contact area and contact flow stress is used in the
models to balance the applied consolidation
force,
errors in the contact area evolution are as important
as the flow stress relationship.
Our objective is to analyze the contact mechanics
of blunting for a representative
non-linear, power-law
creeping hemispherical
contact with laterally imposed
constraint.
We consider
materials
whose uniaxial
stress (atstrain
rate (6) relationship can be described
by a power-law relation
g=go
0
0
-
”
00
where (r. and .6,,are the reference stress and strain rate,
respectively
and n is the stress exponent.
The two
results of greatest practical interest for consolidation
modeling are: (1) the relationship
between the average stress acting normal to a contact and the rate of
contact deformation;
and (2) the relation between the
contact area and the amount of blunting. Obtaining
the appropriate
form for these two relationships
when the matrix is a non-linear
creeping material
plays a prominent
role in the development
of a
blunting theory.
Because the analysis of indentation
experiments
allows mechanical
properties,
such as the elastic
modulus, plastic yield strength and creep exponent,
to be inferred from hardness measurements
[31], the
analysis of indentation
has received significantly
more attention than that of blunting. This has motivated many efforts to analyze indentation
and has led
analytical
and
numerical)
flow
to (empirical,
stressdisplacement
rate and displacement-contact
area relationships
for a wide variety of material
constitutive
behaviors and indenter geometries [32].
Blunting has received less attention. It is also a more
difficult process to analyze because the size of the
deformation
field can be comparable
to that of the
asperity and is therefore significantly influenced by
the proximity and shape of the asperity’s free surfaces.
It will be shown (in Section 2.1) that the form of
contact flow stress relationship
for the blunting of a
power-law creeping material is expected to be
where oE is the average contact stress, h/ago is the
effective strain rate (in which h is the rate of blunting
displacement
and a is the contact radius, Fig. 2), F is
a material non-linearity
(i.e. stress exponent, n) dependent
coefficient and a/r is the contact radius
normalized by the radius of the undeformed
contact.
The analysis of both indentation
and blunting
requires knowledge of the evolution of contact area
with displacement.
It is well established [33] that, for
rate-independent
materials,
the relationship
during
indentation
depends on the work-hardening
rate (i.e.
the hardening exponent), and on the stress exponent
in rate dependent materials described by equation (1).
It will be shown (in Section 2.2) that during the
GAMPALA
1482
blunting of a power-law creeping
radius-displacement
relationship
et al.:
CREEP
contact, the contact
can be described by
-a2
h_!-
0
2c(n)’
r
where c is a material non-linearity
dependent
coefficient.
Together, equations (2) and (3) describe blunting of
power-law creeping contacts in a form that can be
readily incorporated
into existing densification
and
fiber microbending/fracture
models. The unknown
coefficients,
F(n, a/r) and c(n), are determined
by
means of finite element analysis (Sections 3 and 4)
and the resulting solution applied to the consolidation of monolithic
powder and composite preforms
(Section 5).
2. CONTACT
DEFORMATION
2.1. Contact stress-effective
MECHANICS
strain (rate) relations
When the contact stress between a pair of elastic-plastic
bodies is small (typically less than about
1. la,), both indentation and blunting occur by elastic
deformation.
Hertz [34] showed that the average
contact stress, (a,), for an elastically compressed
hemisphere
displaced a distance, h, against a rigid
surface is given by
8
E
-._
Q’c=j&-v2
h
a
where E and v are the Young’s modulus
and
Poisson’s ratio of the hemisphere, and a is the contact
radius. Equation (4) is a linear relation between the
average contact stress and a measure of the effective
strain, h/a. Analogous measures of the effective strain
rate during time-dependent
contact deformation have
also been proposed {for example, Sargent and Ashby
[35] proposed the quantity /ii(&),
where h is the
displacement
rate and A the contact area}. The Hertz
solution (4) applies equally to the complementary
problem of indentation
of an elastic half-space by a
rigid, spherical indenter of radius, r. The contact
radius then refers to the projected area of the contact
and h to the depth of indentation,
Fig. 2. Elastic
behavior
is confined
to small displacements
(i.e.
deformations
a/r < 0.01) and in this regime, blunting
and indentation have identical contact stress-effective
strain behaviors.
For inelastically deforming media, analytical solutions equivalent to (4) are not available. Although
the blunting and indentation
problems may both be
expressed as exact boundary value problems (at least
tThis relationship was obtained by applying conservation
of
volume to an incompressible,
solid hemisphere of radius,
r, compressed uniaxially within a rigid cylindrical die of
constant
radius, r, and initial height, r. The initial
relative density is thus D, = 2/3. Additionally,
the contact radius was related to the contact displacement,
h, by
h = 0.38az/r [see equation (18)].
BLUNTING
OF CONTACTS
for infinitesimal deformations),
the complexity of the
inelastic constitutive relations for plasticity and creep
preclude analytical treatment. Approximate
methods
such as plane strain indentation
or the blunting of
perfectly plastic materials have been performed using
slip-line field analysis. Solutions of this type result in
simple contact plastic flow criteria of the form
Q, = Fo,
(5)
where the flow coefficient, F, is a constant (independent of h/a) and by is the uniaxial yield strength.
Examples are Prandtl’s solution for indentation with
a flat punch [36] (F = 2.97) and Ishlinsky’s result [37]
for a spherical indenter (F = 2.66). The densification
models of Helle et al. [25], Kuhn and McMeeking [28]
and Elzey and Wadley [15] all assume F is a constant
coefficient equal to 3. Treating F as a constant (i.e.
assuming the contact stress to cause flow by plasticity
is a constant multiple of the uniaxial flow stress)
implies self-similarity
of the deformation,
i.e. the
conditions
necessary
for plastic yielding do not
change with continued deformation.
While this has
been found to be approximately
valid for indentation,
it is not a realistic approximation
for blunting bccause of the significant redistributions
of stresses and
strains that are likely to occur with increasing contact
displacement,
h.
Gampala et al. [29] have used finite element analysis to investigate the blunting of hemispherical
asperities. Their results indicated that it was possible to
retain the convenient form of the contact flow stress
relation given by equation (5) if F is allowed to be a
function of the relative density. Their results for the
case of an elastic, perfectly plastic hemisphere
subjected to constrained
uniaxial compression
could be
fitted by a relationship
of the form
F(D) = 340’ - 580 + 26
where the relative
normalized contact
D
density, D, was related
radius, a/r, byt
=[;-$>‘I]‘.
(6)
to the
(7)
Equation (6) indicates F is a strong, non-monotonic
function of effective strain. For an initial relative
density of 213, F has an initial value of about 2.4 and
then decreases with strain before exhibiting a rapid
hardening.
The softening arises because at first the
plastic zone is able to more easily expand against a
diminishing
volume of surrounding
elastically deformed material as the effective strain increases-a
consequence
of the non-self-similar
deformation
characteristic
of the blunting process. Subsequent
hardening arises from the lateral constraint imposed
by adjacent contacts. Analyses with a work hardening
constitutive
law have shown that work hardening
compensates
for the softening
phenomenon,
and
under some conditions,
results in a (fortuitous)
strain-independent
F coefficient with an average value
close to 3 [29].
GAMPALA
et al.:
CREEP BLUNTING
The contact (indentation
and blunting) deformation of work-hardening materials has also been
investigated by Matthews [38]. His analysis began by
assuming that the form of the contact stress solution
for the linear elastic problem [equation (4)] would
also be valid for power-law hardening plastic
materials, i.e. those for which the uniaxial stressstrain response is given by
&= (a/&$.
(8)
By matching the solution to the linear elastic and
perfectly plastic (slip-line) solutions in the limits of
material non-linearity, he obtained an expression for
the contact stress of the form
(9)
Matthews’ relationship (9) is strictly valid only for
indentation (since it was matched to indentation
results), but he proposed its use for predicting the
blunting of a work-hardening sphere as well.
The form of solution (9) agrees with empirical
observations for indentation: for example, Meyer’s
indentation experiments [39] showed that the average
contact stress could be expressed as gc = k(a/2r)““,
where k and n are material constants. Much later, the
experimental work of O’Neill [33] and Tabor [40]
demonstrated that n is just the power-law hardening
Table
Material
behavior
Theory
Elastic
Hertz [34]
Model
basis’
Slip-line
field [37]
I, B
1. Summary
Normalized
contact
stress
2 (1 -
I
“c
02)
OF CONTACTS
1483
exponent in equation (8) and k is related to the
hardening coefficient, cr,,in (8) by k = F(n)q,, with F
dependent only on the material non-linearity.
Combining Meyer’s result for a, with that of
O’Neill and Tabor, it is possible to write the contact
flow stress-effective strain relation in the form
where the empirical results of Norbury and Samuel
[41] have been used to relate the indentation depth,
h, to the contact radius, a [see equation (19) of Section
2.2 below]. While analyzing the case of a spherical
indenter penetrating a work hardening plastic solid,
Hill et al. [42] were able to reproduce Meyer’s law as
well as Tabor’s results, thus validating equation (10)
as the correct form of the contact stress-effective
strain relationship for plastic indentation. Their numerical computations were valid for deformations
upto a/r = 0.8. We see that equation (10) is a generalization of equations (4), (5) and (9) for the indentation of elastic (i.e. n = l), perfectly plastic (n = co)
and power-law hardening materials, respectively. Expressions for the form of the flow coefficient, F(n), for
each case are summarized in Table 1. Equation (10)
provides a valid description of blunting only when the
effective strain is small (u/r < 0.1); to be valid for
large deformation blunting, it would need to be
of contact
mechanics
Effective
strain
(rate)
models
Flow coefficient
Fb
h
_
a
8
3n
1
2.66
Area
coefficient,
cc
1
z 0.7
Js
%
Perfectly
plastic
Gampala
B
1.15
et al. [29]
Plastic/work
Matthews
h
hardening
[381
a
Linear
Lee and
viscous
Radok
6n
16 I/”
2n + 1 0 9n
1431
Elzey/Wadley
V51
Power-law
creep
Matthews
[381
Bower
et al. [45]
aI = Indentation,
bF = normalized
cc = a@z.
B = Blunting.
contact stress/effective
Tabulated
strain (rate)““.
F(n)
Tabulated
c(n)
GAMPALA et al.: CREEP BLUNTING OF CONTACTS
1484
modified by writing F as a function of both n and the
effective strain, a/r [e.g. as represented by equation
(6) for the perfectly plastic case].
Equation (10) is valid for rate-dependent material
behavior as well, but with the effective strain, h/a
replaced by the effective strain rate, h/a& and n now
representing the stress exponent in Norton’s powerlaw creep relation (1). The simplest case to analyze is
that of a linear viscous material [i.e. n = 1 in equation
(l)]. Lee and Radok [43] showed that for this situation, the indentation contact flow stress-effective
strain rate relationship has the form
where et, and &,are the reference stress and strain rate,
as in equation (1). This is just the linear viscous
analogue of the elastic result given by equation (4).
By ensuring that the linear viscous solution was
recovered in the limit, n = 1, and the perfectly plastic
solution in the limit as n -+co, Matthews [38] generalized the Lee and Radok result to obtain the contact
stress-effective strain rate relation for a power-law
creeping material:
By employing Hill’s similarity principle [44] to transform the creep indentation of a half-space by a rigid
indenter into that of a non-linear elastic half-space
indented to a unit depth by a rigid flat punch of unit
radius, Bower et al. [45] have recently shown that the
contact stress-effective strain rate relation during the
indentation of a power-law creeping half-space is
generally of the form
5 = F&-j”“.
00
’ ‘\a$)
(13)
. ,
Lee and Radok’s solution [43] was recovered for the
limiting case of a linear viscous solid (n = 1) and
Prandtl’s slip-line field solution [36] was obtained for
a rigid, perfectly plastic solid (n = CD). Bower et al.
[45] used finite element analyses to determine the
function F(n) for intermediate cases of material
non-linearity. Figure 3 compares Bower et aZ.‘s results for F(n) (given in [45] in tabular form) with
Matthews’ result (12). Although the detailed FEM
analyses show the flow parameter, F, to vary slightly
with indentation depth (i.e. effective strain), they
concluded, in agreement with Hill [44], that F can
be regarded as constant for small deformations
(a/r < 0.4). A recent analysis of creep indentation
due to Storakers and Larsson [46] also leads to a
relation of the form given by equation (13).
Given the simplicity of equation (13) and its compatibility with expressions used in densification
models, it would be very convenient if the results
describing the contact stress-effective strain rate relationship
for blunting
of power-law creeping
materials could be cast in a similar form. A simple,
0.0
0.2
0.4
0.6
0.6
1.0
l/n
Fig. 3. Matthews’[38] interpolated estimate for the contact
Row coefficient, F(n), for creep indentation is a good
approximation to the more exact, numerical results of
Bower et al. [45].
yet accurate relation is important since the model
describing the behavior of a single contact is afterwards inserted into rather complicated expressions
for the relation between applied stress and macroscopic densification rate [l&28].
Because it can be shown that for small displacements, blunting and indentation are both described
by the same set of equations, including boundary
conditions, the results of Bower et al. [45] [equation
(13)] must apply to blunting as well, at least for small
displacements. However, due to the non-self-similar
nature of blunting, it is unlikely that the function F
in equation (13) will be independent of the effective
strain beyond very small displacements. By rewriting
(13) in the form
(14)
[i.e. equation (2)] the approach might be generalized
to describe finite strain blunting. Equation (14)
should recover the result of Gampala et al. [29] in the
perfectly-plastic limit (i.e. as n + co), in which case, F
(n-co, u/r) becomes identical to the plastic flow
coefficient given by equation (6).
2.2. Contact radius-displacement relation
In order to use expressions of the form given in
equation (14) for modeling densification processes, it
is necessary to obtain an expression for the contact
radius, a (Fig. 2). The area of a contact is obviously
related to the displacement, h. Thus the contact
radius, a, identified in equation (14) will not be
independent of h and must be determined. The Hertz
analysis of a hemispherical linear elastic constant [34]
shows that the displacement, h, and contact radius, a,
for a hemispherical body of radius, r, are related by
h,a2.
r
(15)
GAMPALA et al.: CREEP BLUNTING OF CONTACTS
Equation (15) is valid for both indentation
and
blunting, but is restricted to very small displacements
(a/r < 0.01).
Calculation of the displacement-contact
radius
relationship
for inelastic deformations
is more
difficult. Fischmeister and Arzt [24] and subsequently,
Helle et al. [25], Kuhn and McMeeking [28] and Elzey
and Wadley [15], have avoided the detailed calculation by assuming the contact area to be given by a
conservation of volume principle. They took the
volume of material plastically displaced from a contact and uniformly redistributed it over the remaining
(free) surface of the particle. When applied to a
laterally constrained, hemispherical, blunting contact
[29], the resulting contact radius-displacement
relationship is given by
(16)
It is interesting to note that if equation (16) is
expanded and only terms of order h are retained,
,
r
Although this is an approximate result (with an error
of less than lo%), it is similar in form to that of Hertz
[34]. Compared to the linear elastic contact of
equation (1 S), equation (17) predicts twice the contact
area for a prescribed displacement. Nonetheless, the
uniform redistribution model underestimates the true
contact area during blunting of a perfectly plastic
material. Detailed finite element calculations [47]
show that the relationship between laterally constrained blunting displacement and contact radius for
a perfectly-plastic material is (to within 5%) given by
h ~0.385
r
(18)
Therefore earlier consolidation models based upon
the redistribution approximation
have significantly
underestimated the rate of growth of contacts.
Extensive experimental observations of contact
area during indentation have led Norbury and Samuel [41] to suggest the existence of a displacementcontact
radius relationship of the form
1485
refers to the radius of a section through a hemispherical indenter located a distance, h, from the point of
initial contact (cf. Fig. 2). The results above are
consistent with a more rapid “piling-up” of material
adjacent to the contact than predicted by the uniform
redistribution model. A similar physical meaning can
be ascribed in the case of blunting (Fig. 2), (with some
subtle differences as discussed below).
Based on Norbury
and Samuel’s data [41],
Matthews [38] proposed that the displacementcontact radius relation during the indentation of a
power-law hardening material could be written in the
form
(20)
Equation (20) in the limit as the material non-linearity n --*co, gives h -m’/er = 0.37a2/r, which is almost
identical to the recent result for blunting [equation
(18)]. The elastic result [equation (IS)] is also recovered when n = 1. Equation (20) is also valid when n
refers to the stress exponent of a power-law creeping
material [38]. Then, if the factor, [2n/(2n + l)]2@‘-‘) is
set equal to 0.5, the uniform redistribution model (17)
will give a correct prediction for the contact area
when n = 3.8 (which is in the middle of the range
observed for many power-law creeping metals and
alloys). Matthews’ result suggests that the coefficient,
c, in equation (19) is a function of n (see Table 1).
Bower ef aZ.‘s numerical results [45] for the evolution
of contact area during indentation of a power-law
creeping material were also found to obey equation
(19) with c = c(n), (see Fig. 7 later).
From the preceding discussion (and from Table l),
it is evident that regardless of the material behavior
(linear or non-linear), the contact radius, a, is related
to displacement, h, by an equation of the form
h=
1
2c(n)2? 0
a2
(21)
While equation (21) has been shown to hold for the
blunting of linear elastic and perfectly plastic contacts, it has not yet been shown to be a valid
representation for the blunting of a power-law creeping material (although this is implied by Matthews
[38]). Below, we will assume its validity, calculate the
contact radius coefficient, c(n), using finite element
analysis and demonstrate that (21) well approximates
power-law blunting for a/r < 0.8.
(19)
where c is a material constant. Equation (19) is a
general result, describing the elastic and perfectly
plastic indentation behavior with values of the contact radius coefficient, c, given in Table 1.
Rearran ement of (19) to give an expression for
c( = a/ Jg-2rh) allows a physical interpretation of the
coefficient during indentation: it represents the ratio
of the true to nominal contact radii, where “nominal”
3. DETERMINATION OF F AND c COEFFICIENTS
Micromechanics-based
approaches to the prediction of densification in both powder compacts and
composite preforms relate behavior at individual
contacts to the overall behavior by invoking equilibrium between applied and contact stresses, and by
requiring that the densification rate be determined by
the sum of the deformation rates of all the contacts.
GAMPALA et al.: CREEP BLUNTING OF CONTACTS
1486
In general, the applied stresses lead to macroscopic
deformations
comprised of dilatational
(volumechanging) and shear (shape-changing) components,
resulting in both normal and shearing displacements
at individual contacts. Except for the very earliest
stages of densification, in which rearrangement of
particles often plays an important role, densification
(i.e. the volume-changing strain component) during
consolidation occurs primarily by displacements normal to interparticle contacts. Our analysis considers
a single hemispherical contact subjected only to normal stresses and displacements. Since the zone of
deformation at the contact is considered to be much
larger than any microstructural
features, no size
dependence enters into the problem. The single contact model may therefore be regarded as a unit cell
analysis which can be incorporated
within a
micromechanics model for an aggregate containing a
spectrum of contact sizes.
We could identify a representative hemispherical
asperity and analyze its isolated behavior when flattened between two frictionless parallel plates. Howencountered
in particulate
ever, the contacts
consolidation problems are not free to laterally deform in this way (except perhaps in the very earliest
stages of densification). Instead, the deformations of
neighboring particle contacts constrain lateral flow as
densification proceeds. The severity of this will depend on the number and area of the adjacent contacts. Since this varies from particle to particle, it
results in varying degrees of constraint on the flow
associated with a given contact. In addition to the
constraint of neighboring contacts, monotape surface
asperities are also laterally constrained by their connection with the monotape [see Fig. l(c)]. Because of
the stochastic nature of the internal geometric features of these aggregates like those in Fig. 1, the
precise conditions under which any given contact
deforms is not known and may at best be specified
only as a probability. To include the influence of this
“imposed” constraint on the creep blunting behavior,
L
Fig. 4. The representative
contact blunting problem is taken
to be a hemispherical,
power-law creeping solid subjected to
uniaxial compression
within a rigid cylindrical die.
we consider a hemispherical solid compressed uniaxially within an encircling, rigid cylindrical die (Fig. 4).
We realize this is a significant simplification; in
reality, some contacts will experience a more severe
and some a lesser constraint than this. The simpler
problem formulated here incorporates a representative contribution of the increasing incompressibility
as the relative density of the cell approaches unity
(a/r + 1). The analysis of the unconstrained
problem
would substantially underestimate u, (which would
approach (TVand not co, as a/r -+l). Subsequent
studies should explore this issue in more detail.
Our objective is to apply the power-law creep
blunting theory, as given by equations (2) and (3), to
determine the contact stress-effective strain rate response and the relationship between blunting displacement (h) and the contact area for the situation
shown in Fig. 4. Our approach uses the finite element
method to calculate the evolution of blunting displacement (h) and contact radius (a) with increasing
unit cell density. Equation (3) is then used to determine the dimensionless contact radius coefficient, c,
for each value of it. Since the temporal evolution of
h and a is then known, h’can be calculated and the
flow coefficient, F(n, a/r), determined from equation
(2). The results of c(n) and F(n, a/r) are then fitted
to polynomial functions, which can be reinserted
back into equations (2) and (3) to provide approximate closed form solutions to the laterally constrained creep blunting problem.
The axisymmetry of the FEM problem allows the
solution to be obtained by analysis of a plane radial
quadrant of the cell shown in Fig. 4. Contact deformation is assumed to occur by power-law (i.e. steady
state) creep with a uniaxial stress-strain behavior
given by equation (1). Elastic loading contributions
are included [their significance is addressed separately
(Section 4.3)], but since contacts typically undergo
large inelastic strains during consolidation processing, the elastic deformations are negligible and moreover, do not contribute to permanent densification
(although their calculation might be important in
determining residual stresses in incompletely densified materials).
The FEM calculations were conducted by first
elastically loading the hemisphere and then allowing
it to creep under a constant applied load, L. The
strain-displacement relations used are those for large
deformations. The elastic deformations were determined assuming an isotropic, linearly elastic solid
(with Young’s modulus, E = 70 GPa and a Poisson’s
ratio, v = 0.3). The particular values of E and v have
no effect on the results for the flow and contact radius
coefficients, F(n, a/r) and c(n) and are in that sense
arbitrary. The creep strain rate components, iii, were
calculated using a power-law creep relation for a
material under a general state of stress
GAMPALA
et al.:
CREEP
where S, ( =aij - ~,,6,~/3) are the deviatoric stress
stress
components,
a, is the Mises equivalent
and 6, and o,, are the reference strain
(=J&%&
rate and stress, respectively. The functions, F(n, a/r)
and c(n), are independent of the reference stress and
strain rate, a,, and $, and so they were assigned
arbitrary values of 10.0 MPa and 0.01 SK’, respectively
Two frictional
analyzed:
contact
@,, = Is23= 0
i, = tii, = 0
conditions
(Fig. 4) were
Ir,l < a (frictionless)
(23)
1~~1
<a (no slip).
(24)
Frictionless interface elements for a sliding contact
were chosen for the lateral contact (these solid elements undergo a large relative sliding). The hemisphere was modeled using 780 second order,
rectangular, reduced-integration
(2 x 2) axisymmetric solid elements. The mesh was developed using a
mesh refinement scheme that enforced multi-point
constraints; elements farthest from the contact were
relatively coarse and become increasingly refined as
the region of contact was approached (see Fig. 3 in
Ref. [29]). Second order axisymmetric interface elements were used at the contact interface. The analy-
t
n=i
BLUNTING
1487
OF CONTACTS
ses were performed using the commercially available
finite element code ABAQUS [48].
4. RESULTS
AND DISCUSSION
Contours of accumulated Mises strain are shown in
Fig. 5 for creep stress exponents, n = 1,2, 5 and 10
following a blunting displacement of h/r = 0.1. These
results were obtained assuming frictionless conditions
at the contact. Results for no-slip conditions are quite
similar. The strain is seen to be intensified at the edge
of the contact. However, the extent of this localization is very sensitive to n and decreases markedly
with decreasing stress exponent. This has two major
consequences: firstly, because of the increased homogeneity of the stresses and strain rates at low n, the
presence of neighboring contacts more strongly
affects the contact stress-deformation
response of
materials having a low stress exponent. Secondly, as
n --f cc, the zone of most concentrated strain near the
contact becomes surrounded (and constrained) by
increasingly slower deforming material remote from
the contact. Thus greater contact stresses are required
to blunt materials with a high stress exponent. A
third, less significant effect of the concentration of
I
n=
101
Fig. 5. Contour plots of von Mises strain showing increasing concentration
of strain at the blunting
contact with increasing stress exponent, n. (The blunting displacement
was h/r = 0.1 for all the cases.)
GAMPALA et al.: CREEP BLUNTING OF CONTACTS
1488
Table 2. Numerical solutions for the coefficients c and F
Max
HT0r
l/n%
Fb
Cb
(%)
15
air’
0.2
1.0
0.867
0.7
0.898
0.5
0.923
2.2-4.49(t)+
0.2
1.059
2.83 -4.41(f)
0.1
1.112
2.86 - 3.92(;)
0.05
1.139
2.95-3.46($+2.81~)
-
-
0
1.1691
2.95 - 3.46(;y
2.0
0.2
0.86 + 12.95
-
not determined
4.4
0.43
+ 5.28(;)
2.0
0.62
+ 3.84(f)
2.2
0.7
11.94(;)
+ 2.81(f)
“For all n > 20, changes in the dependence of Fan a/r are negligible.
bAverage of frictionless and no-slip contact results.
cNormalized contact radius at which the maximum error occurs.
deformation at the contact with increasing n, is the
piling-up of material at the contact. This resulted in
a higher rate of increase in contact radius (for a given
h/r-value as n + co). For example, when the blunting
displacement is small (h/r = 0.08), the normalized
contact radius a/r = 0.32 for n = 1 whereas for the
same displacement, a/r = 0.43 when n = 10. These
three effects are all reflected in a dependence of the
coefficients F and c on the stress exponent.
4.1. Contact stress relations
The normalized mean contact stress (a,/a,) and
normalized effective strain rate (h/a&) were determined from the FEM analysis, and their ratio computed to obtain the flow coefficient, F, at various
values of effective strain, a/r. These data were then
fitted to a quadratic function of the effective strain for
each n-value (best fit expressions are given in Table
2). For the n + cc case, equation (2) agrees with the
results of Gampala et al. [29] for perfectly plastic
blunting, i.e F(n+oo, a/r), as given in Table 2, is
identical to the plastic flow coefficient, fi(a/r), in Ref.
[29]. A similar check in the n = 1 limit cannot be
made since there are no available solutions for the
constrained linear viscous blunting problem (though
it does tend to the solutions of Matthews [38] and
Bower et al. [45] as a/r-r0 for indentation).
Figure 6 shows a plot of the flow coefficient,
F(n, a/r) versus the effective strain, a/r for a wide
range of creep exponents. It is apparent that F is a
strong function of both n and the normalized contact
radius, a/r. For small effective strains (a/r < 0.4), F
is lower for materials with a low stress-sensitivity (i.e.
low value of n). These materials will experience a
greater blunting velocity (for a given contact stress)
than their higher-n counterparts. We also see that the
curves for n-values greater than 1 are all characterized by a minimum. Initial deformation in these cases
is accompanied by strain softening. This is a geo-
metric or “shape” effect, in which the “material”
constraint imposed by surrounding, elastically deformed or more slowly creeping material on the
deforming region near the contact is lowered as the
effective strain increases. This occurs because as a
hemisphere is deformed by blunting, its geometry
gradually approaches that of a right circular cylinder
which, in the absence of friction and lateral constraint, deforms uniaxially [i.e. e = &,(c/o,,)n for
diswhich F equals unity] with homogeneously
tributed stress and strain rate.
The subsequent increase in F with effective strain
(the start of which depends sensitively on the value on
n) is a direct manifestation of the lateral (externally
imposed) constraint. As material fills the void between the hemisphere and its surrounding constraint,
the system becomes less compressible. In the limit as
a/r + 1, incompressibility requires F+ co. Materials
with low n-values (for which deformation is the most
homogeneous, see Fig. 5) almost immediately experience the imposed constraint (note that for the n = 1
case, lateral constraint hardening is evident even at
the smallest deformations). As n-rco (i.e. in the
perfectly plastic limit), deformation is concentrated at
the contact and is little affected by the constraining
die wall until the blunting deformation is quite large
(a/r > 0.8). Overall, the influence of the external
constraint is very strong: note that all the curves in
Fig. 6 would approach 1 as a/r + 1 in the absence of
this constraint. External constraint of this type is
inevitable during consolidation and its consequences
must be included if a model is to be obtained which
is reasonably accurate at large deformations. The
error in predicted response if external constraint is
not included will be most severe when modeling the
behavior of linear viscous (low-n) materials.
Predictions for F based on Matthews’ indentation
model (12) are shown for comparison as dotted lines
in Fig. 6. The indentation creep results of Bower et al.
3’5------l
...............................................
I .. .... ... ..ml
3.0
......................................
............,o
:y
t
e
z
E
...
2.5
t
2.0
2
.g
z
P
LL
1.5-
y”
. =’ ..
l.O-
0.5
1,
0.0
5
.............
/
...
..... Matthews
t
0.0
4
...............
.hh_._.2?.
.......
......
E
$!
“‘z..
0.2
0.4
. ...
(equation
0.6
”
=
12)
,
1
i
0.8
Normalized contact radius, a/r
Fig. 6. The contact flow coefficient, F(n, a/r), characterizing the power-law
creep blunting
response
of a single
contact for various creep stress exponents.
GAMPALA et al.:
CREEP BLUNTING
Relative density, D
0.667
0.8
m
0.70
0.75
0.30
0.85
0.30
0.6
5
e
F
G
sc
0.4
8
z
2
62
0.2
z
o.ov
0.0
I
0.1
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
I
0.7
+s-E
Fig. 7. FEM results for the relationship between the contact
radius and blunting displacement. The curves may be approximated as straight lines which, according to the blunting
theory [equation (3)], should have constant slope, equal to
the contact radius coefficient, c.
[451 are very similar to those predicted by Matthews less
accurate, but closed form solution and could equally
well have been shown. We observe that the flow
coefficient for indentation
(dashed curves in Fig. 6) is
independent
of a/r because there is no loss of material
constraint during indentation
(a semi-infinite body of
material always surrounds the deformation zone). It can
be seen that for perfect plasticity (n --f co), the Matthews
solution (i.e. the slip-line field result for indentation with
F (and hence the contact
a flat punch) overestimates
flow stress) required to cause blunting, whereas for
n = 1 or 2, the flow stress is underestimated.
OF CONTACTS
1489
The difference in the predictions
of blunting and
indentation
theory can be understood
as follows: at
high n, indentation is more difficult than blunting (i.e.
indentation underestimates
the rate of creep blunting)
because of the softening associated with diminishing
material (shape) constraint during blunting. (Otherwise, if no softening occurred, F would be about the
same for both indentation
and blunting.) At low n,
where deformations
are more uniform, blunting is
more difficult than indentation
because of hardening
caused by the lateral constraint
imposed by neighboring contacts; in the absence of lateral constraint
the behavior during blunting and indentation
would
again be quite similar. Thus the softening due to
diminishing
material constraint
dominates
at high
n and hardening due to external constraint
at low
n. Without
these effects, F would exhibit little
dependence
on a/r and indentation
models would
apply equally well to blunting.
As shown below
(Section 5), these differences in the predicted flow
coefficient can, depending on the value of n, lead to
substantial error in the expected overall densification
rate.
4.2. Contact
radius
Equation (3) predicted that a lot of normalized
contact
radius, a/r, against P- 2h/r, results in a
straight line for each n-value) with a slope, c. Both
from the finite
a/r and /- 2h/r can be determined
element analyses for each creep exponent, n. Figure
7 shows that roughly straight line behavior is exhibited by the numerical results when n -+ cc. It is a less
accurate description
of materials with low n-values.
The results show that the contact radius grows most
slowly for materials with low n-values. The shaded
region of Fig. 7 indicates that regardless of n, the
Piling up
Sinking in
Piling up N-
Blunting
4
Sinking
in
Hunting
Indentation
indentation
4a
1
-i
0.0
0.2
0.4
0.6
0.8
‘r
_
LEX
1.0
l/n
Fig. 8. The contact radius coefficient, c, as a function of creep stress exponent. The c > 1 case corresponds
to “piling up” of material near the contact and occurs when n is high; c < 1 indicates “sinking in”. The
behavior during blunting is very similar to that of indentation except at low n-values.
1490
GAMPALA
et al.:
CREEP
coefficient, c (representing the average slope), always
lies in the narrow range between 0.8 and 1.2.
The best straight line has been fitted through the
results shown in Fig. 7 and the average slope, c,
determined. These results are given in Table 2. The
dependence of c on n during blunting can be approximated (to within 2%) by
c(n) = 1.17 -0.64@
+ 0.34($
(25)
The observation that c depends negligibly on the
displacement (i.e. depth of penetration) during indentation is not surprising in light of the self-similar
nature of the indentation process, but it is at first
sight surprising that the same is true for blunting
where finite size effects are so significant.
Figure 8 shows a plot of c against l/n. For
comparison, the predictions of Bower et al. [45] and
of Matthews [38] [equation (20)] for indentation are
also shown. When c > 1, the true contact radius
exceeds the nominal, indicating the “piling up” of
material adjacent to the contact. From Fig. 8, this is
seen to occur for materials with a high n-value
(n > 3-4) and is a result of the stress and strain
concentrations near the contact (Fig. 5). Figure 8
shows a slightly greater tendency for this pile up to
occur during indentation. For intermediate n-values,
the indentation results of Bower et al. [45] and those
for blunting are seen to be almost identical. However,
when n < 3, c < 1, indicating that “sinking in” occurs, i.e. material is pressed into the blunting asperity
(Fig. 8). Bower et al.% [45] and Matthews’ [38] work
suggest that this phenomenon occurs more easily
during indentation than blunting.
predominantly
BLUNTING
OF CONTACTS
4.3. Infiluence of elasticity
The ratio of true to nominal contact radius, c, and
the flow coefficient, F, relating the contact stress, u,,
with blunting velocity, /i, were calculated above for
the case in which the elastic displacements are much
smaller than the inelastic deformation. When applying these results to consolidation, it has been assumed
that elastic contributions to the contact area can
always be ignored. The likelihood that this assumption will lead to appreciable error can be assessed
using an idea of Bower et al. [45]. The normalized
blunting displacement of an incompressible linear
elastic hemisphere, he/a, can be expressed as
h’
qF(n,a/r)
Qc
a
-=EF(l,a/r)=EF(l,a/r)
h’ ‘in
0z
(27)
Thus A can be interpreted as the ratio of the stress
required to cause blunting of an elastic cylinder of
gage length, a, by an amount, h, to the stress required
to blunt a cylinder by power-law creep at a rate, h’.
Large values of ,4 indicate that the elastic strains are
small relative to the total deformation. Elastic effects
are negligible for A & 1 and can be ignored when
A > 50.
0.4
0.5
Effective strain, a/r
Fig. 9.
(26)
Following Bower et al. [45], we can define a parameter, A = F(n, a/r)h/F(l, a/r)he, which is a
measure of the significance of elasticity (note here h
is the actual, i.e. elastic plus inelastic, displacement).
Using the definition of F(n, a/r) as given by equation
(2) and solving equation (26) for F(l, a/r), A can be
written as
Plastic
0.3
.
The factor, A, which reflects the importance of elastic effects during blunting, is plotted against
a/r for various values of the stress exponent, n; A increases quickly with a/r in all cases, indicating the
insignificance of elastic effects during high temperature blunting.
GAMPALA
et al.:
CREEP
Figure 9 shows a plot of A as a function of
normalized contact radius, a/r, for various power-law
exponents. Elastic contributions to the effective strain
(or contact radius) are seen to be most significant
when a/r is very small and deformation is perfectly
plastic. For this case, deformation is concentrated
near the contact, and much of the blunting solid has
not exceeded the yield strength, i.e. it is elastically
deformed. The effective strain (u/r) beyond which
elastic effects are negligible, even for the perfectly
plastic case, is about 0.12. For a hemisphere within
a cylindrical die, this corresponds to a relative density
of 0.67. This is only slightly above the initial relative
density of 0.66, and so for all practical purposes the
elastic strains can be neglected during densification
by power-law creep blunting.
This detailed analysis of blunting leads to the
conclusion that the contact stress-effective strain rate
relationship for blunting is quite different from that
of indentation, even though similar relations can be
used to define the flow coefficient, F [cf. equations (2)
and (13)J The contact mechanics of blunting are
strongly influenced by the finite size (material constraint) and lateral constraint imposed on the blunting body, whereas during indentation, deformation is
remote from all but the indented surface and is thus
predominantly free from edge effects. For this reason,
the contact flow coefficient is a function of the
effective strain during blunting, and must be included
in any accurate effective constitutive model describing
densification by contact blunting.
5. APPLICATIONS
The blunting model developed above accounts for
two effects which could not be considered by an
analysis based on indentation: (i) the effect of the
finite size (i.e. the diminishing materal constraint) of
the blunting body as a/r -+ 1; and (ii) the influence of
externally imposed lateral constraining forces (arising
either from nearby contacts or from the substrate to
which the blunting body is attached). The significance
of incorporating these blunting mechanics results into
consolidation models can be evaluated by using these
new blunting results to predict densification and fiber
fracture and compare the new predictions with those
of previous indentation-based
models [15, 16,281.
BLUNTING
OF CONTACTS
1491
were assumed to bc statistically distributed and the
resulting development of new contacts, together with
the continued deformation of ones formed earlier,
were incorporated into the model. By realizing that at
any instant the externally applied consolidation force
must be in equilibrium with the sum of forces acting
on all existing contacts, a relation between the applied
stress, C, and the contact forces, L, was developed
X
s
Oc’
1 exp( --Ir)L(H,
r, z, i) dr
dH
(28)
0
where, z, the compacted monotape thickness, is related to the relative density of the monotape, D, by
z = zoDo/D, z. is the initial (undeformed) monotape
thickness, I is the area1 density of asperities, r and H
are the radius and undeformed height of a particular
asperity, respectively, and a,, Z?and 1 are statistical
parameters characterizing the distribution of asperity
sizes.
The force, L, acting on a contact can be obtained
for the case of power-law creep from either equation
(13) (indentation model) or from equation (2) (blunting model), by noting that the contact stress,
u= = LIza’. For the blunting case, equation (3) can be
used to eliminate the contact radius, a, in equation (2)
giving
L = nuOF
[2rhc(n)2]((2”-‘)‘2n) -I; I’“. (29)
0 80
The overall rate of compaction, i, (equal to the
asperity displacement rate, h’), is obtained by substituting the contact force (29) into the statistical model
(28) and solving for I;. Since i = zo(Do/D2)d, the densification rate obtained using the blunting model is
d,=
a(n)Z”D2
ZODO
x exp[ -fr$)]dH
s
--n
a,
X
3.r’ -w’~)
exp( - Ar) dr
(30)
0
5.1. MMC monotape derkjication
relative densities below about 0.9, densification
of spray-deposited MMC monotapes during either
hot isostatic or vacuum hot pressing has been modeled to occur by contact deformation of surface
asperities [15]. The size and location of the asperities
At
TThe factor a, given in Ref. [15] for the indentation-based
model contains
an error and should read a = 0.34
(@)‘-“2”*-“.
All calculations
reported here are based
on the corrected factor.
where H - z is just the displacement, h, and TV
is given
by
~o(2c(n)2)“2-n
cr(n) =
(%)n
.
Examples of the ratio of the densification rates
obtained using the blunting (8,) and indentation
{equation (19) in Ref. [15]t} (d,) models are plotted
as a function of relative density in Fig. 10 for two
example materials (Cu and Ti-24Al-1 lNb, an a2 + fi
titanium alloy) chosen to have widely different
GAMPALA
1492
et al.:
CREEP
Monotap
S-Lamha
Power-lawCreep Mechanism
’
0.50
0.4
I
0.5
0.6
I
I
I
0.7
0.6
0.9
Relative Density, D
Fig. 10. Ratio of densification
rates for spray-deposited
MMC monotapes
as predicted by blunting and indentation
analyses of contact deformation.
For materials with high
n-values
(e.g. Cu with n = 4.8), the indentation
model
underestimates
the densification
rate, while for materials
with low n-values (n = 2.5 for Ti-24Al-1 lNb), indentation
theory overestimates
the densification
rate.
creep properties, Table 3. The steady state creep
behavior of Cu [49] was described by
BLUNTING
OF CONTACTS
materials like Cu for all densities greater than about
0.47. On the other hand, the densification rate of
materials like Ti-24Al-1lNb
is always overestimated
on the basis of indentation. The differing results for
these two materials can be understood in terms of the
single-contact behavior: Fig. 6 shows that for materials with a relatively high n-value, such as Cu, the
coefficient, F, for indentation is greater (implying
lower densification rate for a given stress) than that
for blunting when a/r > 0.25. For Ti-24Al-llNb,
which has an n-value of around 2, the opposite is
true: F for indentation is generally lower (greater
densification rate for a given stress) than that for
blunting. Both curves are also seen to exhibit a
maximum. This arises because at first &/B, increases
due to the softening associated with blunting (the
shape effect associated with loss of material constraint). This is followed by a decreasing &/d, as the
blunting response hardens due to the increasing
influence of neighboring contacts (laterally imposed
constraint) at large strains.
5.2. Fiber fracture
g = A {D,,,,exp[ - Q,/RT]) FT
i
”
0
(32)
with material parameters as defined in Table 3. The
temperature-dependent
shear modulus was taken to
be G,(l -0.54(T -300/T,)),
where GO, the room
temperature shear modulus, was 42.1 GPa.
The power-law creep behavior of the a* + /I titanium alloy was described by
8 = A % ‘exp(-QJRT)
0
with the values of material parameters as defined in
Table 3. The relative density shown in Fig. 10 refers
to that of the surface roughness layer (designated
“S-1amina”) and not to the composite laminate as a
whole (see Fig. 3 in Ref. [15] for its definition). Hence,
depending on the standard deviation of the surface
roughness distribution, the starting density is around
0.4 (as compared to the initial overall density of
around 0.65). Since both densification models have
the same dependence on pressure and temperature,
these terms cancel and the ratio of densification rates
is a function of only the relative density.
Figure 10 shows that, depending on the material’s
creep parameters and relative density, earlier indentation-based models in some cases overestimate, and in
others underestimate, the densification rate. Indentation analysis underestimates the blunting rate for
Table 3. Creep narameters
Parameter. svmbol (units)
Creep constant,
stress exponent,
A (b-l)
n
Activation energy, Q, (kJ moleI)
Young’s modulus, E (GPa)
Pre-exp volume diffusion, D,,, (III* sr’)
Melting temperature,
T,,, (K)
Bureers vector. b (ml
during densljication
of monotape
Ceramic fibers in metal matrix composite monotapes are susceptible to microbending and fracture
during consolidation processing [12]. During consolidation, bending occurs because the asperity contact
stress, uc, results in localized forces distributed randomly along the length of the fibers. The evolution of
the bending and fracture has been modeled by considering a statistical distribution of unit cells, each
consisting of a segment of fiber undergoing threepoint bending due to forces imposed by contacting
hemispherical asperities [16]. The contact force was
obtained using an approximate, indentation-based
model. By substituting the power-law creep blunting
prediction for the contact force into the fiber bending
unit cell of Ref. [16], the evolution of fiber damage
can be compared with that of the original model.
Previous fiber fracture models [16] assumed the
asperity response to be given by a uniaxial creep
response (i.e. F = 1):
where y is the deformed height of the asperity,
(r - h). The average contact stress, oc = L/na’, where
L is the applied force. The force exerted on the fiber
by the asperity must be balanced by the bending
reaction so that L = 2k,(z - y), where k, is the fiber
bend stiffness and z is the monotape thickness [so that
used for model medictions
cu 1491
3.4 x 106
4.8
197.0
2.0 x 10-5
1356
2.5 x 10-l’
Ti-14Al-2lNb
1151
6.0 x 10”
2.5
285.0
140.0 - 0.12 T (“C)
GAMPALA et al.:
CREEP BLUNTING
2(z - y) represents the fiber deflection]. By approximating the contact area as 27rr(r -v) [cf. equation
(17)], equation (34) can be written as a first order
ordinary differential equation in y
A similar differential equation for blunting can be
obtained by rearranging equations (2) and (3)
3 = -$J2r]y
’
- rlc(n)’
%(z -Y)
( y - r)/r)
2xra,( y - r)c(n)‘F(n,
1’
(36)
where h’= 3.
The number of fibers fractured per unit length of
fiber can be calculated for any consolidation cycle by
incorporating the unit cell model [either equation (35)
or (36)] into the macroscopic model given in Ref. [16].
Figure
.c
‘i
11 shows the cumulative
number
of fractures
during densification of composite materials with
matrix properties of either Cu (n = 4.8) or Ti-24Al11Nb (n = 2.5) and reinforcements with SCS-6 (SIC)
fiber properties (the fiber properties given in Ref.
[ 161). The model based on blunting mechanics predicts a higher number of fractures in both cases, with
the correction being more significant in the case of
Ti-24Al-11Nb.
The unrealistically low value of F
(= 1) used in the earlier model for this case in
particular resulted in an underestimate of the fiber
damage probability of around 40%.
5.3. Consolidation of metal powders due to power-law
creep
The model of Kuhn and McMeeking [28] is widely
used for predicting the consolidation behavior of
metal powders by power-law creep. Starting with the
average normal stress at an interparticle contact, cr,
(given by the indentation solution of Fischmeister
and Arzt [24]), they write the energy dissipation rate
per unit area of contact for a pair of particles as u,h’,
where h’is now the relative normal velocity of the two
contacting particles. Similarly, the creep dissipation
rate for a pair of blunting particles [with a, given by
equation (2)] is:
u,ti=o,F
80-
1493
OF CONTACTS
n,f
(37)
r
Y
c
-
3
Kuhn and McMeeking [28] assumed F = 3 and also
used the result of Helle et al. [25] to relate the average
contact radius to the overall (average) relative density, D:
a/r = [(D - Do)/(3(1 - ~oW12,
60-
5
5
2
40-
LL
20 -
0
0.4
0.5
0.6
0.7
0.8
0.9
Relative density, D
(b)
240
I
I
I
(b) n = 4.8 (Copper)
200-
where Do is the initial relative density.
However, the material non-linearity influences the
relationship between contact size and density, with a
growing faster with D as n increases. For the case of
blunting, combining equation (3) with the expression
D = 2r/(3(r - h)) (which can be obtained from conservation of mass for a hemispherical contact subjected to constrained uniaxial compression) gives
T = 543K (0.4Tm)
a/r =,(,)&(I
z.
7
c
160 -
120-
8
5
ti
F
(38)
where values of c(n) are given in Table 2.
The next step is to relate the creep dissipation for
a single contact to the total dissipation occurring per
unit volume in the powder aggregate. Following
Kuhn and McMeeking [28], this is
u.
-
-2)
60-
IL
40 -
+ 2&/3l’ + ‘insin 4 d4
0.5
0.6
0.7
0.8
0.9
Relative density, D
Fig. 11. Comparison of fiber damage during consolidation
of spray-deposited composite monotapes predicted
the basis of indentation
and contact blunting
Ti-24Al-11 Nb/SCS-6; (b) Cu/SCS-6.
on
(a)
(39)
where l? and fi are the macroscopic deviatoric and
dilatational strain rates, 4 locates the angular position of a contact relative to the (cylindrical) coordinate system, and K is given by:
K = f(,,‘?)“n&n,
D)D2
2c(r~)~(D - Do) 1- I’*”
DO
(40)
>
1494
GAMPALA et al.:
CREEP BLUNTING OF CONTACTS
where P = F(n, a/r) with a/r given by equation (38).
For a given material creep exponent,
n, the coefficients c and F may be obtained from Table 2.
With the exception
of the parameter,
K(n, D),
equation (39) is identical to the creep dissipation rate
obtained by Kuhn and McMeeking
[28] based on
indentation analysis of single contact behavior. In the
blunting approach, this parameter replaces their coefficient, C(n, D) [not to be confused with c(n) given
in equation (3)]
2.5
r
I
I
Powder Stage I Consolidation
Power-law Creep Mechanism
2.0 aa”
.s
E
1.5 -
” = 4.6
(Copper)/‘I
p
5
‘Z
z
1.0
E
t
$
The relation between macroscopic
stress and strainrate can be expressed
in potential
form so that
or inversely, _Ei,= atr/az,,
C, = n/(n + i)awvjaE,j
where C, is the applied mean stress, & the applied
effective stress and the lower bound estimate for Y is
Y
5
=z,A+z,B-
W”(fi>8).
(42)
The structure of the macroscopic constitutive relation
(as expressed by the potentials) is unaffected by the
choice of a blunting- or indentation-based
analysis of
a single particle contact. The only difference is in the
magnitude of the factors K and C (which are independent of C,j and gij). Therefore the shape of the creep
contours (i.e. constant values of !P plotted in stressspace) is the same for both models, but their magniin stress-space
may differ
tudes at any point
significantly, depending on the value of the stress-exponent and relative density. The factors, K(n, D) and
C(n, D), given by equations (40) and (41) represent
the magnitude or strength of the potential (increasing
to increasing
creep
values of K or C correspond
resistance). Thus a comparison
of the macroscopic
behavior
predicted
on the basis of blunting
and
indentation
is obtained
from
the
ratio
of
K(n, D)/C(n,
D). Figure 12 shows K/C as a function
of relative density for two n-values: for low n < 2,
1.6
1.4
c
\
tndentation overe*timates
creep rate
blunting
A
I
n = 10 tndentat~on underestimates
blunting creep rate
0.75
Relative
0.80
density,
0.85
0.0
0.65
n = 2.5 (Ti-24AI-11
I
I
0.70
0.75
I
I
0.65
0.60
Relativedensity,
Nb)
0.90
D
Fig. 13. Examples of the ratio of powder densification rates
predicted using contact blunting and indentation for copper
(n = 4.8) and Ti-24Al-11Nb (n = 2.5).
K/C > 1, indicating that the blunting model predicts
a greater creep resistance, as expected from singlecontact behavior (Fig. 6). As n increases, the softening associated with the shape effect during blunting
lowers the contact creep resistance. Thus, for n = 10,
the blunting-based
prediction
leads to lower creep
resistance
than on the basis of indentation
(i.e.
K/C < 1) for all D > 0.73. To evaluate the consequence of this for the densification rate, we note that
the dilatational
component
of strain
rate
fi
(= -d/D),
obtained
by differentiating
the creep
potential
(42) with respect to macroscopic
mean
stress is
IC
fi = -2d,{2~(2/3)@+
n/n -n
I[(
1
(n + 1)/n
*
+(EJ”“l”~I(%)ii.
>
(43)
Figure 13 shows the ratio of densification
rates due
to blunting (6,) and indentation
(6,) for Cu and
Ti-24Al-1lNb
powders,
whose power-law
creep
properties are given in Table 3. The densification rate
of a low n-value material such as Ti-24Al-llNb,
is
overestimated
on the basis of indentation whereas for
high n-value materials like Cu, indentation underestimates the densification rate (in this case for densities
greater than about 0.76). The differences can clearly
be very significant (depending
on the values of n
and D).
I
n = 2
0.5
0.90
D
Fig. 12. The ratio of creep resistance coefficients, K and C,
indicating the relative strengths of creep dissipation during
blunting and indentation, respectively; if the stress exponent
is low (e.g. n < 2), then K/C > 1, indicating that the creep
resistance during blunting is greater than that predicted by
indentation theory. Indentation therefore overestimates the
overall densification rate of powders for low n. The opposite
trend is observed for high n (e.g. n = 10).
6. CONCLUSIONS
The availability of models for analyzing indentation has led to their widespread use for estimating the
response of deforming contacts in situations where
blunting is a more physically accurate description of
the deformation.
A relatively simple, yet accurate
model for blunting has been proposed and a finite
GAMPALA
et al.:
CREEP
element analysis of a laterally constrained,
power-law
creeping hemispherical
contact used to calculate the
strain and stress exponent
dependence
of a flow
coefficient, F and the contact radius coefficient, c, as
a function
of blunting displacement.
The contact
stress-displacement
rate relationship
during blunting
is controlled by the degree of constraint imposed on
the flow of material near the contact: initially, the loss
of internal (material) constraint
as the deformation
field becomes more homogeneous
dominates, leading
to a strain softening; this is followed by hardening in
which the influence of externally
imposed
lateral
constraint
(due to the presence of neighboring
contacts) become dominant. The model has been used to
revise existing consolidation
models for powders and
spray deposited monotapes.
It has been shown that
the earlier models can either substantially
over- or
underestimate
the densification
rate. They also failed
to incorporate
lateral constraint
hardening
which,
depending
on the value of the creep exponent, can
exert a strong influence on the contact flow stress
even when the densification
strains are low.
Acknowledgements-The
authors
are grateful
to M. F.
Ashby, R. M. McMeeking,
N. A. Fleck and J. M. Duva for
helpful discussions. The financial support of the Advanced
Research Proiects Agency (Program Manager, W. Barker)
and the National
Aeronautics-and
Space Administration
(Program Manager, D. Brewer) through grant NAGW 1692
is gratefully acknowledged.
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