-
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
Pergamon 0956-7151(95)00279-O
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 stress- displacement 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 indentation- based models can
substantially over- or underestimate the true rate of
densification.
1. INTRODUCTION
Many of today’s high performance structural ma- terials (e.g.
high strength tool steels [l], titanium and nickel-base alloys [2,
31, refractory metals [4], cer- amics [5] and metal/intermetallic
matrix composites [6,7]) are manufactured to near-net shape in a
two- step process. First, monolithic powders [8], (tape cast)
powder/ceramic fiber preforms [9] or spray-deposited monotapes
containing unidirectional fibrous re- inforcements [lo] are
produced (Fig. 1). These are then encapsulated, evacuated and
consolidated to full density using processes such as hot isostatic
or vac- uum 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 recog- nizing 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
dis- places material from the contact into adjacent (inter-
particle) 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 redis- tribution” 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 ma- terial
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
predomi- nantly 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 mech- anics results from analyses of perfectly
plastic or creep indentation have been used to develop densifi-
cation and fiber damage models.
Wilkinson and Ashby [21] were the first to propose a creep
contact model for analyzing the densification
1479
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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 repre-
sentative 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 liken- ing 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 sub- sequently 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 kin- etics 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 pro- cess 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 pen- etration once
fully inelastic deformation has been established]. This leads to
simple indentation depth- independent relationships between the
average con- tact 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 esti- mate 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, ad-
ditional lateral (neighboring) contacts are estab- lished, material
flow at the primary contact extends through the asperity or
particle and becomes more constrained, requiring higher contact
stresses to sus- tain deformation. Indentation-based models do not
incorporate the effect of lateral constraint upon a contact’s
deformation. Recent models for the blunt- ing of hemispherical
contacts by plasticity have ac- counted 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).
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GAMPALA et al.: CREEP BLUNTING OF CONTACTS
Plastic indentation Plastic blunting of a half-space 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]. How- ever, these ignore the “piling-up” of a
matrix near the contact and are expected to give inaccurate predic-
tions 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
0 ” g=go - 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 aver- age 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.
1481
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 motiv-
ated many efforts to analyze indentation and has led to (empirical,
analytical and numerical) flow 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 sur-
faces.
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) de- pendent 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
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1482 GAMPALA et al.: CREEP BLUNTING OF CONTACTS
blunting of a power-law creeping contact, the contact
radius-displacement relationship can be described by
h_!- - a2 0 2c(n)’ r where c is a material non-linearity
dependent co- efficient.
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 consolida- tion of monolithic powder and
composite preforms (Section 5).
2. CONTACT DEFORMATION MECHANICS
2.1. Contact stress-effective strain (rate) relations
When the contact stress between a pair of elas- tic-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 h -._ Q’c=j&-v2 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 sol- utions
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 con- tact radius was related to
the contact displacement, h, by h = 0.38az/r [see equation
(18)].
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 (indepen- dent 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 bc- cause 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 analy- sis to
investigate the blunting of hemispherical asper- ities. 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 sub-
jected to constrained uniaxial compression could be fitted by a
relationship of the form
F(D) = 340’ - 580 + 26 (6)
where the relative density, D, was related to the normalized
contact radius, a/r, byt
D =[;-$>‘I]‘. (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 de- formed 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].
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GAMPALA et al.: CREEP BLUNTING OF CONTACTS 1483
The contact (indentation and blunting) defor- mation 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 stress- strain 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
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 nu- merical computations were valid for
deformations upto a/r = 0.8. We see that equation (10) is a gener-
alization of equations (4), (5) and (9) for the indenta- tion of
elastic (i.e. n = l), perfectly plastic (n = co) and power-law
hardening materials, respectively. Ex- pressions 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
Material behavior Theory
Table 1. Summary of contact mechanics models
Normalized Effective Model contact strain Flow coefficient
basis’ stress (rate) Fb
Area coefficient,
cc
Elastic Hertz [34]
Slip-line field [37]
Perfectly plastic
Gampala
et al. [29]
Plastic/work hardening
Linear
viscous
Power-law creep
Matthews
[381
Lee and
Radok 1431
Elzey/Wadley
V51
Matthews
[381
Bower et al. [45]
I, B 2 (1 - 02) h _ a
I “c 1 %
B
h
a
8 3n
1 - z 0.7 Js
2.66
1.15
6n 16 I/”
0 2n + 1 9n
Tabulated F(n) Tabulated c(n)
aI = Indentation, B = Blunting. bF = normalized contact
stress/effective strain (rate)““. cc = a@z.
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1484 GAMPALA et al.: CREEP BLUNTING OF CONTACTS
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 power- law 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 situ- ation,
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]
general- ized 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 trans- form 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”“. (13) ’ ‘\a$)
. , 00
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 re- sults 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 com- patibility
with expressions used in densification models, it would be very
convenient if the results describing the contact stress-effective
strain rate re- lationship 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 after- wards inserted into
rather complicated expressions for the relation between applied
stress and macro- scopic densification rate [l&28].
Because it can be shown that for small displace- ments, 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)
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GAMPALA et al.: CREEP BLUNTING OF CONTACTS 1485
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 calcu- lation
by assuming the contact area to be given by a conservation of
volume principle. They took the volume of material plastically
displaced from a con- tact 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 re- lationship 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 con- strained 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 Sa- muel [41] to suggest the
existence of a displace- mentcontact radius relationship of the
form
(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 con- tact 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”
refers to the radius of a section through a hemispheri- cal
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 displacement- contact 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-linear- ity 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 recov- ered 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
1 h=
a2 2c(n)2? 0
(21)
While equation (21) has been shown to hold for the blunting of
linear elastic and perfectly plastic con- tacts, it has not yet
been shown to be a valid representation for the blunting of a
power-law creep- ing 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.
3. DETERMINATION OF F AND c COEFFICIENTS
Micromechanics-based approaches to the predic- tion of
densification in both powder compacts and composite preforms relate
behavior at individual contacts to the overall behavior by invoking
equi- librium 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.
-
1486 GAMPALA et al.: CREEP BLUNTING OF CONTACTS
In general, the applied stresses lead to macroscopic
deformations comprised of dilatational (volume- changing) 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 nor- mal to interparticle contacts. Our analysis
considers a single hemispherical contact subjected only to nor- mal
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
con- tact 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 flat- tened between two
frictionless parallel plates. How- ever, the contacts encountered
in particulate consolidation problems are not free to laterally de-
form 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 de- pend on the number and area of the
adjacent con- tacts. 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 con- nection with the monotape [see Fig.
l(c)]. Because of the stochastic nature of the internal geometric
fea- tures 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 uniax- ially 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 representa- tive
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 (TV and 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 re- sponse and the relationship
between blunting dis- placement (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 dis- placement (h)
and contact radius (a) with increasing unit cell density. Equation
(3) is then used to deter- mine 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
approxi- mate closed form solutions to the laterally con- strained
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 defor- mation 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 process- ing, the elastic deformations are negligible
and more- over, do not contribute to permanent densification
(although their calculation might be important in determining
residual stresses in incompletely den- sified 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 deter- mined
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 BLUNTING OF CONTACTS 1487
where S, ( =aij - ~,,6,~/3) are the deviatoric stress
components, a, (=J&%&
is the Mises equivalent stress and 6, and o,, are the reference
strain
rate and stress, respectively. The functions, F(n, a/r) and
c(n), are independent of the reference stress and strain rate, a,,
and $, and so they were assigned arbitrary values of 10.0 MPa and
0.01 SK’, respect- ively
Two frictional contact conditions (Fig. 4) were analyzed:
@,, = Is23 = 0 Ir,l < a (frictionless) (23)
i, = tii, = 0 1~~1
-
1488 GAMPALA et al.: CREEP
Table 2. Numerical solutions for the coefficients c and F
Max HT0r
l/n% Cb Fb (%) air’
1.0
0.7
0.5
0.2
0.1
0.05
0
0.867
0.898
0.923
1.059
1.112
1.139
1.1691
0.86 + 12.95
not determined
2.2-4.49(t)+ 11.94(;)
2.83 -4.41(f) + 5.28(;)
2.86 - 3.92(;) + 3.84(f)
2.95-3.46($+2.81~)
2.95 - 3.46(;y + 2.81(f)
15
4.4
2.0
2.2
-
2.0
0.2
-
0.43
0.62
0.7
-
0.2
“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 deter- mined from the FEM
analysis, and their ratio com- puted 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 character- ized by a minimum. Initial deformation in
these cases is accompanied by strain softening. This is a geo-
BLUNTING OF CONTACTS
metric or “shape” effect, in which the “material” constraint
imposed by surrounding, elastically de- formed 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 con- straint, deforms uniaxially [i.e. e =
&,(c/o,,)n for which F equals unity] with homogeneously dis-
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 be- tween 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 experi- ence 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 3.0
t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . I . . . . . . . . . . . ml
e 2.5
z t
...................................... : y
............ ,o ... “‘z.. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 5
2 .hh_._.2?. 4 ....... ...... E 2.0 .g E $! 1.5-
z P LL l.O- y”=’
/ . . . . . . . . . . ” = , 1
0.5 t
. . . . . Matthews (equation 12) i
0.0 1, 0.0 0.2 0.4 0.6 0.8
Normalized contact radius, a/r
Fig. 6. The contact flow coefficient, F(n, a/r), characteriz-
ing the power-law creep blunting response of a single
contact for various creep stress exponents.
-
GAMPALA et al.: CREEP BLUNTING OF CONTACTS 1489
Relative density, D 0.667 0.70 0.75 0.30 0.85 0.30
0.8
m 0.6
5 e F
G s c 0.4
8
z 2
2 6 0.2
z
o.ov I I I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
+s-E
Fig. 7. FEM results for the relationship between the contact
radius and blunting displacement. The curves may be ap- proximated
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 a flat punch)
overestimates F (and hence the contact flow stress) required to
cause blunting, whereas for n = 1 or 2, the flow stress is
underestimated.
Piling up
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. (Other-
wise, 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 neigh- boring
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 a/r and /- 2h/r can be determined from the
finite element analyses for each creep exponent, n. Figure 7 shows
that roughly straight line behavior is exhib- ited 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
Sinking in
Blunting Piling up N- 4 Sinking in Hunting
Indentation indentation
4a
‘r 1 _
-i LEX
0.0 0.2 0.4 0.6 0.8 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 BLUNTING OF CONTACTS
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
approxi- mated (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 inden- tation 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” oc- curs, 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.
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 apply- ing these results to consolidation, it has
been assumed that elastic contributions to the contact area can
always be ignored. The likelihood that this assump- tion 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’ Qc qF(n,a/r) h’ ‘in -=EF(l,a/r)=EF(l,a/r) z . 0
(26) a
Following Bower et al. [45], we can define a par- ameter, 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
(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.
predominantly Plastic
0.3 0.4 0.5
Effective strain, a/r Fig. 9. 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 BLUNTING OF CONTACTS 1491
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 con- straint) and
lateral constraint imposed on the blunt- ing 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.
5.1. MMC monotape derkjication
At relative densities below about 0.9, densification of
spray-deposited MMC monotapes during either hot isostatic or vacuum
hot pressing has been mod- eled to occur by contact deformation of
surface asperities [15]. The size and location of the
asperities
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.
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 re- lated 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) (blunt- ing 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) 0
I; I’“. - (29) 80
The overall rate of compaction, i, (equal to the asperity
displacement rate, h’), is obtained by substi- tuting the contact
force (29) into the statistical model (28) and solving for I;.
Since i = zo(Do/D2)d, the den- sification rate obtained using the
blunting model is
d,= a(n)Z”D2
ZODO
x exp[ -fr$)]dH
s a, --n X 3.r’ -w’~) exp( - Ar) dr (30) 0 where H - z is just
the displacement, h, and TV is given by
cr(n) = ~o(2c(n)2)“2-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
-
1492 GAMPALA et al.: CREEP BLUNTING OF CONTACTS
Monotap S-Lamha Power-law Creep Mechanism
0.50 ’ I I I I 0.4 0.5 0.6 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
g = A {D,,,, exp[ - Q,/RT]) FT 0
i ” (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 tita- nium alloy was
described by
8 = A % 0
‘exp(-QJRT)
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 indenta- tion-based models
in some cases overestimate, and in others underestimate, the
densification rate. Indenta- tion analysis underestimates the
blunting rate for
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 ma-
terials 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 con-
straint). 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 during densljication of monotape
Ceramic fibers in metal matrix composite mono- tapes are
susceptible to microbending and fracture during consolidation
processing [12]. During consoli- dation, bending occurs because the
asperity contact stress, uc, results in localized forces
distributed ran- domly along the length of the fibers. The
evolution of the bending and fracture has been modeled by consid-
ering a statistical distribution of unit cells, each consisting of
a segment of fiber undergoing three- point 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
Table 3. Creep narameters used for model medictions
Parameter. svmbol (units) cu 1491 Ti-14Al-2lNb 1151
Creep constant, A (b-l) stress exponent, 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
3.4 x 106 6.0 x 10” 4.8 2.5
197.0 285.0 140.0 - 0.12 T (“C)
2.0 x 10-5 1356
2.5 x 10-l’
-
GAMPALA et al.: CREEP BLUNTING OF CONTACTS 1493
2(z - y) represents the fiber deflection]. By approxi- mating
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)
2xra,( y - r)c(n)‘F(n, ( y - r)/r) 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 11 shows the cumulative number of
fractures
‘i .c 80- Y
c -
3 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 - T = 543K (0.4Tm)
7 z. 160 - u.
c -
8
120-
5 ti F 60-
IL
40 -
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 on the basis of
indentation and contact blunting (a)
Ti-24Al-11 Nb/SCS-6; (b) Cu/SCS-6.
during densification of composite materials with matrix
properties of either Cu (n = 4.8) or Ti-24Al- 11Nb (n = 2.5) and
reinforcements with SCS-6 (SIC) fiber properties (the fiber
properties given in Ref. [ 161). The model based on blunting
mechanics pre- dicts 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 n,f r (37)
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 den- sity, D:
a/r = [(D - Do)/(3(1 - ~oW12, 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 con-
servation of mass for a hemispherical contact sub- jected to
constrained uniaxial compression) gives
a/r =,(,)&(I -2) (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
+ 2&/3l’ + ‘in sin 4 d4 (39)
where l? and fi are the macroscopic deviatoric and dilatational
strain rates, 4 locates the angular pos- ition of a contact
relative to the (cylindrical) coordi- nate 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 co- efficients 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
co- efficient, C(n, D) [not to be confused with c(n) given in
equation (3)]
The relation between macroscopic stress and strain- rate can be
expressed in potential form so that C, = n/(n + i)awvjaE,j or
inversely, _Ei, = atr/az,, where C, is the applied mean stress,
& the applied effective stress and the lower bound estimate for
Y is
Y =z,A+z,B- 5 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 indepen- dent of
C,j and gij). Therefore the shape of the creep contours (i.e.
constant values of !P plotted in stress- space) is the same for
both models, but their magni- tudes at any point in stress-space
may differ significantly, depending on the value of the stress-ex-
ponent 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 values of K or C correspond
to increasing creep 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 I n = 2 tndentation overe*timates c \ blunting creep rate
1.4 A
I n = 10 tndentat~on underestimates
blunting creep rate
0.75 0.80 0.85 0.90
Relative density, 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).
2.5 r I I
Powder Stage I Consolidation Power-law Creep Mechanism
2.0 - a-
a”
.s E
1.5 - ” = 4.6 (Copper) /‘I p 5
‘Z z 1.0 E t
$ 0.5 n = 2.5 (Ti-24AI-11 Nb)
0.0 I I I I 0.65 0.70 0.75 0.60 0.65 0.90
Relative density, 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 single- contact behavior
(Fig. 6). As n increases, the soften- ing 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 conse- quence 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 1 (n + 1)/n fi = -2d,{2~(2/3)@+ n/n -n
I[( > *
+(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 underesti- mates 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).
6. CONCLUSIONS
The availability of models for analyzing indenta- tion 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 BLUNTING OF CONTACTS 1495
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 con- tacts) 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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