Acta mater. Vol. 45, No. 3, pp. 973-986, 1997
Copyright 0 1997 Acta Metallurgica Inc.
Pergamon
PII: S1359-6454(%)00242-X
Published by ElsevierScience Ltd
Printed in Great Britain. All rights reserved
1359X%54/97
$17.00+ 0.00
A FIBER DAMAGE MODEL FOR EARLY STAGE
CONSOLIDATION
OF METAL-COATED FIBERS
J. WARREN,?
D. M. ELZEY and H. N. G. WADLEY
Department of Materials Science and Engineering, School of Engineering and Applied Science,
University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 8 March 1996; accepted 27 June 1996)
Abstract-Recent
studies of the high temperature consolidation of titanium alloy coated c+alumina fiber
tows and BC monofilaments have both revealed the widespread occurrence of fiber bending and fracture
during early stage consolidation. This damage was shown to arise from the bending of unaligned fibers
during consolidation and was found to be affected by the mechanical behavior of the metal-metal contacts
at fiber crossovers. To predict the incidence of fiber fracture during early stage high temperature
consolidation, a time-temperature dependent micromechanical model incorporating the evolving contact
geometry and mechanical behavior of both the metal matrix and the ceramic fibers has been combined
with a statistical representation of crossovers in the pre-consolidated layup. The damage predictions are
found to compare favorably with experimental results. The model has subsequently been used to explore
the effects of fiber strength, matrix constitutive properties and the processing conditions upon the incidence
of fiber fracture. It reveals the existence of a temperature dependent pressurization rate below which
fracture is relatively unlikely. This critical pressure rate can be significantly increased by the ‘enhanced’
superplasticity of the initially nanocrystalline coating. Copyright 0 1997 Acta Metallurgica Inc.
1. INTRODUCTION
Continuous
fiber reinforced
titanium matrix composites (TMCs) possess combinations
of specific
modulus, strength, and creep resistance that are well
suited for a variety of aerospace applications
[l, 21.
However, the pursuit of many of these applications
is impeded by the high cost of TMCs. Significant
fractions of this cost originate in the very expensive
single monofilament
silicon carbide fibers used for
reinforcement and with the considerable difficulty of
synthesizing high quality composites from them [3].
To reduce the contribution
of the fiber to the
composite cost, relatively inexpensive (sol-gel synthesized) cc-alumina fiber tows (e.g. the 112 or 410,
12 pm fiber filament Nextel 610TM tow) are being
explored as an alternative TMC reinforcement [4,5].
One promising
approach
to composite
synthesis
using either fiber type involves first coating the fibers
with thin protective
layers to both reduce the
deleterious chemical reactions between the fiber and
the matrix alloy during subsequent high temperature
processing and promote a low fiber matrix sliding
stress. When sliding is able to occur at a relatively low
stress
(say
- 100 MPa),
relatively
weak stress
concentrations
develop in the fibers adjacent to fiber
breaks, and a global load sharing model predicted
strength can be achieved [5,6]. Recent work has
tPresent address: General Electric, Coolidge Laboratory,
4855 W. Electric Avenue, W. Milwaukee, WI 53219,
U.S.A.
explored several coating concepts for this including
sacrificial TiB2 coatings as well as more complex (and
more costly) diffusion inhibiting Nb/Y*O, duplex
layers [_5,71. To synthesize a composite from these
protected fibers, the fiber tows or monofilaments are
coated with the titanium matrix alloy using a physical
vapor deposition (PVD) process such as electron
beam evaporation [2,8-lo] or sputtering [ll, 121.The
alloy coated tows/monofilaments
are then packed
into a suitably shaped container and consolidated to
theoretical density using either hot isostatic or
vacuum hot pressing [ 131.
The consolidation process has been shown capable
of causing significant damage to the fibers, to the
matrix and to the engineered fiber-matrix interface
[5, 131. For example, if the cc-alumina fibers are not
totally protected, the extended high temperature
exposure during consolidation can result in local
chemical reactions between the fiber and matrix. This
can release a very large (matrix embrittling) flux of
oxygen as well as weaken the fiber [5]. In addition,
because the individual metal coated fibers within the
a-alumina tows (or the SIC monofilaments in a
random packing of metallized monofilaments) are
not perfectly aligned, individual fibers can bend
between crossover contacts and result in fiber fracture
during consolidation (Fig. 1) [13-151. The bending
and fracture are likely to reduce the effective strength
of the fiber bundle [3, 51 and thus degrade the
performance of the composite. The breaks also
provide an additional pathway for chemical reactions
913
e.*
#=--__
t
c’
.*’
--__
(4
_I’
:
’ .-
..*.
“---_____
F
L
Fig. 1. Fiber bending and fracture resulting from consolidation of misaligned metal coated fiber (PVD
Ti-6Al4V/a-alumina) aggregates: (a) polished section of a consolidated sample (HIP’ed 9OO”C,100 MPa,
4 h); (b) view of damaged fibers after partial removal (etching) of matrix; (c) a schematic cross-sectional
view of a typical metallized fiber tow.
WARREN
et al.:
CONSOLIDATION
between the matrix and the fibers that may further
embrittle the matrix.
Experimental studies of the fiber fracture have
shown that failure occurs early during consolidation
by excessive bending of crossing fiber segments that
span contacts with other fibers, Fig. 1 [13, 151. These
regions of fiber bending are established by the fiber
packing geometry. As densification progresses, large
stresses develop at the contacts between the crossing
fibers. While this leads to desirable (densifying)
inelastic matrix flow at the contacts, it also results in
elastic bend stresses in the fiber span. For some spans,
the stresses can exceed the (statistical) strength of the
fibers before sufficient inelastic matrix flow occurs to
infiltrate the void region between the fiber contacts
and eliminate further bending [13].
The solution to this problem involves reducing the
incidence of fiber crossovers (i.e. better fiber
alignment) and decreasing the matrix resistance to
inelastic flow. The latter can be accomplished if rapid
creep at low stress is promoted either through the use
of high temperatures or by exploiting the very small
grain size frequently created in PVD alloys when
deposited at intermediate temperatures [12]. The
initial grain size depends upon the conditions used for
vapor deposition (i.e. deposition rate, substrate
temperature, etc.) of the matrix. The microstructure
will subsequently coarsen at a rate that depends on
the thermal cycle used for consolidation [13, 151.
Since concurrent changes in relative density (which
affects the contact stress and the bend cell span
lengths), the matrix creep rate (due to rapid grain
growth) and fiber strength (associated with diffusion
through or dissolution of the protective fiber coating)
and fiber strength (associated with diffusion through
or dissolution
of the protective fiber coating)
accompanying consolidation, the fiber damage is
likely to be a quite complex function of the initial
matrix microstructure/fiber geometry and the consolidation process conditions.
Improved alignment of the metal-coated fibers
shifts the distribution of fiber bend spans to longer
lengths. The resulting improvement in processibility
would have to be weighed against the additional
effort of this approach, arising from the difficulty of
producing tows containing aligned filaments, or in
handling monolithic fibers and the need to control
residual stresses (caused by nonuniform
coating
thickness) during matrix deposition. Our approach
here is to explore processing strategies which would
allow the production of high quality composite
material using low cost (e.g. relatively poorly aligned)
fiber preforms.
The goal is to develop a preliminary model that
establishes the dominant
‘trends’ between fiber
fracture and the consolidation conditions for PVD
coated Nextel 610TM alumina fibers. In solving the
problem, the geometry and micromechanics
are
simplified so that reasonably tractable solutions can
be obtained. The model combines an analysis of the
OF METAL-COATED
FIBERS
915
bending of a representative span due to viscously
deforming contacts with a measured initial span
length distribution.
Time/temperature
dependent
creep properties for the microstructurally
evolving
matrix are then used to compute a time dependent
distribution of bend stresses within the layup during
the early stage consolidation (where most damage
occurs). These stresses are combined with Weibull
fracture statistics for the fiber to determine an overall
fiber fracture probability and thus, a fiber fracture
density (i.e number of fractures/meter of fiber). This
is compared with earlier experimental results [ 131.
The verified model is then used to identify the
trade-offs between pressurization rate and consolidation temperature that minimize the fracture problem.
The model predicts that provided good chemical
protection of the fibers can be achieved, acceptable
levels of fiber fracture can be accomplished during
densification of these tow-based materials. We find
this to be enabled by the unusually high creep rate of
the fine grain size PVD matrix alloy. The model is
equally applicable to the consolidation of metallized
SIC monofilament arrays and indicates that similar
consolidation strategies could also be applied there.
2. FIBER FRACTURE MODEL
The goal of consolidation is to cause the metal
matrix to laterally flow and fill the void regions in (see
Fig. l(c)), while avoiding fiber crossover contact
stresses high enough to cause a significant probability
of fracturing the alumina fiber (or SIC monofilament). The model is needed to explore how the extent
of fiber damage (the model output) depends upon the
packing of the metal coated fibers, the resistance of
the matrix to flow, the bend stiffness and strength of
the fibers, and the process conditions (i.e. the
temporal variation of the consolidation pressure and
the temperature). A similar statistical micromechanits approach was used by Elzey and Wadley [16] to
model the evolution of fiber damage during the
consolidation
of MMC monotapes produced by
induction coupled plasma spray deposition. In that
system, fiber microbending resulted from surface
roughness, a characteristic of the plasma sprayed
tape. In the case, as the MMC monotapes were
pressed together during consolidation,
localized
stresses developed where asperities contacted adjacent plies and caused fiber bending. While changes in
the matrix microstructure of plasma spray deposited
metals/alloys has not been incorporated
in the
previous models, this evolution cannot be ignored in
the present case of PVD matrix coated fibers.
2.1. Model formulation
Experiments have shown that the initial relative
densities of metal coated Nextel 610TMtow or SIC
monofilament
layups prior to consolidation
lie
between 0.45 and 0.55 [13, 151, significantly less than
the 0.9 1 relative density of a hexagonal close packing
976
WARREN et al.:
CONSOLIDATION
a) Idealized architecture
M$,~~,$$wo!$d
I
Fiber
\
Void
Z
b) Representative cell
-“F
L
Y
c) Cell deformation
Consolidation
pressure
Matrix plane
strain flow
Fig. 2. Idealized cellular solids architecture used to simulate
early stage densification and fiber fracture during consolidation processing.
of uniform diameter fibers. This arises from fiber
crossover caused by misalignment created during
synthesis of the tow (and its random packing for
consolidation), and fiber bending upon cooling after
metallization (due to non-uniform
metal coating
thickness). This creates a preconsolidation
fiber
architecture of the form shown schematically in
Fig. l(c).
Our objective is to predict the number of fiber
fractures which occur per meter of fiber length as
these metal coated fiber aggregates densify during
consolidation. We take a micromechanics approach
in which a unit cell is first identified whose behavior
can be used to respresent the overall aggregate. This
unit cell (Fig. 2(b)) is chosen to consist of a segment
of coated fiber and three matrix contacts with
neighboring fibers. The random nature of the
structure illustrated in Fig. l(c) (with fibers crossing
at various angles) leads to a type of cellular structure
characterized by a distribution of cell lengths. Since
the force required to deflect a fiber depends on the
fiber’s span length, a unique stress-strain response is
associated with each of the cells in the distribution.
OF METAL-COATED
FIBERS
The overall response will therefore be the sum of the
responses of the cell distribution.
The macroscopic structure of the aggregate can be
approximated by allowing the unit cells to repeat in
a three-dimensional
space, creating a ‘cellular’
structure similar to that discussed by Gibson and
Ashby [17]. Analysis of each unit cell predicts the
forces required for early stage densification and also
captures the interaction between the creeping matrix
and the (predominantly) elastic fibers that bend and
fracture.
Considerable
simplification
could be
achieved if the variously sized cells were able to be
replaced by the same number of identical (average)
cells (i.e. the usual micromechanics
approach).
However there exist two obstacles to this. Firstly, the
elastic-nonlinearly viscous behavior of a cell like that
shown in Fig. 2(b) results in a path-dependent
stress-strain response. An array of different span
length cells deforming in parallel cannot be modeled
using a single cell. Secondly, no single ‘average’ cell
could describe both the densification and fiber
fracture behavior since these two processes exhibit
different (nonlinear) dependencies on cell length
[16, 181.
Our approach is to therefore discretize the cell size
distribution, model the behavior of a finite number of
cells with different span lengths deforming in parallel,
weight each cell by its probability of occurrence in the
cell population, and obtain a weighted sum of the
responses. The resulting model structure in two
dimensions is shown in Fig. 2(a), which represents the
y-z plane in Fig. l(c). Analysis of the problem is
complicated for a general state of consolidation
stress. Rather than attempt a multiaxial constitutive
model for the overall behavior, we restrict the
development to uniaxial macroscopic deformations
(in the z-direction in Fig. l(c)). The resulting model
is expected to provide a reasonably
accurate
simulation of densification and fiber damage during
vacuum hot pressing (VHP) and to be approximately
valid for the HIPing of a sheet-like geometry where
most deformation occurs through the thickness.
The overall fiber fracture evolution
during
consolidation
is obtained by: (i) applying the
processing pressure (at the elevated processing
temperature) to a unit cell array; (ii) allowing a
consolidation strain to develop in each cell (with
stresses satisfying equilibrium with the applied stress);
(iii) determining the probability of fiber fracture in
each cell; (iv) weighting each cell by its relative
frequency; (v) summing the weighted fracture
probabilities of all the cells in the ensemble.
2.2. Macroscopic relations
Suppose a time dependent processing pressure,
P(t), is applied to a metallized fiber layup like that
shown in Fig. l(c). If the layup is contained in a die
(as for example in vacuum hot pressing), a
macroscopic strain occurs in the direction of the
applied stress (with the lateral strains being zero).
WARREN et al.:
CONSOLIDATION
Although local variations in uniaxial strains can
occur, we assume that macroscopically, isostrain
conditions apply (i.e. planes perpendicular to the
macroscopic strain remain plane). For such a uniaxial
deformation,
the macroscopic behavior can be
inferred from the response of any of the 01-z) planes
such as that shown in Fig. 2(a), provided it contains
a reasonably large (representative) number of unit
cells. The analysis can be further simplified to a single
row of cells if the length, L, of the plane under
consideration is much greater in the fiber direction
than the average cell length, again to ensure a
representative cell population. This condition being
satisfied, the analysis of just a single row of cells will
be representative of the behavior of the three-dimensional array.
Suppose a total of N cells exist in a row. If the
probability of finding a cell of length between 1 and
I+ dl is 4,(l) dl, then the distribution of cell lengths
can be described by a probability density function
(PDF), 4,(l), and the average cell length, 1, will be
defined by its first moment
If the cells are taken to be of width, && (where dr is
the fiber diameter and 19is a packing factor chosen
such that the cells are volume-filling
in the
x-direction, Fig. l), the average area occupied by the
cells is
OF METAL-COATED
FIBERS
where ho and D,, are the initial plane thickness (i.e. cell
height) and density, respectively. Since the plane is
assumed to deform under isostrain conditions, the
height of any particular unit cell, h, must be the same
as for all others; therefore h, = h. With the unit cell
height, h, thus related to the composite density, it can
be seen that the equilibrium equation (4) also
provides the overall constitutive response, i.e. a
relation between the applied stress and composite
density. With the applied force in equation (4) given
and the cell length distribution, 4,, measured [13], the
next step is to relate the force acting on a unit cell,
F,, to its length, 1 and height, h.
2.3. Unit cell analysis
The time-dependent deformation occurring at the
contact between two metal coated fibers which cross
at some angle during consolidation is a complicated
inelastic flow problem which has not as yet been
investigated. The unit cell, shown in Fig. 2(b), is
clearly an idealized representation of the geometry
present in the actual material. The contact has been
idealized as a rectangular asperity with dimensions,
z,y,&. As force is applied to the cell, the cell height,
h, decreases because of contact deformation (decrease
in asperity height, zC)and due to deflection, A, of the
fiber. From Fig. 2(b), it can be seen that the unit cell
height, h, depends on the extent of contact
deformation and fiber deflection, and is given by
h(t) = 22,(t) + df - A(t), which, when differentiated
with respect to time leads to
h = 22, -
The force acting on the cells whose area is given by
equation (2) is related to the applied stress by
F(ct) = P(t);l.
(3)
Enforcing equilibrium between the applied force and
the local forces acting on the cells (1c,) gives
F(t) =
Lq,(l) . F,(f, h, t) dl
s II
(4)
where h is the unit cell height.
The cell height is related to the relative density, D,
of the composite, which is a function of the applied
stress and can be expressed as D = G(P, t), Since all
planes (made up of unit cells) perpendicular to the
applied stress are identical, the relative density of any
one of these planes is the same as that of the
composite. The macroscopic relative density is then
related by conservation of volume (due to the
incompressibility of both fiber and matrix) to the
thickness (in the z-direction),
h,, of a single
(arbitrary) (x - y) plane subjected to constrained
uniaxial compression by
h
P
=!%!a
D
911
A.
(6)
The rate of change of contact height, i,, is
determined by the contact stress, a(t), and by the
constitutive response of the matrix. Since significant
pressure is normally applied only after the layup has
been heated into a regime of rapid creep, it is assumed
that the strain-rate of the matrix can be represented
by a power law constitutive relation for steady-state
creep. Since the matrix exhibits significant grain
coarsening during consolidation, we use a constitutive model that explicitly incorporates the grain size.
Warren et al. [12] found that for uniaxial deformation, the strain rate of the PVD matrix coating
could be well represented by
i =
&,
e-Qi(Rn.
!?
dp
where Q is an activation energy for creep, B,, is a
temperature independent creep parameter, n is the
creep stress exponent, d(T, t) is a temperature,
time-dependent function which describes the instantaneous matrix grain size (see Table l), p is a grain
size exponent, T the absolute temperature and
R( = 8.315 J/(mol K)) is the universal gas constant.
The contribution of plastic deformation and transient
creep are ignored in such an analysis. They effectively
reduce the deformation resistance of the contacts for
a fixed loading rate so the steady state creep
WARREN et al.: CONSOLIDATION
978
assumption equation (7) is likely to result in a
(conservative) overestimate of the predicted fiber
damage.
Approximating the uniaxial contact strain rate as
i = ie/zo equation (7) can be used to write the contact
deformation rate
Q”
OF METAL-COATED
FIBERS
The fiber is assumed to experience only elastic
deformations, although the elastic modulus, Er, is
treated as a function of temperature. The fiber
deflection (Fig. 2(c)) is related to the cell force (from
elementary beam theory) by
F, = k, A
(11)
where k, is the bend stiffness
Since the contact and the fiber are in series (i.e. the
applied force is transmitted through the contact to
the fiber), the force acting on both elements is just the
force applied to the cell. The stress acting over the
contact area, a,, is related to the force by cr = F,/u,.
As the contact deforms in the z-direction, incompressibility requires that lateral (x and y) deformations conserve volume. Again in the interest of
arriving at the simplest mathematical formulation
which preserves the essential physical phenomena,
lateral strains are considered to take place only in the
y-direction, Fig. 2(c). Conservation of volume then
leads to
yczc = y,z,
(9)
where y% and z,~ are the initial asperity width and
height, respectively. The contact area, given by
a, = y,(t)& can then be expressed in terms of z,
a, = y, 2 df.
k,(t) = s
(12)
s
where /$(t) is the length of the fiber span subject to
bending, which changes with time because of lateral
spreading of the contact. The bending span length is
expressed as IS(t) = I- ye(t) = I- yszs/ze(t). The
contribution of the metal coating to the fiber bend
stiffness will usually be small and has therefore not
been included in equations (11) and (12).
2.4. Composite densiJication model
The composite stress-strain response can now be
specified by the equilibrium equation (4), the contact
constitutive equation (8) and equation (6) for relating
the rate of change of cell height to the fiber deflection
rate. These can be expressed as a system of three,
nonlinear differential equations in F,, z, and h.
(10)
d L
P = Z o (q,(l) . Fc(f, h, 0) df
s
With equation (lo), equation (8) becomes an ordinary
differential equation in z,.
Table 1. Mechanical properties of the NEXTEL a-alumina fiber, the PVD and conventionally processed Ti-6Al-4V alloy.
a-Alumina fiber properties
Symbol
Diameter (pm)
Young’s modulus 1 (MPa)
Reference strength* (MPa)
Weibull modulus
dr
El
Temperature independent creep parameter
(m (MPa)-” s-i)
Activation energy for superplastic flow
(kJ mol-r)
Creep stress exponent
Grain size exponent
Grain growth exponent at 76O”Cp
Grain growth exponent at 840°C
Grain growth exponent at 900°C
Grain growth constant at 760”Ct (pm sp)
Grain growth constant at 840°C (pm SK’)
Grain growth constant at 900°C (pm s-‘)
Initial grain size at 760°C (pm)
Initial grain size at 840°C (pm)
Initial grain size at 900°C &m)
Creep stress exponent
Grain size exponent
Temperature independent creep parameter
(m (MPa)-” SK’)
Activation energy for superplastic flow
(kJ mol-I)
00
m
Value
12
390,000
2380
9
PVD Ti-6Al-4V matrix propertiesf
BO
Q
n
P
(I
a
:
k
:
1
Conventionally processed Ti-6Al-4Vf
n
P
BO
Q
tPrior to consolidation processing
$A11material parameters shown apply to equation (8) in the text.
gApplies to a grain growth relationship of the form, d[t] = do + kta.
0.00003
140
1.4
Ref.
[51
t:j
151
WI
WI
WI
1
0.24
0.20
0.20
0.14
0.23
0.23
0.11
0.20
0.50
1.67
0
19.9
153
[ii;
P21
WI
t:;\
WI
WI
tt;;
[2X 241
W241
P, 241
W241
(13)
WARREN
et al.:
CONSOLIDATION
OF METAL-COATED
979
FIBERS
in a time-dependent fiber survivability, Y(t), which
for an arbitrary cell is then given by [20]
”
Y.[t]=exp[-rci(,>-]
O<$,<l
for
cf > 0
(18)
where I0 is the reference length and CT,,
is the reference
strength of the fiber, m is its Weibull modulus and K
is a factor which is a measure of the uniformity of the
stress distribution in a beam of circular cross-section
[21] and is given by
Equation (13) is an integro-differential equation
obtained as the time derivative of equation (4).
Equation (14) is obtained from equation (8) with the
substitution, cr = FJac = Fcz,/y,z,,df, and equation
(1.5) was obtained by differentiating equation (6) and
making appropriate substitutions using equations
(8Hl2).
Their solutions are constrained by the
isostrain condition that requires the cell height, h, be
independent of cell length, 1. This is satisfied if all cells
deform at the same rate for all t > 0:
^
;/;=o.
(16)
Equations (13)( 16) provide a basis for calculating
the densification rate of an aggregate of misaligned,
metal coated fibers during consolidation at elevated
temperatures. The model requires constituent material parameters, BO, Q, p, n, Ef and df (which may
be treated as time- and/or temperature-dependent
quantities if data or models for their evolution are
available) and the fiber span length distribution, $,,
controlled by the process used to manufacture the
coated fibers and by the manner of their packing.
2.5. Fiber damage model
The overall deformation (as given by the cell
height, h(t)) for a given applied stress is obtained by
solution of equations (13)-( 16), which also yields the
contact heights, z,(t). The fiber’s deflection, A, within
any given cell of length, 1, is then found from
equation (6), which is subsequently used to determine
the stress in the fiber. Many different bending
geometries could occur. We calculate the maximum
tensile fiber stress (of) by analyzing a cylindrical,
elastic beam subjected to symmetric three-point
bending assuming fixed end constraints [19]
(17)
The time dependence of the stress arises from both
the time dependence of the deflection and the cell
length. The ceramic fibers used for metal matrix
composities typically exhibits time independent
fracture strengths in tension which can be described
by a Weibull distribution.
However, the time
dependence of the span length and fiber stress results
1
w=---=
L"J
(19)
where I is the gamma function. If the fibers are
chemically attacked, CJ~may become a function of
process temperature and time.
Finally, the number fraction of undamaged cells
during processing is the overall survivability, Y(t),
expressed as
L
Y(t) =
‘R(l)’ YY,(l,t) dl.
(20)
Since the cumulative probability of fracture for a
given cell is just 1 - Yy,,the overall damage, given by
the number of breaks per meter of fiber is
I.
~(0 . (1 - Yc(l, t)) dl
s0
P(f) =
L
(21)
with the average cell length, E, defined in equation
(1). With the densification response for an arbitrary
applied stress and temperature
cycle known
(equations (13)-(16)), equations (17)-(21) are sufficient to calculate the time-dependent accumulation of
damage. The only additional material data needed is
the distribution of fiber strengths characterized by the
Weibull parameters, m, u,, and lo.
3. NUMERICAL
IMPLEMENTATION
Solving the differential equations (13Hl6)
to
obtain the overall densification response for a given
applied stress yields the cell height, h(t), which is a
macroscopic quantity and is thus the same for all
cells, a distribution
of time dependent contact
heights, z,(Z, t), and a distribution of time dependent
cell forces, F,(I, t). From this, the distribution of fiber
deflections, A(!, t), can be found from equation (6),
which also evolves with time. Given the deflections,
the peak tensile stress in each deflected fiber segment
is determined from equation (17), which is then
980
WARREN et al.:
Consolidation pressure
CONSOLIDATION
OF METAL-COATED
p
iii = A.
P
Fig. 3. The macroscopic densification/damage behavior is
modeled as an ensemble of unit cells, each of which is
described by a single Maxwell element. Springs represent the
elastic deflection of the fibers, while the dashpots model the
viscous (power-law creep) deformation of contacts at fiber
crossovers.
inserted
into equation
(18)
probability
of survival within
FIBERS
to
determine
(23)
The 3N + 1 system of equations (equations (14), (15),
and (23) for each of the N cell types plus the
equilibrium equation (22)) were programmed using
MathematicaTM [22] and solved with an adaptive
stepsize, fourth order Runge-Kutta ODE integration
scheme.
The overall fiber survivability is obtained by
replacing the integral in equation (20) with a discrete
summation
Y(t) =
2“A$&
i=,
t).
(24)
Similarly, the discrete approximation
of the fiber
fracture density, p, obtained from equation (21) is
the
each fiber segment
(cell). Finally, the overall fiber damage is calculated
by integrating
the probabilities
of fracture
(1 - survivability) over all cells (equation (21)). A
numerical solution procedure has been developed as
described below.
3.2. Unit cell distribution
3.1. Discretization
The densificatiomdamage
model embodied in
equations (13x21)
is based on a continuous
distribution of cell lengths. A numerical implementation will be used in which c#@) is replaced by a
discrete distribution containing N unit cell lengths.
The composite unit cell behavior given by equations
(6x12) is that of a Maxwell element in which a linear
spring (bending fiber) is placed in series with a
nonlinear viscous dashpot (matrix creep at contacts).
The overall behavior can then be modeled as an array
of N Maxwell elements, all undergoing the same
uniaxial displacement rate, 8, Fig. 3. While all cells
are required to have the same height, the load they
each support as well as their fiber deflection and
contact deformation will all be different.
Replacing the integral in equation (13) with a
summation, thereby reducing the analysis to a finite
number of fiber bend cell lengths, gives
It has been assumed that a composite layup will
contain a distribution of cells like the ones described
in Section 2.1. The cell length distribution used for
this simulation
was determined
by a detailed
metallographic analysis of preconsolidated
specimens, Fig. 4 [13]. This is a very broad distribution
requiring many cells to be included in the analysis
and thus extensive computation. To simplify, we note
that when the matrix of consolidated samples was
dissolved by acid etching, roughly 60% of the fiber
segments in the sample were of length ~70 fiber
diameters, suggesting that the bending mechanism
responsible for fiber breaks is active over a
distribution of cell lengths much narrower than the
one depicted in Fig. 4 [ 131. The very long cell lengths
C
.G
‘i,
0.07
0.06
s
where N is the number of different cell lengths
considered (i.e. the number of bins into which the
entire range of cell lengths are divided), J; represents
the number fraction of cells of length, Z,,and &, is the
rate of change of contact force acting on a cell of
length, Zi. The contact height, ze, and cell height, h,
are still given for each cell type by equations (14) and
(15), respectively. In addition, the isostrain condition
(16) is satisfied by having the rate of change of height
of each cell be equal to that of the plane containing
all the cells, fi
iii
0.05
f
0.04
z’
0.03
0.02
0.01
0.00
I
i
.
loo
200
300
400
500
800
700
e ldf
Fig. 4. The measured distribution of unit cells (fiber bend
segment lengths) in a sample prior to consolidation.
WARREN
et al.:
CONSOLIDATION
OF METAL-COATED
981
FIBERS
Table 2. Physical dimensions of each unit cell type
Cell type
lldr
1
2
3
4
5
6
7
8
9
10
11
12
8.50
10.00
12.50
15.00
18.75
24.00
28.00
36.25
41 .oo
45.00
54.00
70.00
L(O)/dr
ywldr
shown in the distribution will likely interact with
nearest neighbor fibers during consolidation,
rearrange and form additional contacts so that few
fractured segments of this length are found after
consolidation. Thus experiments indicate (and our
subsequent model analysis reveals) that fiber segments of length, I> 7Odf, do not appear to be very
important. In addition, very short fiber segments, say
I < 4-6dr, must be excluded from the analysis since
they possess very high stiffness in bending and are
unlikely to fracture by a bending mechanism.
Therefore we truncate the distribution of cell lengths,
only cells within
the limits,
$I(09 consider
8.5dr < I< 7Odr and obtain
a computationally
efficient solution.
z=oidt
1.3
1.5
1.9
2.3
2.8
3.6
4.2
5.4
6.1
6.8
8.1
10.5
6.2
8.5
10.6
12.7
16.0
20.4
23.8
30.8
34.9
38.2
45.9
59.5
survivability (equation (18)) and the cumulative
ensemble survivability (equation (24)).
The model in its present form can be used for
predicting early stage fiber damage, to densities
D m 0.75. This corresponds to the density value when
some of the fiber segments deflect beyond the
geometric bounds of their respective cells. The elastic
deflection of fibers into the void space between
crossover contacts is interpreted here as stage 1
consolidation behavior. If needed, a subsequent stage
2 consolidation model could treat the formation of
new contracts along the bending fiber segment
resulting in cell division.
4. SIMULATION
3.3. Process simulation methodology
In practice, consolidation process schedules are
usually used in which temperature is first increased,
followed by the application of pressure once the
temperature has reached a constant (soak) value.
Although more complicated cycles could be used as
input, the isothermal process schedule is adequate for
exploring the factors affecting fiber damage evolution. The fiber fracture simulations were carried out
in the following steps: (1) selection of a consolidation
temperature, T, and pressurization cycle, P(t); (2)
determination of the lengths of the cell types, l,, and
the number fraction of each type in the ensemble, f;,
(from experimental
data); (3) selection of the
appropriate matrix and fiber properties (i.e. the
modulus and Weibull parameters for the fiber and
the temperature dependent matrix creep and grain
parameters); (4) simultaneously solving the loading
rate relationship, equation (22) and, for each cell
type, the three governing differential equations that
determine their coupled, micromechanical response,
equations (14), (15) and (23) (the contact height
relationship, the cell height relationship and the
isostrain condition, respectively). The simulations
presented below are based on the simultaneous
output response of 12 unit cell types ranging in
length from 8.5dr to 70df (see Table 2). A system of
37 (3N+ l), first order, ordinary
differential
equations were solved to determine, for each cell type,
the fiber bending stress (equation
(17)), fiber
1;
0.046
0.039
0.036
0.072
0.128
0.056
0.171
0.020
0.095
0.027
0.023
0.283
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
OF FIBER FRACTURE
To investigate the model validity, a detailed
simulation has been performed for two previously
reported process cycles, designated VHP-1 and -2
(Fig. 5) conducted at a consolidation temperature of
840°C [13]. For these experiments, the metal coating
around the fiber was a uniform 4.2 pm in thickness
for a resultant matrix volume fraction of 0.65. The
cycle VHP-1 had been designed (using this model) to
have a loading rate that would not significantly
fracture fibers. The VHP-2 cycle used a more rapid
loading rate, typical of current consolidation
practice. The fiber fracture densities were measured
experimentally for both tests and are described in
IO
840%
I
I
I
/I
Time
(set)
Fig. 5. Vacuum hot press consolidation
process cycles:
VHP-2 is typical of conditions used to consolidate metallic
powders, while VHP-1 is a cycle designed for consolidation
of metal coated fibers.
982
WARREN
Time
et al.:
CONSOLIDATION
(set)
Fig. 6. Calculated fiber bending stresses developed during
the (a) VHP-2 and (b) VHP-1 process cycles.
[13]. The initial and final densities of the specimen in
each simulation were identical (0.48 and 0.73),
respectively.
The initial contact height, z;, = 0.7dr (8.4 pm),
represented the initial thickness of matrix material
located between adjacent crossing fibers. An initial
contact length of YiO= 0.151,was selected to obtain an
initial cell density of 0.50 which was similar to the
initial packing density of the preconsolidated
specimen. The grain size dependent, constitutive
properties of the matrix material were those
measured for the superplastic PVD Ti-6A14V alloy
coating and are given in Table 1 [12]. The measured
mechanical properties of the fiber are also given in the
table. It has been assumed for each simulation that
the fiber reference strength, rrO, was a function of
temperature only [5]. This is equivalent to assuming
the fiber remains fully protected from the matrix.
Consider first the bending stresses generated in the
fibers due to the VHP-2 simulation shown in
Fig. 6(a). Since each cell in the ensemble experiences
the same compressive strain (i.e. the isostrain
assumption imposed by equation (23)), the applied
processing load is supported initially by the stiffest
cells (the cell types of length 8.5dr and 1Odr in the
figure). These relatively short, stiff cells contain fiber
spans that do not readily deflect resulting in relatively
low bending stresses in the fibers. The contact stresses
developed in these cell types are consequently large
and result in extensive viscoplastic contact strain,
densification of the cell, and thus rapid additional
stiffening. As the shorter cells stiffen, they become
progressively capable of supporting larger loads
OF METAL-COATED
FIBERS
without breaking the fibers they contain. The longest
cells (of length 54df and 70df in Fig. 6(a)) have a low
initial bending stiffness, allowing the fibers to deflect
readily when loaded. At the start of the process cycle
these long cells support virtually no load and
experience a low fiber bending stress. As the process
pressure increases, the load in the long cells begins to
increase but at a much slower rate than other shorter
cells in the population. The longest and shortest cells
in the ensemble have the lowest bend stresses,
Fig. 6(a) and are therefore least likely to contain
breaks Fig. 7(a).
Reducing the loading rate decreases the densification rate and thus results in smaller cell contact
forces, reduced fiber bend stresses and less fiber
fracture. For example, in the VHP-1 simulation, the
processing pressure was applied much more slowly,
allowing significant time for creep densification to
occur. The maximum fiber bending stresses predicted
using VHP-1 (shown in Fig. 6(b)) are seen to be less
than half the bending stresses predicted using the
VHP-2 pressure cycle and resulted in a significantly
higher fiber survivability, Fig. 7(b). The nonlinearity
of the fiber strength distribution amplifies this. The
ensemble survivability (i.e. the cumulative number
fraction of cells containing breaks) indicates a tenfold
reduction in fractures, Fig. 8. Each figure shows the
instantaneous
number fraction of cells in the
ensemble likely to contain breaks. The VHP-2
simulation
predicts approximately
48% of the
ensemble members are likely to contain breaks after
processing to a final relative density of 0.75 compared
to 2% for the VHP-1 simulation. This equates to 950
0.6
a0
60
1 .o
$
0.8 -
.s
;
0.6 -
f/d,=41
-/
.r
2 0.4 z
=
0al 0.2 0.0
’
0
I
200
I
400
Time
I
I
I
600
800
1000
1
(set)
Fig. 7. Fiber survivability
for various fiber bend (unit cell)
lengths for the (a) VHP-2 and (b) VHP-1 process cycles.
CONSOLIDATION
WARREN et al.:
OF METAL-COATED
983
FIBERS
0.7
0.6
0.5
Fiber reference strength (GPa) = 1.2
0.4 1
0
I
40
I
20
1
60
1
60
400
200
o9
.c
(b) VHP-I
0.7 t
0.41
0
1
I
200
I
600
I
400
Time
I
800
I
1000
(set)
Fig. 8. Cumulative number fraction of cells containing
breaks for (a) VHP-2 and (b) VHP-1 process cycles. When
multiplied by the number of cells, the VHP-1 cycle is
predicted to result in 88 fractures/m (120/m measured) while
VHP-2 was predicted to have 950 breaks/m (compared with
600/m measured).
fiber fractures per meter of fiber in
VHP-2
and VHP-1, respectively and compares
relatively favorably with the 600 and 120 measured
breaks per meter for VHP-2 and VHP-1 [13].
The model appears to overpredict the degree of
damage caused by the process cycle VHP-2. The
discrepancy may be due, in part, to the onset of stage
2 deformation
early in the densification cycle
resulting in limited bending fractures, and neglect of
the deformation associated with both matrix plasticity and transient creep. These reduce the deformation resistance of the contacts and are likely to
result in less severe fiber deflection. The model also
and 88 predicted
somewhat underpredicted the damage caused by the
VHP-1 processing cycle. The inconsistency may be
due to a gradual reduction in the fiber reference
strength as a result of chemical attack by the matrix.
Fibers removed from consolidated specimens by acid
etching of the surrounding matrix exhibit surface
pitting, indicative of reactivity with the matrix [13].
The effect of fiber reference strength is considered
further in Section 5.
5. THE EFFECT OF PRESSURIZATION
1.0,
10 MPalh
150 MPalh
100
200
400
300
Time
500
600
(set)
Fig. 9. Cumulative overall survivability (i.e. processibility)
for four constant process pressurization rates at a
consolidation temperature of 840°C and a final relative
density of 0.75.
RATE
From the results presented above, fiber failure
during consolidation is seen to be sensitive to the rate
of loading which governs the overall densification
rate, local contact pressures, and thus fiber bend
stresses. Figure 9 shows the predicted cumulative
survivability (i.e. the processibility) for four constant
pressurization rates of 150, 100,50, and 10 MPa/h, all
imposed at 840°C. The initial and final relative
densities of each simulation were identical (0.48 and
0.75, respectively), and the material properties of the
matrix and the fiber were those given in Table 1. The
figure shows that rapid pressurization rates result in
extensive fiber fracture.
n
0
1000
(set)
Fig. 10. The influence of the fiber’s reference strength on
cumulative overall fiber survivability for the VHP-1 process
cycle.
Simulation: S8 fractures/m
Experiment: 120 fracturBS/m
0.8
800
600
Time
*
.z?
x
0.9 t
.$
0.7 -
5
0.6 -
ii
E
E
0.4 -
$
0.3 -
w
L(
\’
Critfcal pressurization
rates for no damage
J,‘J
I
0.8 -
0.5 -
0.2 1
1
I
10
Pressure ramp
I
100
(MPa/h)
Fig. 11. Cumulative overall fiber survivability as a function
of (constant) pressurization rate for selected fiber reference
strengths.
984
WARREN et al.:
i.o.,,,.-_P
2
2
-\
VHP-1
PVD alloy
‘\
D=
‘1
\
0.9 -
2
$
,
‘\
T = 040%
E
z
cd
CONSOLIDATION
Cmmtlonal
\
\
\
\
0.75
\
“S
SUperplaStiC:
0.8 -
TI-BAI-4Valloy
-
0~0.72
Iii
0.7
0
1
I
200
400
I
I
800
600
Time (set)
I
I
1000
1200
Fig. 12. Processibility (i.e. fiber survivability) is enhanced
by using the PVD Ti-6Al4V matrix due to its ultrafine
grain size which gives rise to enhanced superplastic
behavior.
We can use the densification/damage
model to
explore the influence of the fiber’s reference strength
on the susceptibility
to fiber failure, as shown in
Fig. 10. The cumulative ensemble fiber survivability
for the process cycle VHP-1 is shown as a function
of time for various fiber reference strengths. (Here,
the cell distribution, the mechanical properties of the
fiber, and the visco-plastic properties of the matrix
are unchanged.) In Fig. 11 the cumulative fiber
survivability for a range of fiber reference strengths,
eO,is shown as a function of (constant) pressurization
rate. We note the existence of a critical pressurization
rate below which negligible fiber damage occurs. This
pressurization rate is a strong function of the fiber
strength (doubling the strength allows a ten-fold
increase in pressurization rate). From this it is clear
that: (1) high fiber reference strengths are necessary
to avoid fiber fractures (and efforts to develop
stronger fibers will be beneficial from a processibility
standpoint), (2) fiber damage is very difficult to avoid
with the fibers used in this simulation unless low
consolidation loading rates are maintained during the
early stages of densification. Finally, for a fixed
pressurization rate, a 20% drop in the fiber reference
strength (from 2.0 to 1.6 GPa) results in a 20%
increase in the number of fractures, again emphasizing the need for high strength, damage resistant
fibers.
OF METAL-COATED
The constitutive properties of a conventionally
processed Ti-6Al-4V
alloy in the 750-900°C
temperature range can be determined from the
combined experimental data of Arieli et al. [23] and
Pilling et al. [24] (see Table 1). Two separate
simulations using the VHP- 1 process conditions were
then conducted; one with a conventional (superplastic) Ti-6Al4V
alloy and the other with the PVD
alloy. The cumulative fiber survivability for each
simulation are compared (Fig. 12) together with the
final density reached. The figure shows that the PVD
matrix alloy achieved a higher density in a shorter
time and with significantly less fiber damage. This
arises from the submicron grain size of the PVD alloy
since the strain-rate varies inversely with grain size
and is significantly enhanced for the PVD microstructure [12]. Although rapid coarsening accompanies
consolidation at 840°C [12], the grain size of the PVD
alloy still remains about a factor five less than that of
the conventionally processed material.
Critical pressurization
rates for no damage
I * I111111 ,
9 0.9 0
% 0.0 .g
’
0.7 -
z
0.6 -
al
E
E
0.4 -
w3
WITH A PVD MATRIX
The process simulations above indicate that fiber
damage can practically be eliminated even in poorly
aligned fiber tows if optimal process conditions are
chosen. However, the flow properties of the metallic
matrix, which determine the relative ease of contact
flow (leading to densification)
or fiber deflection
(leading to fracture), are clearly very important. It is
informative to substitute the constitutive response of
a conventional Ti-6A1-4V alloy for that of the PVD
matrix and to explore the extent to which the
nanocrystalline
structure
of the vapor deposited
matrix affects processibility.
‘\
‘\ ‘\
‘\
‘\
‘\
‘\
Cwwentional superptasti~*‘.,
0.5 -
0.3 - (a) T= 760°C
0.2.
* ~~‘~~‘~(
1.0
0.1
’ “‘,a,,’
’ “‘~~*~’
10.0
100.0
1.0
9 0.9
E
z
0.8
.g
0.7
3’
0.6
a
B
E
0.5
158
0.3
Conventional supefplastic~‘~,~
0.4
0.2
-
(b) T=
040°C
1
I
I
10
100
1.0
*
6. PROCESSING
FIBERS
0.9
z
0
0.0
.g
0.7
’7
0.6
a,
B
5
0.5
g
0.3
W
0.4
0.2
1
(0)
T= 900°C
1
I
10
100
Pressure ramp
(MPa/h)
Fig. 13. Overall survivability as a function of pressurization
rate for three different processing temperatures. The two
matrices are comparable at the highest processing
temperature due to rapid grain growth in the PVD matrix.
WARREN et al.:
CONSOLIDATION
E
14
g
TMAI-4V
12- D= 0.75
1
a
3
.$
'C
0
1
lo-
E
In
8?
FIBERS
985
is minimized
;
a
OF METAL-COATED
P"D/
I':
,'
,'
I'
a'
Conventional
supeiplastic
*’
__.*
2-
___--______----750
800
_
when the pressurization rates are low
and the processing temperature is high, (conditions
favoring matrix flow at contacts as opposed to fiber
bending). It must be kept in mind that although
consolidation temperatures promote matrix deformation, the accelerated reaction kinetics at the
fiber-matrix interface may result in unacceptable loss
of fiber strength unless good protective coatings have
been previously applied to the fibers.
__-'
o
I
850
Temperature
900
(“C)
Fig. 14. Critical pressurization rate (defined as the rate for
which no damage occurs) increases with processing
temperature and is a factor of 2-10 greater for the PVD
TiAAl4V
matrix.
In Fig. 13(aHc) the ‘processibility’ of the PVD
alloy is compared to that of the conventional alloy for
a range of consolidation temperatures and pressurization rates. The model predicts that improved
processing behavior can be achieved by using the
PVD matrix material, which readily creeps under the
consolidation conditions simulated here. The critical
pressurization rate for avoiding damage also improves. In Fig. 14 the critical pressurization rate for
both the PVD and the conventional processed alloy,
is plotted as a function of the consolidation
temperature.
The figure again shows that the
enhanced creep behavior of the PVD alloy allows
significantly higher pressurization rates to be used
during consolidation. As the processing temperature
is increased, the difference between conventional and
PVD alloy processibility disappears due to concurrent grain growth in the PVD material.
The effects of loading rate and consolidation
temperature are summarized in Fig. 15. Fiber damage
+----
Increasingdamage
7. CONCLUSIONS
A micromechanical
model for predicting the extent
of fiber damage during the elevated temperature
consolidation
of PVD metallized fibers has been
developed and applied to the consolidation of PVD
metallized NEXTEL 610 a-alumina tows. The model
incorporated creep contact deformation, elastic fiber
bending/fracture and grain growth during consolidation. It predicts levels of damage (and a dependence
on loading rate) which are similar to those of the
experimental results. The model has been used to
investigate the sensitivity of the damage to the
consolidation processing conditions and fiber/matrix
properties. It has been found that the rapid
pressurization rates typically employed to consolidate
metal powders are unsuitable for consolidating metal
coated ceramic fiber composites and result in
extensive fiber fracture even when the compacts are
fully heated before application of pressure. The
simulations indicate that fiber damage can be avoided
by increasing the temperature and lowering the
pressurization (densification) rate during the early
stages of consolidation: the optimal conditions will
depend on the fiber’s strength, its protective coating
integrity and the elevated temperature creep properties of the matrix alloy. A critical pressurization rate
has been found, below which damage is unlikely. The
relatively low amount of fiber fracture predicted for
this system arises from the enhanced superplasticity
of the PVD alloy matrix.
Acknowledgements-The
authors would like to thank
Messrs H. Deve, J. Storer, R. Kieschke, and P. DeBruzzi of
the 3M Metal Matrix Composites Center for their advice
and assistance. This work has been supported by the
Defense Advanced Research Projects Agency (W. Barker,
Program Manager) and the National Aeronautics and Space
Administration (D. Brewer, Program Monitor) and the
DARPA URI through UCSB.
I
800
Temperature
I
850
(“C)
Fig. 15. Contours of constant overall fiber survivability as
a function of consolidation processing conditions. Low
pressurization rates and higher temperatures favor matrix
flow at contacts rather than fiber deflection (and fracture).
REFERENCES
1. T. M. F. Ronald, Adv. Mater. Proc. 6, 29 (1989).
2. P. G. Partridge and C. M. Ward-Close, Intern. Mater.
Rev. 38 1 (1993).
3. P. Cantonwine and H. E. Deve, in Recent Advances in
Titanium Metal Matrix Composites (edited by F. H.
Froes and J. Storer), p. 201. TMS, Warrendale,
(1995).
PA
986
WARREN et al.:
CONSOLIDATION
4. D. M. Wilson, D. C. Lueneburg and S. L. Lieder, Cer.
Engng Sci. Proc. 14, 609 (1993).
5. R. R. Kieschke, H. E. Deve, C. McCullough and C. .I.
Griffin, in Recent Advances in Titanium Metal Matrix
Composites (edited by F. H. Froes and J. Storer), p. 3.
TMS. Warrendale. PA (1995).
6. H. Y: He, A. G. Evans and W. A. Curtin, Acta metall.
mater. 41, 871 (1993).
I. D. F. Paul, K. D. Childs and W. F. Stickle, in Recent
Advances in Titanium Metal Matrix Composites (edited
by F. H. Froes and J. Storer), p. 167. TMS, Warrendale,
PA (1995).
8. C. M. Ward-Close and P. G. Partridge, J. Mater. Sci.
25, 4315 (1990).
9. C. M. Ward-Close and C. Loader. in Recent Advances
in Titanium Metal Matrix Composities (edited by F. H.
Froes and J. Storer), p. 19. TMS, Warrendale, PA
(1995).
10. J. F. Groves and H. N. G. Wadley, Camp. Engng, in
press (1996).
11. C. McCullough, J. Storer and L. V. Berzins, in Recent
Advances in Titanium Metal Matrix Composites (edited
by F. H. Froes and J. Storer), p. 259. TMS, Warrendale,
PA (1995).
12. J. Warren, L. M. Hsiung and H. N. G. Wadley, Acta
metall. mater. 43, 2773 (1995).
13. J. Warren, D. M. Elzey and H. N. G. Wadley, Acta
metall. mater. 43, 3605 (1995).
OF METAL-COATED
FIBERS
14. J. F. Groves, D. M. Elzey and H. N. G. Wadley, Acta
metall. mater. 42, 2089 (1994).
15. H. E. Deve, D. M. Elzey, J. Warren and H. N. G.
Wadley, in Advanced Structural Fiber Composites
(edited by P. Vincenzini), p. 313. Techna, Florence, Italy
(1995).
16. D. M. Elzey and H. N. G. Wadley, Acta metall. mater.
42, 3997 (1994).
17. L. J. Gibson and M. F. Ashby, in Cellular Solids,
Structure & Properties, p. 143. Pergamon Press, Oxford
(1988).
18. D. M. Elzey and H. N. G. Wadley, Acta metall. muter.
41, 2297 (1993).
19. T. Baumeister, E. A. Avalone and T. Baumeister III
(editors), Mark’s Standard Handbook for Mechanical
Engineers, Chap. 5, p. 24. McGraw-Hill, New York
(1978).
20. W. Weibull, Trans. ASME: J. appl. Mech., p. 293, Sept.
(1951).
21. A Seimers, R. L. Mehan and H. Moran, J. mater. Sci.
23, 1329 (1988).
22. Wolfram Research Inc., Muthematica, version 2.2,
Wolfram Research Inc., Champaign, IL (1992).
23. A. Arieli and A. Rosen, Metall. Trans A 8, 1591
(1977).
24. J. Pilling, D. W. Livesey, J. B. Hawkyard and N. Ridley,
Metal. Sci. 18, 117 (1984).