On the tensile properties of a fiber reinforced titanium matrix composite—I. Unnotched behavior

Pergamon

Aeta metall, mater. Vol.42, No. 10, pp. 3443-3450, 1994 Copyright © 1994ElsevierScienceLtd 0956-7151(94)E0135-4 Printed in Great Britain.All rights reserved 0956-7151/94$7.00+ 0.00

ON THE TENSILE PROPERTIES OF A FIBER REINFORCED TITANIUM MATRIX COMPOSITE--I. U N N O T C H E D BEHAVIOR C. H. WEBER, X. CHEN, S. J. CONNELL and F. W. ZOK Materials Department, University of California, Santa Barbara, CA 93106, U.S.A. (Received 19 January 1994)

Abstract--An investigation of the ultimate tensile strength and fracture strain of a fiber-reinforced Ti-matrix composite has been conducted. Comparisons have been made between experimental measurements and predictions of two micromechanical models: one assumes that the fibers behave independently of the matrix, i.e. as in a dry fiber bundle, and the other assumes frictional coupling between the fibers and the matrix, characterized by a constant interfacial sliding stress. To conduct such comparisons, a number of constituent properties have been measured, including the fiber strength distribution, the thermal residual stress and the interfacial sliding stress. In addition, the effects of gauge length on the tensile properties of the composite have been studied. The comparisons indicate that the model prediction based on frictional coupling provide a good representation of the experimental results. In contrast, predictions based on the dry fiber bundle approach strongly underestimate both the ultimate strength and the fracture strain and predict a gauge length dependence that is inconsistent with the experiments.

1. INTRODUCTION

results and comparisons with model predictions are

There has been considerable interest in fiberreinforced Ti matrix composites for use in advanced aerospace propulsion systems, motivated by the attractive stiffness, strength and creep resistance characteristics of this class of composite at low and intermediate temperatures (to ~ 600°C) [1]. The present article focuses on the room temperature tensile properties of a unidirectionally reinforced composite, A study of the notched strength characteristics of this material is presented in a companion paper [2]. In unidirectional materials, the tensile response is characterized by two, approximately linear regimes [2-4]. In the first, the material is elastic, with a modulus given by the rule-of-mixtures. In the second, the matrix is yielded whereas the fibers remain essentially elastic. In the latter regime, fiber fracture also occurs, resulting in a slight deviation from linearity and ultimately causing composite fracture. The main objective of this work is to assess the utility of micromechanical models in predicting both the ultimate tensile strength (UTS) and the fracture strain of this class of composite, The paper is organized in the following way. Section 2 reviews the existing models pertaining to the tensile properties of unidirectional MMCs. The models identify the relevant constituent properties that need to be measured in order to rationalize the properties of the composite and assess the utility of the models. Section 3 describes the various experimental measurements made in this study, with the results being presented in Section 4. Analysis of the

presented in Section 5. 2. THEORETICAL BACKGROUND The axial tensile stress-strain response, trc(E), of a unidirectional MMC can be written as ac (E) = fo'f (E) + (l --j00"m (6), (l) where f is the fiber volume fraction, and at(e) and am(E) represent the volume-average axial stresses in the fiber and the matrix, respectively. If both phases respond elastically, then a~(E) = Ere

(2a)

and t~m(E) = Eme, (2b) whereupon the composite response becomes linear, with a modulus E = f E f + (1 - f ) E m,

(3)

with Efand Em being the Young's moduli oftbe fibers and the matrix, respectively. In the absence of residual stress, non-linearity in the stress-strain curve occurs at a strain equal to the matrix yield strain, EYm.Assuming the matrix to be elastic-perfectly plastic, the subsequent tensile response (E > Ey) is given by at(E) =ftrf(E)+ (1 -f)trYm, where cr~ is the matrix yield stress (=EmE~).

3443

(4)

WEBER et al.: PROPERTIES OF A Ti MATRIX COMPOSITE--I

3444

The fiber contribution, of(e), is governed by the statistical distribution in strengths generally c h a r a c - ~ 0 terized by the Weibull function [5] ~

-L{°'~m , L0 \o0}

Pf = 1 -- exp - - / - - /

(5)

where Pf is the cumulative failure probability, a is the tensile stress acting on the fiber, L is the fiber length, % and L0 are reference values of stress and length, and m is the Weibull modulus. In some cases, the in situ strength characteristics of the fibers differ from those of the pristine fibers and therfore require independent measurement following composite consolidation. This can be achieved by extracting fibers from the composite and measuring their tensile strengths [4, 6, 7]. Moreover, it has been recognized that the fiber bundle within a composite behaves differently from that of a dry fiber bundle, a result of the sliding resistance of the fiber/matrix interface[8-11]. The coupling between the fibers and matrix in combination with the fiber strength characteristics represent dominant features in the fiber response, Traditionally, the approach to evaluating of(E) has been based on the behavior of a dry fiber bundle, assuming that no coupling exists between the fibers and the matrix (the so-called "rule-of-mixtures" ap-

101 :

~ ~ = l 10 "

~ _o ~

l

I llllll

I

lllnll

I

l

l llllll

I

1

l llllll

I

l

llllllll

l

llllll

46 2

10°

~, tr. 1°~10"4

10"a

10"a

10"1

100

Gauge Length, L/L°

101

10a

Fig. 1. Trends in normalized tensile strength, au/f%, with gauge length, 1/lo, and Weibull modulus, m, for a y = 0 based on dry fiber bundle behavior. shear stress, z, acting along the debonded interface. Such debonding occurs in regions adjacent to the fiber failure sites and leads to the development of shear tractions at the fiber ends, allowing stress to be transferred from the matrix to the fibers. The length over which this transfer occurs is the slip length, d, given by d = a_fiR 2z '

(10)

proach) [12]. In this case, the fiber response is simply

where a is the tensile stress acting on the fiber when

of(E) = (1 -Pf)EEf. (6) Combining this result with equation (5) and recognizing that o = EEr yields the result

it failed and R is the fiber radius. Outside of this region, the fiber stress remains unchanged. Upon further loading, the fiber may fail at numerous other points along its length, with each failure event occurring essentially independently of all otherst. Consequently, the stress-strain response of a fiber bundle embedded within a ductile matrix is essentially identical to that of a single fiber embedded in the same matrix [9, 10]. The tensile response of a fragmenting, embedded fiber can be expressed as

Or(e) ( ~ ) exp - L ( E E f y ' (7) % = ~ \ ao/ ' The composite tensile strength ou and the corresponding fracture strain Eu are evaluated by setting do¢(E)dE=

f dof(E) dE = 0,

(8)

whereupon

EEl

[ \lmeLI -l/m + (1 --f)OYm Ou =fa0\ L0 /

Of(E) = EEf(1 -- ~) + ~ ~,

and

/'mL'~-l/m Eu = °°Ef| - -L0 l\]

"

(1 1)

(9a)

(9b)

Trends in the normalized strength, ou/fao, with gauge length, L/Lo, and Weibull modulus, m, are plotted in Fig. 1. A key feature predicted by this model is the reduction in strength with increased gauge length, Recently, models have been developed to account for the frictional coupling between the fibers and the matrix and its effect on the statistics associated with fiber fracture [9-11]. At the simplest level, the coupling can be characterized by a constant interfacial fProvided the fragment length remains large in comparison to the slip length.

where ~ is the fractional length of the fiber contained within the slip zones adjacent to fiber breaks. The factor of 2 in equation (11) arises because the average fiber stress within the slipped region is one-half of that in the unslipped regions (provided that none of the fragments are shorter than 2d). The parameter can be interpreted as the cumulative fiber failure probability within a gauge length, l = 2d, at a stress, a. Consequently - 2d/" a ' ~ " = Pf = 1 - exp ~ ~),

(12)

which, combined with equation (10), gives ~ = 1 --exp(°R~(a---~ m .

\zLo,]\%,]

(13)

et al.:

WEBER

PROPERTIES OF A Ti MATRIX COMPOSITE--I

This result can be re-written as

(a) 0,9

e=l-exp-

-\0"

(14) */

if,=

(15)

0.8

~ •

= "

the Taylor series expansion e x p - - ~**

~1--

\o',/

l

l

I

I

I

I

~

U B o u n dp

0.5

~

f

i

r

I

4

I

" Lower Bound (Eqn. 22)

J

/

e

~

J 0.6

p

l

6

L

I

i,

,

. . . . .

8

I

10

I

I

I

I

t

12

14

(b)

(17)

1.0

. . . . .

UpperBound (Eqn. 20)

0.9

~

0.a

to

._~ ~n

0.7

(18)

~

0.6

The maximum average stress trB that the fragmenting fiber can support is evaluated by setting dtrr/dE = 0, whereupon

"

0.s

where 2 is the normalized strain

2 = EEr/a,.

or.

s

(16)

(7,

cr~

I

~

'

Combining this result with equations (11) and (14) yields the stress-strain response [9, 10] O'f(E) = /~,[1 -- l~m + 1],

I

~

0.4

/oVl --

I

="

Provided the argument within the exponential term in equation (14) is sufficiently small compared with unity, the term can be approximated by the first two in

l

6 ~

where a , is a characteristic stress defined by

terms

3445

= { 2 X~ll(m+l){m + 1"] \-2~m}

\-m--~]'

'

I

4

i

I

6

l

I

s

i

I

~o

i

~2

~4

Weibul, Modulus, m

(19)

Fig. 2. Predictions of the upper and lower bounds to (a) the

fracture stress and (b) the fracture strain, resulting from the fiber fragmentation model.

with a corresponding strain of

tr,(

0.4

Bound

2 ~,/(m+,) (20)

differing from equation (11) by the contribution

Trends in both the ultimate strength and the failure strain with Weibull modulus predicted by this model are presented in Fig. 2t.

EB= ~ k , ~ - 2 }

associated with the slipped region (EEr~/2). The local stress maximum is again evaluated by setting daf/dE = 0, whereupon the fracture stress becomes

It should be recognized that equations (19) and (20) respresent upper bound estimates of the UTS and fracture strain. This is a consequence of the implicit

a~ [ 1 ~l/(m+l)[m -~-1\ -/ / l / or, = k2~-mm/ \-m-~/"

assumption that all volume elements within the cornposite are equivalent. However, for statistical reasons, the process of fiber fragmentation may occur more rapidly in some regions than in others, leading to strain localization prior to the stress maximum predicted by equation (19). A set of lower bound conditions for fracture can be obtained by assuming all the fiber failure sites within a section of length 2d to be aligned in a plane perpendicular to the loading direction. Along this plane, the response of the fibers is given by

This estimate differs from the upper bound [equation (19)] by a factor of 21/("+ ~). The fracture strain is then evaluated by substituting equation (22) into the constitutive law, equation (17). Comparisons of the upper and lower bound estimates of the tensile strength and fracture strain are shown in Fig. 2. Following similar arguments, the average fiber fragment length, 7f, during tensile loading can be expressed as

ar(E ) = EEf(1 -- ~),

(21)

I"A more rigorous solution to the fiber fragmentation prob-

lem has recently been developed, accounting for the potential overlap of slip lengths of adjacent fiber breaks and the "shadowing" of defects that occurs within the slipped regions [11]. The predictions of ultimate strength resulting from this solution are essentially the same as those of the approximate solution, differing by < 5% for m/>4.

-/f = 2d pf'

(22)

(23a)

which, combined with equations (10), (12) and (15), gives '/r /',Er~-l/,~ - = [---=1 (23b) 10 \ a0 / " Another factor influencing the composite response is the residual stress, arising from thermal expansion mismatch and phase transformations. In most metal matrix systems, the matrix thermal expansion

3446

WEBER et al.: PROPERTIES OF A Ti MATRIX COMPOSITE--I l ~"

I

: i

Fiber

Composite

OmY "E'E-m~I ore' . f /i

Matrix ~ Ef

,/"" Strain,e,

i

gmr

EaY

%"

the difficulties in accurately ascertaining both the relevant misfit strain and the flow and creep characteristics of the matrix over the entire processing cycle, experimental methods for evaluating residual stresses are preferred. The effects of residual stress on the composite response can be understood with the aid of the schematic in Fig. 3. Provided a~ < a~, the material initially behaves elastically with a modulus, E, given by equation (3). Yielding of the matrix occurs at a strain, Ey, at which the total matrix strain (thermal plus mechanical) reaches the (unconstrained) matrix yield strain, Ey. This result can be expressed as

~----O'(

Y- Y -

E c -- 6 m

Fig. 3. Schematic diagram showing the effects of thermal residual stress on the composite response, coefficient, ~m, exceeds that of the fibers, ~f. Consequently, after cooling from the processing temperature, the matrix experiences an axial tensile stress, a~, whereas the fiber experiences an axial compression, -cr[. Neglecting relaxation effects due to matrix creep or plasticity, the residual stresses due to a misfit strain, f~, are [13] ¢7rm I E m --- fl ~

(24a)

and tTrf/Ef= --oJflfl,

(24b)

where

~

Ern.

Upon further loading (E > E~), the slope (or tangent modulus) of the curve is dictated by the fiber properties. Provided the extent of fiber failure is small and the matrix is elastic-perfectly plastic, the slope in this regime is ~ f E r . Fiber bundle failure then occurs when the average fiber stress reaches aB. This occurs at a strain, E*, of e* = Ea--o'[/Ef.

(24c)

and (1

O ) ~ - - --f )- E- m,

(24d)

fEf with v being the Poisson's ratio (assumed to be the same for the fibers and the matrix). The misfit strain is = (~f - 0~m)AT+ ~p, (24e) where AT is the temperature change and tip is the (unconstrained) linear strain associated with phase transformations, In general, Ti matrix composites are consolidated at temperatures at which the matrix can readily creep, and thus the relevant temperature governing the residual stresses is ill-defined. Furthermore, Ti alloys undergo a phase transformation from the high ternperature b.c.c. (fl) phase to a low temperature h.c.p, (~) phase. For the Ti alloy of interest in this study, the transformation temperature lies in the range 950--1000°C [14]: lower than the processing temperature ( ~ 1100--1200°C). The volume expansion associated with the fl--.~ transformation is ~2.2%, resulting in a linear strain, ~p ~ 0.7% [14]. Because of tSigma fiber, produced by British Petroleum.

(26)

The corresponding stress, ~r*, is a * = f a B + (1 - f)aYm,

(27)

where aB is given by either equations (20) or (22), and the matrix yield stress, a y, is related to the composite yield strain through aYm= (E~ + E~)Em.

f(1 + El~E) fl = 1 + (1 - 2 v ) E / E f

(25)

(28)

The axial residual stress in the fibers also influences the average fiber fragment length. This effect can be incorporated into equation (23b) by adding the thermal component of stress to the mechanical one, yielding the result

~f ('Ef~-¢~tf~-lira -= - (29) l0 \ a0 / ' The preceding background identifies a number of key measurements that are required in order to assess the models. The measurements include: (i) the tensile stress-strain response of the composite, (ii) the in situ fiber strength distribution, (iii) the interfacial sliding resistance, T, and (iv) the axial residual stresses. In addition, tensile tests conducted on specimens of various gauge lengths should provide additional evidence, supporting either the dry fiber bundle model (which predicts gauge length dependent behavior) or the fiber fragmentation model (which predicts gauge length independent behavior). The experimental portion of this study is based on this insight. 3. EXPERIMENTS The material used in this study was a Ti-6AI-4V matrix reinforced with unidirectional, continuous SiC fiberst, 100 pm in diameter. The composite panel was comprised of six plies, with a total thickness of

WEBER et al.: PROPERTIES OF A Ti MATRIX COMPOSITE--I I

i

200 pm (a) Ti 4

_l_

-i-

TiB TiBz C _1 I~_1= -~ i - - r

SiC

2pro (b) "---'---Fig. 4. Transverse section through composite, showing (a) the spatial distribution of fibers and (b) the fiber/matrix interfacial region, 1.0 mm. The fiber volume fraction was 32%. Prior to consolidation, the fibers had been coated with ~ 1 lam of C, followed by ~ 1/am of TiB2. The TiB2 coating serves as a diffusion barrier between the fiber and the matrix. During consolidation, the TiB2 reacts with the matrix to form a layer of TiB needles, ~0.7/am thick. Micrographs of a transverse section through the composite showing the distribution of fibers and the fiber coatings are shown in Fig. 4. Uniaxial tensile tests were conducted at ambient temperature. The speciments were ~ 6 mm wide and cut parallel to the fiber axis. Aluminum or steel tabs with a 10° bevel were bonded to the specimen ends. Tests were conducted in a servohydraulic testing machine, using hydraulic wedge grips to load the specimen. The specimen gauge length was varied between 12 and 220 mm. Axial strains were monitored using a 10 mm contacting extensometer. Tests were conducted at fixed displacement rates, corresponding to a nominal strain rate of 0.5%/rain. The average fiber fragment length of one of the long specimens was measured following fracture, using a two step process. First, the tabbed end of the broken specimen and a small portion of the gauge length were masked with an epoxy adhesive, and the matrix material within the remaining portion of the composite dissolved using HF acid. The length of the dissolved section was ~ 140mm. During this process, the broken fibers within the gauge section were extracted and discarded. The remaining "brush"

3447

of fibers protruding from the masked section was then impregnated with epoxy and the epoxy allowed to cure. The brush was subsequently cut along the edge of the mask and the fiber fragments extracted from the epoxy by dissolving the epoxy in a solvent. The lengths of the individual fragments were measured using vernier calipers. Provided that none of the remaining fragments spans the entire length of the dissolved section, the average fiber length measured in this fashion corresponds to exactly onehalf of the average fragment length within the composite. In the present case, only ~ 5% of the fibers spanned this length. The axial residual stresses were measured using a matrix dissolution technique [15, 16]. A long, thin strip was cut from the composite panel parallel to the fiber axis. Both the composite strip and a reference steel strip of known length were mounted adjacent to one another on a glass plate. Scratches were then scribed on the sample near both ends of the steel strip. The distances between the scratches and the ends of the steel strip were measured in an optical microscope. The ends of the composite strip (including the scratches) were masked with epoxy and the matrix in the central region dissolved using HF acid. The length of the dissolved section, l, was 32 mm. The masks were subsequently removed and the specimen again mounted on the glass plate adjacent to the reference strip. The distances between the scratches and the ends of the reference strip were re-measured. These measurements were combined with those taken prior to dissolution to obtain the length change 6 in the fibers due to the relaxation of residual stress. The residual axial stresses in the matrix tr r and in the fiber tr[ prior to dissolution are related to 6 by [15, 16] tr[ = - 6 E r / L

(30a)

a~ = - f a [ / ( 1 - f ) = 6 E r f / L ( l - f ) .

(30b)

and

An independent measure of the matrix yield stress was obtained using micro-hardness measurements. Vickers indentations were made in the matrix rich regions between fibers on a composite section cut perpendicular to the fibers. Following several preliminary tests, it was found that the indentation size could be kept to within ~<1/3 of the edge-to-edge fiber spacing for an indentation load of 200 g (Fig. 5). This load was used for all subsequent measurements. The matrix yield stress was estimated using the relation [17] a~ -- H / C (31) where H is the Vickers hardness (expressed in units of MPa) and C is a plastic constraint factor, taken to be ~2.5. The in situ strength characteristics of the fibers were measured on individual filaments that had been extracted from the composite. The tests were

3448

WEBER et al.: PROPERTIES OF A Ti MATRIX COMPOSITE---I

~. Fig. 5. Optical micrograph of the indentation produced using a 200 g load.

conducted in a dedicated fiber tensile testert. Four gauge lengths were used: 5, 12.7, 25 and 265 mm. For each length, a minimum of 50 fibers were tested, The strength distribution was characterized using the Weibull function [equation (5)]. For comparing the strength characteristics of fibers of different gauge lengths, it is convenient to re-write equation (5) as I n [ - In(1 - Pf)] - In L / L o = m In a - m In tr0

(32)

such that a plot of l n [ - l n ( l - Pf)] - In L / L o vs In a can be used to evaluate m and a 0. The results were compared with those obtained on pristine fibers with a length of 25 mm, provided by the fiber manufacturer, The sliding resistance of the fiber-matrix interface was measured using fiber pushout tests [18]. Specimens for pushout testing were prepared by cutting sections ~ 500 # m thick transverse to the fibers, followed by grinding and polishing to a final thickhess of ~ 400 #m. The matrix on one side of the specimen was then carefully etched to a depth of ~ 30 #m, leaving the fibers protruding above the matrix surface. A 300#m tall cylindrical indentor was placed on top of a selected fiber. (The indentors had been machined from the Sigma-SiC fibers themselves.) The fibers were subsequently pushed out of the section and the load-displacement characteristics measured. Additional details of the testing apparatus can be found elsewhere [18]. In calculating the interfacial sliding resistance, z, the shear stress was assumed to be uniform along the interface and given by P

Z = "2~R(t -- u)

(33)

where P is the applied load, t is the section thickness and u is the amount of sliding displacement. 4. MEASUREMENTS AND OBSERVATIONS

response was elastic with a modulus, E = 201 + 11 GPa. This value is consistent with one calculated from the rule of mixtures, E = 203 GPa, using a matrix modulus, E m = I 1 0 G P a [19], and a fiber m o d u l u s , Ef = 400 GPa:~. The onset of yielding occurred at a tensile strain of ~0.45%. This strain corresponds to a nominal matrix stress (neglecting the residual stress) of ~ 500 MPa. Fracture occurred at an average tensile stress of 1590 + 100 MPa and a tensile strain of 0.94 + 0.05%. There was no apparent effect of the specimen gauge length on the shape of the stress-strain curve, the ultimate tensile strength or the fracture strain (Fig. 7). The apparent fiber length distribution following dissolution of a broken specimen is shown in Fig. 8. The average length in this distribution is T = 39 mm, corresponding to an /n situ fiber fragment length, 7r = 27"= 78 mm. No correction was made for the fibers that had remained intact over the entire length of the dissolved section, though the number of these was small ( ~ 5%) and is not expected to substantially alter the result. The dissolution experiment to measure residual stress yielded a length change of 85/~m in a gauge length of 32 mm. Combining this result with equation (30) yields axial residual stresses of tr~ =490 MPa and tr[ = - 1040 MPa. These results have been cornbined with the measured composite yield strain along with equation (28) to obtain the (unconstrained) matrix yield stress, tr y = 1000 MPa. This value is in reasonable agreement with the range of values inferred from the microhardness measurement: tr~ = 940 MPa. In addition, both of these values are consistent with the yield strengths of similar Ti-rAI-4V alloys, which range from ~ 900 to ~ 1050 MPa [19]. The results of the fiber tensile tests, presented in the form suggested by equation (32), are shown in Fig. 9. In this form, the data collapse onto essentially a single band. This result confirms the scaling of failure probability with gauge length via equation (32). Though the curve exhibits some non-linearity at low values of strength, the majority of the data can be

2000

~

is00

~

1000

~0o 0 0.0

The tensile stress-strain curves exhibited the features shown in Fig. 6. Upon initial loading, the tMicropull Science. :~Provided by fiber manufacturer,

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Strain(%) Fig. 6. Typical tensile stress-strain response of the Ti/SiC composite. Also shown is the prediction from the fragmentation model, with the two thin arrows corresponding to the lower and upper bond estimates of the fracture point.

WEBER et al.:

PROPERTIES OF A Ti MATRIX COMPOSITE--I

(a) 2200

60 ;

~

,

~

1800

i

i

j

~.-

~)

-l-Tq

o

Z

~

1200

-

Dry Fiber Bundle Model "~-- -._.

:~

10(30

,

I 50

40

u.

/ 1400

50 co

"~

N

-g ~ I--

i

Fragmentation Model: / Upper Bound #j/~ Lower Bound

2000 o~

3449

i

10 0 10

30

50

/ -" -i-" - • - - - - - - r - - "~- - -i 150 200 250 300

1O0

70

90

110

130

Fragment Length (mm)

Fig. 8. Distribution of fiber fragment lengths following tensile fracture.

Gauge Length, L (mm)

(b) 1.6

,

t

I

i

z

1.4

I

/

z

z

i

t

.

~

1.2

'@

~/

1.0 0.8

-2-

0

~

-~L -1 "1

_T_

o

-.

LL

""

DryFiberBundleModel .................

"--..__

0.6 0.4

/

Lower Bound

//// E

I

Fragmentation Model: / Upper Bound

0

'

~

50

'

~

100

'

~

150

'

~

200

'

~

250

'

300

Gauge Length, L (ram)

Fig. 7. Influence of gauge length on (a) the ultimate tensile strength and (b) the fracture strain. Also shown are the upper and lower bounds predicted by the fiber fragmentation model, and the prediction of the dry fiber model,

r e a s o n a b l y well a p p r o x i m a t e d by the two p a r a m e t e r Weibull function, using values a0 = 1.47 _ 0.2 G P a a n d m = 5.3t. It should also be n o t e d t h a t the strengths of the extracted fibers are substantially lower t h a n those o f the pristine fibers, indicating t h a t additional flaws are i n t r o d u c e d into the fibers d u r i n g composite consolidation. The results of the p u s h o u t tests are presented in Fig. 10. In all cases, sliding initiated at a stress, z ~ 9 0 - 1 2 0 M P a . U p o n further loading, the sliding resistance increased a n d reached a s a t u r a t i o n level of r ~ 130 M P a following ~ 0 . 5 # m of fiber displacement. The same sliding resistance was m e a s u r e d for fiber displacements up to ~ 2 0 / ~ m .

In Fig. 7, the predicted u p p e r a n d lower b o u n d s to the fracture stress a n d fracture strain from the fiber f r a g m e n t a t i o n model are plotted as solid horizontal lines. The error bars represent the range o f predicted values, resulting from the uncertainty in the fiber reference strength (a 0 = 1.47 _ 0.2 GPa). The d a s h e d lines c o r r e s p o n d to the predictions of the dry fiber b u n d l e model. It is evident t h a t the dry fiber b u n d l e model strongly underestimates b o t h the fracture stress a n d fracture strain, a consequence o f the a s s u m p t i o n t h a t n o coupling exists between the fibers a n d the matrix. In addition, the model predicts a gauge length dependence t h a t is inconsistent with the experimental measurements. Conversely, the fiber f r a g m e n t a t i o n model predicts a fracture stress a n d a fracture strain t h a t are b r o a d l y consistent with the experimental measurements, with the better correlation being established with the lower b o u n d estimates. F u r t h e r m o r e , the f r a g m e n t a t i o n model predicts gauge length independent behavior, consistent with the experiments. The predicted average f r a g m e n t lengths for the relevant range of values of a 0 are also in good agreement with the experimentally m e a s u r e d value (Fig. 11), providing yet additional s u p p o r t for the f r a g m e n t a t i o n model.

10

, ,

,

,

, , , , , , o =1.27GP~ . ~ , 1.47GPa ~ 1.BZGPa

% ~

6

/

_~C

4

~--~,~[~/

5. C O M P A R I S O N OF EXPERIMENT WITH THEORY

~-.= 2

C o m p a r i s o n s of the experimental results with the m o d e l predictions are s h o w n in Figs 6 a n d 11. T h e relevant c o n s t i t u e n t properties used in the models are s u m m a r i z e d in T a b l e 1.

-= -=

0

,.1 Extracted F i b e r s _ E t ~ ' ~

.tL_j

©

-2 -4 -1.5

tBased on linear regression analysis, neglecting the five lowest strength values. These five constitute ~2s/o of the total number of tests. A more detailed discussion of the effects of gauge length on the strength distribution can be found in Ref. [7].

, , , , ,

8

o ' ' -1.0

o ' ' -0.5

//// '"/' '

'

0.0

0.5

'

~ , ~ , 1.0

1,5

~ , 2.0

2.5

In (o/GPa) Fig. 9. Results of the fiber tensile tests, presented in the form suggested by equation (32). Also shown are the results for the pristine fibers, derived from > I000 fiber tests conducted by the fiber manufacturer.

WEBER et al.: PROPERTIES O F A Ti M A T R I X C O M P O S I T E - - I

3450 ....

2oo

, ....

, . . . I ' ....

I

~" B_ t,..,

15o

Table 1. Summary of constituent properties

Room Temperature

. - ~

_0._ r ~ . . . . ~b-- - _O_ 0-<2__ _o _0_-o- _0_ o- ¢ [~oat:~t-e,'~a a ~x ~x a ,, ~- -~ o)

c ='° co "~ 1: -=

100

Fibers

Young's modulus, Ef Axial residual stress, tr~ Weibull modulus, m Reference strength, tro Volume fraction, f Diameter, 2R

400 GPa -1040 MPa 5.3 _ 0.2 GPa 1.47 + 0.32 100/zm

Matrix Young's modulus, Em Axial residual stress, trm r Yield stress, try Volume fraction, 1 - f Interface Sliding stress, z

50

0

.

.

.

.

i

0

.

.

.

.

i

0.5

.

.

.

.

i

1

.

.

.

110 GPa 490 MPa 1000 MPa 0.68 130 MPa

.

1.5

Fiber Displacement (Ixm) Fig. 10. Results o f the fiber pushout tests,

since this b o u n d is c o n s e r v a t i v e , it w o u l d be a p p r o p r i a t e for use in the design o f c o m p o s i t e structures.

Acknowledgement--Funding o f this work was supplied by the D A R P A University Research Initiative Program of UCSB under O N R contract N-0014-92-J-1808.

6. CONCLUDING REMARKS T h e p r e s e n t w o r k d e m o n s t r a t e s t h a t t h e axial tensile p r o p e r t i e s o f u n i d i r e c t i o n a l T i / S i C c o m p o s i t e s c a n be d e s c r i b e d u s i n g m o d e l s b a s e d o n fiber fragm e n t a t i o n , t a k i n g into a c c o u n t t h e effects o f t h e t h e r m a l residual stress, t h e in situ fiber s t r e n g t h d i s t r i b u t i o n , t h e m a t r i x yield stress a n d the sliding r e s i s t a n c e o f t h e f i b e r - m a t r i x interface. Clearly, m o d e l s b a s e d o n t h e b e h a v i o r o f a dry fiber b u n d l e

model are inadequate. Moreover, the conclusion that t h e s t r e n g t h o f t h e c o m p o s i t e is independent o f g a u g e

length has important implications regarding t h e design o f large s t r u c t u r e s , p a r t i c u l a r l y since m a t e r i a l c h a r a c t e r i z a t i o n is generally c o n d u c t e d o n relatively

small coupons. A n i m p o r t a n t a d d i t i o n a l f e a t u r e i n c o r p o r a t e d into t h e f r a g m e n t a t i o n m o d e l is t h e l o w e r b o u n d c o n d i t i o n f o r fiber b u n d l e failure. T h e present results i n d i c a t e t h a t this b o u n d is c o n s i s t e n t w i t h the m e a s u r e m e n t s o f b o t h t h e f r a c t u r e stress a n d the f r a c t u r e strain in t h e T i / S i C c o m p o s i t e . M o r e o v e r ,

10~

E~,

100

_9

10-~

i,~

10-a

'

'

~

'

'

'

'

'

oo=1.27--

~.47~: •

10-s 0.4

~

i 0.6

~

t 0.8

~

i 1.0

~

i 1.2

~ 1.4

Strain (%) Fig. 11. Comparison o f the predicted average fiber fragment length with the measured value.

Note added in proof--The

discrepancy between the measured fracture stress and fracture strain and the corresponding upper bound estimates from the fragmentation model may be due to the relatively small number o f fibers within a single tensile specimen (~200), as described in the

articlebyW. A. Curtin, J. Mech. Phys. Solids41,217(1993). REFERENCES 1. J. Doychak, J. Metals 44, 46 (1992). 2. S. J. Connell, F. W. Zok, Z. Z. Du and Z. Suo, Acta metall, mater. 42, 3451 (1994). 3. S. Jansson, H. Drve and A. G. Evans, Metall. Trans. 22A, 2975 (1991). 4. S. L. Draper, P. K. Brindley and M. V. Nathal, Metall. Trans. 23A, 2541 ( 1 9 9 2 ) . . 5. G. J. DeSalvo, Theory and Structural Design Applications of Weibull Statistics. General Westinghouse, Pittsburgh, Pa (1970). 6. J. R. Porter, Mater. Res. Soc. Syrup. Proc. 273, 315 (1992). 7. F. W. Zok, X. Chen and C. H. Weber, J. Am. Ceram. Soc. Submitted. 8. B.W. Roxen, Mechanics of Composite Materials:Recent Advances, p. 105. Pergamon Press, Oxford (1983). 9. W. A. Curtin, J. Mater. Sci. 26, 5239 (1991). 10. W. A. Curtin, J. Am. Ceram. Soc. 74, 2837 (1991). 11. J. Neumeister, J. Mech. Phys. Solids 41, 1383 (1991). 12. K. K. Chawla, Composite Materials, Science and Engineering, Chap. 12. Springer, New York (1987). 13. B. Budiansky, J. W. Hutchinson and A. G. Evans, J. Mech. Phys. Solids 34, 164 (1986). 14. C. Barrett and T. B. Massalski, Structure of Metals, 3rd edn, p. 361. Pergamon Press, Oxford (1980). 15. B. N. Cox, M. R. James, D. B. Marshall and R. C. Addison Jr, Metall. Trans. 21A, 2701 (1990). 16. D. Beyerle, S. M. Spearing, F. W. Zok and A. G. Evans, J. Am. Ceram. Soc. 75, 2719 (1992). 17. T. H. Courtney, Mechanical Behavior of Materials, Chap. 1. McGraw-Hill, New York (1990). 18. P. D. Warren, T. J. Mackin and A. G. Evans, Acta metall, mater. 40, 1243 (1992). 19. Metals Handbook, 9th edn, Vol. 3, Properties and

Selection: Stainless Steels, Tool Materials and Special Purpose Metals, pp. 388-91. ASM, Metals Park, Ohio (1990).