Prediction of Creep-Rupture Life Matrix Composites Subjected to

of Unidirectional Titanium Transverse Loading

REJI JOHN, M. KHOBAIB, and PAUL R. SMITH

Titanium matrix composites (TMCs) incorporating unidirectional fiber reinforcement are considered as enabling materials technology for advanced enginr which require high specific strength and elevated temperature capability. The resistance of unidirectional TMCs to deformation under longi- tudinally applied sustained loading at elevated temperatures has been well documented. Many in- vestigators have shown that the primary weakness of the unidirectional TMC is its susceptibility to failure under very low transverse loads, especially under sustained loading. Hence, a reliable model is required to predict the creep-rupture life of TMCs subjected to different transverse stress levels over a wide range of temperatures. In this article, we propose a model to predict the creep-rupture life of unidirectional TMC subjected to transverse loading based on the creep-rupture life of unidi- rectional TMC subjected to transverse loading based on the creep-rupture behavior of the corre- sponding fiberless matrix. The model assumes that during transverse loading, the effective load-carrying matrix ligament along a row of fibers controls the creep-rupture strength and the fibers do not contribute to the creep resistance of the composite. The proposed model was verified using data obtained from different TMC fabricated using three matrix compositions, which exhibited dis- tinctly different types of creep behavior. The results show that the creep-rupture life of the transverse TMC decreases linearly with increasing ratio of the fiber diameter to the ply thickness. The creep- rupture life is also predicted to be independent of fiber spacing along the length of the specimen.

I. INTRODUCTION

TITANIUM matrix composites (TMCs) incorporating unidirectional continuous fiber layup are considered as en- abling materials technology for advanced engines and may be potentially useful for aerospace structural applications in hypersonic aircraft that require high specific strength and elevated temperature capability.m The TMC is usually re- inforced with ceramic fibers which offer significant resis- tance to deformation under longitudinal sustained loading at elevated temperaturesY 5] Due to the weak nature of the fiber/matrix interfacial bond, a primary weakness of the un- idirectionally reinforced TMC is its susceptibility to failure under very low off-axis loads, especially under sustained transverse loading.[4-131 Previous results [4-111 have indicated that the creep-rupture life under transverse loading is sig- nificantly lower than that of the unreinforced matrix. Uni- directionally reinforced TMCs are targeted for use in applications such as ring-type components in engines with the fibers aligned in the hoop direction. During service, such composites will be subjected to centrifugally induced sustained loads acting perpendicular to the direction of the fibers. Hence, a reliable tool is required to predict the creep- rupture life of unidirectional TMC subjected to sustained transverse loads. In this article, we propose a new model to predict the creep-rupture life of the unidirectional TMC subjected to different transverse stress levels over a wide

range of temperatures. The proposed model is also com- pared with the model proposed by Wright.t4]

1I. BACKGROUND

Crossman e t al. [9] analyzed the deformation behavior of a metal matrix reinforced with ceramic fibers subjected to sustained transverse loads. A finite element model was used E91 to analyze the effect of different arrays of fiber layup and the degree of bonding between the fiber and the matrix on the steady-state creep strain rate of the composite. The finite element analysis showed that the creep of the com- posite subjected to transverse loads could be described by a function similar to that for the unreinforced matrixJ 9j For unreinforced matrix, the steady-state creep strain rate, era, is given by a function as

em= f(O-m, T) [1]

where ~r m is the stress applied to the matrix and T is the temperature. Knowing the matrix creep behavior, the steady-state creep strain rate of the composite, ec, under transverse loading can be calculated using[9]

ec : f ( ~ , T) [2]

where ~ is the equivalent matrix stress in the transverse composite. The value of ~ was related[9] to the farfield composite stress, ergo , as

REJI JOHN, Associate Research Engineer, and M. KHOBAIB, Research Engineer, are with the Advanced Materials Characterization Group, Structural Integrity Division, University of Dayton Research Institute, Dayton, OH 45419-0128. PAUL R. SMITH, Materials Research Engineer, is with the Write Laboratory, Materials Directorate, WL/MLLN, Wright-Patterson AFB, OH 45433-7817.

Manuscript submitted January 30, 1995.

09O = - - [ 3 ]

cgVf

where V r = volume fraction of the fibers and g is a constant. In this article, "90" is used to identify the composite sub- jected to transverse loading and [90], implies that the corn-

U S GOVERNMENT WORK 3074~VOLUME 27A, OCTOBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

log o" m

(or)

log (~90

Unreinforced Matrix

[90] n (p:l)

Strength Reduction Factor, F~

f

P = T [ C + log t r ] (K,hr)

Fig. 1--Schematic of the Larson-Miller parameter (P) approach to predict the creep-rupture life of the [90], layup of titanium matrix composites.

posite contains n rows of fibers. Using finite element analysis, Crossman et al. [9] determined g for various fiber arrangement arrays and the degree of bonding between the fiber and the matrix. The average value of g was equal to -2.02 for a composite with fully separated fibers arranged in a square or diamond array. When the modulus of the fiber was set equal to zero, g decreased to -2.28 and -4.00 for fibers arranged in square and diamond arrays, respec- tively. Thus, for unidirectional MMCs with fully separated fiber-matrix interface, Eqs. [1] through [3] can be used to determine the creep strain rate of a transversely loaded composite under steady-state conditions. In Eq. [3], g < 0 results in 00e > 0090, implying that the composite deforms faster than unreinforced matrix tested at the same stress level (00m = 0090) and temperature consistent with experi- mental observations, t4-11]

W r i g h t [4] used Eq. [3] to predict the creep-rupture life of SCS-6/Ti-24Al-llNb loaded in the transverse direction with g = -2.02. The schematic of the procedure is shown in Figure 1. The Larson-Miller paramete# 14] has been com- monly used to represent the creep-rupture life of unrein- forced materials and composites tt,4~l subjected to different stress levels over a wide range of temperatures. The Lar- son-Miller parameter, P, is defined as

P = T [ C + log tr] [4]

where T is the temperature in kelvin, t. = time to rupture in hours, and C is a constant, typically about 20. When the applied stress is plotted vs P in a log-linear scale, the data can be represented using a single curve, usually a straight line, as shown by trend lines in Figure 1. The trend line for the unreinforced matrix can be expressed as

log 00m = C, + C 2 P [5]

where C, and C2 are the intercept and slope, respectively, obtained by fitting the matrix data. A similar approach can be used to predict the composite behavior under transverse

loads by relating the composite stress, 0~ to an equivalent matrix stress using a strength reduction factor, Fo, as

0090 = F~00m [6]

Note that " 9 0 " is used to identify the [90] layup of the unidirectional composite subjected to transverse loading. Using Eqs. [5] and [6], we get the behavior of the [90] composite as

log 0090 = log F~ + C1 + C2P [7]

Thus, the trend line for the [90] layup can be obtained as shown in Figure 1. Eqs. [5] and [7] indicate that the slope of the trend line for the [90] layup is the same as that of the unreinforced matrix. Using Eqs. [3] and [6], Wright's strength reduction factor would be given by

F~, , C = eggs [8]

Assuming that the fibers were fully separated from the ma- trix, g = -2.02 was usedt41 to predict o-9o. For example, if V s = 0.30 and g = -2.02, then F~.w< = 0.55. Hence, o'90 = 0.55 00m, for a given Larson-Miller parameter, P.

The Wright-Crossman (WC) model[4] predicted tr of plasma-sprayed SCS-6/Ti-24Al-11Nb (atomic percent) sat- isfactorily, while t r was overestimated for foil/fiber/foil SCS-6/Ti-24Al-11Nb. The difference between the data and the prediction for foil/fiber/foil SCS-6/Ti-24AI-11Nb was attributed to the possible differences between the creep be- havior of the unreinforced matrix and that of the matrix in the composite.t4a Recent studies[10~ have shown that the steady-state creep rate of a Ti-22A1-23Nb unreinforced ma- trix fabricated by foil/fiber/foil was dependent upon the orientation tested, i .e., longitudinal direction (parallel to the foil rolling direction) exhibited a higher creep rate than transverse direction (perpendicular to the foil rolling direc- tion). This result suggests that texture induced during roll- ing, whether microstmctural or crystallographic, may play a role in the minimum creep rate. Since Ti-24Al-11Nb is similar to Ti-22AI-23Nb, the texture-induced differences in the matrix behavior could possibly explain the differences between the data and predictions observed by Wright.ta] Note that texture-induced differences are more likely to oc- cur in foil/fiber/foil composites than in plasma-sprayed composites. Recently, Smith et al. t~5] also concluded that the Crossman et al. [91 model overestimates the creep life of [90]4 SCS-6/Ti-21AI-22Nb subjected to sustained trans- verse loading.

During this study, we used the WC model to predict the creep-rupture life of [90]4 SCS-6/TIMETAL*21S. The

*TIMETAL is a trademark of Timet Corporation, Henderson, NV.

specimens for unreinforced Timetal21S were obtained from plates which were fabricated using the procedure identical to that used for SCS-6/Timetal21S. t51 The model predictions overestimated the creep-rupture life of the composite. The V s values for SCS-6/Timetal21S, foil/fiber/foil SCS-6/Ti- 24A1-11Nb, and plasma-sprayed SCS-6/Ti-24AI-11Nb were 0.33, 0.35, and 0.28, respectively. The WC model appears to yield good predictions at lower V i and overestimate life at higher V s. This article discusses the development of a new model to predict the creep-rupture life of [90] TMCs for any V s and the applicable range of the WC model.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, OCTOBER 1996~3075

Table I. Data Used for Verification of Transverse Creep-Rupture Life Prediction Models

Volume Fiber Average Radius Fraction Spacing, Thickness, of Fiber, Number

Processing of Fibers, s B R I- of Plies, Matrix Fiber Technique V I (ram) (mm) (mm) n Reference

Ti-22A1-23Nb* SCS-6** foil/fiber/foil 0.32 0.197 0.97 0.070 4 10 Ti-24A1-11Nb* SCS-6 foil/fiber/foil 0.35 0.197 1.80 0.070 8 6 Ti-24Al-11Nb* SCS-6 plasma-sprayed 0.28 0.190 1.13 0.070 4 4 Timetal 21St SCS-6 foil/fiber/foil 0.33 0.197 0.95 0.070 4 2 Timetal 21St SCS-9** foil/fiber/foil 0.23 0.110 0.79 0.040 4 11

*Composition in atomic percent. **Silicon carbide (SIC) fibers manufactured by Textron Specialty Materials Division, Lowell, MA.

tTi-15Mo-2.6A1-3Nb-0.3Si, composition in wt pct.

Table II. Constants in the Larson-Miller Parameter Equation for the Unreinforced Matrix and the Average Values of Composite Strength Reduction Factors Predicted by the Different Life Prediction Models

Composite

Volume Fraction of

Processing Fibers, Technique Vj-

Constants in the Larson-Miller Parameter

Equation for the Unreinforced Matrix

Predicted Average Strength Reduction Factor, F~

Proposed WC Model WC Model C~ C2 Model (g = -2.02) (g = -2.28)

SCS-6/Ti-22A1-23Nb foil/fiber/foil 0.32 SCS-6/Ti-24Al-11Nb foil/fiber/foil 0.35 SCS-6/Ti-24Al-11Nb plasma-sprayed 0.28 SCS-6/Timetal 21S foil/fiber/foil 0.33 SCS-9/Timetal 21S foil/fiber/foil 0.23

5.00 - 0.1376 0.423 0.521 0.479 4.38 -0.1064 0.378 0.496 0.453 4.28 -0.1075 0.505 0.568 0.528 6.35 -0.2211 0.408 0.513 0.471 6.35 -0.2211 0.593 0.627 0.590

III . EXPERIMENTAL RESULTS

Available data t~6,1~ on different types of titanium ma- trices reinforced with SiC fibers were collected as reported in Table I. The matrices examined included a metastable beta composition Timetal21S and two titanium aluminide compositions, Ti-24Al-11Nb and Ti-22A1-23Nb (both in at. pct). The composite stress levels ranged from 21 to 138 MPa, temperature from 593 ~ to 871 ~ and the time to rupture from 0.5 to 150 hours. All the tests were conducted in laboratory air. The relevant properties of the composite and that of the constituents are also shown in Table I. There were two types of fibers, namely, SCS-6 and SCS-9, with Vf ranging from 0.23 to 0.35. The primary difference be- tween the two fibers is their diameters. The difference in diameter influences the longitudinal tensile strength of the fiber (since they are deposited on the same size carbon monofilament) and does not have any effect on the trans- verse load carrying capacity of the fiber.

The constants C1 and C2 in Eq. [5] for the unreinforced matrices are reported in Table II. The wide range of Cl and C2 indicates the significantly different creep behavior of the titanium matrices. For example, the rupture lives of unrein- forced Timetal21S, Ti-24Al-11Nb, and Ti-22A1-23Nb un- der identical creep loading conditions (760 ~ MPa) were 0.6, 142, and 151 hours, respectively.r161

The fracture profiles of a [90]4 SCS-6/Timetal21S and [90]4 SCS-6/Ti-22A1-23Nb are shown in Figure 2. At ele- vated temperatures, Timetal21S exhibits near-elastic-plastic behavior, while Ti-22A1-23Nb exhibits relatively brittle behavior. Even though the behaviors of the unreinforced matrices are distinctly different, the fracture profiles of the transverse layups show strikingly similar behaviors. There

is clear evidence of full separation between all the fibers and the matrix. The carbon cores of the SCS-6 fibers also appear to have been burnt away. The failure of the four- ply composite layup occurred along the weakest ligament, i.e., the location containing four fibers in a row. Such fail- ure is clearly evident in the case of SCS-6/Ti-22A1-23Nb, where the failure ligament occurred below two layers con- taining only three fibers in a row.

IV. PROPOSED MODEL---DEVELOPMENT AND COMPARISON W I T H EXPERIMENTAL RESULTS

Based on the observed fracture profiles of most of the [90] composite layup, a net-section-based unit cell model is proposed. A representative fracture profile was shown in Figure 2 along with the proposed unit cell. The schematic of the proposed model is shown in Figure 3. In the figure, R r = fiber radius, s = center-to-center fiber spacing parallel to the loading direction, B = thickness of the specimen, n = number of plies in the composite, and B , = B / n = av- erage ply thickness. Assuming that the transverse behavior of the composite is identical to that of the unreinforced matrix with holes, the matrix stress in the net section can be related to the composite stress based on load equilibrium as

Using Eqs. [6] and [9], we obtain a new strength reduction factor as

3076---VOLUME 27A, OCTOBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

(a)

I 1' ( 9o

Matrix

G'rn

2Rf I_, i -,qlL

Bp ,.._1 "-I

s/

Fig. 3--Schematic of a unit cell simulating a [90] layup of a titanium matrix composite (neglecting the fiber).

1000 Vf = 0.32

�9 . . . , Data: matrix

�9 .. " ' " . , Fit: matrix t/)

O o o o o o , . �9 �9 ~, (n Prediction: 100 ~ . .... "-.. W/C, g=-2.02

._ Data: [90]4 "'I'" Q. <: Prediction:

This study 1 0 i i , I , l , l , I , I ,

17 18 19 20 21 22 23 24 25 26

P = T [ 2 0 + l o g t r ] , (K,hr) (x 1/1000)

Fig. 4 - -Compar i son of measured and predicted creep-rupture lives of the [90]4 layup of foil/fiber/foil SCS-6/Ti-22A1-23Nb.

(b)

Fig. 2- -Frac ture profiles of (a) [90]4 SCS-6/TImetal21S tested at 41 MPa and 650 ~ in laboratory air and (b) [90]4 SCS-6/Tl-22A1-23Nb tested at 69 MPa at 650 ~ in laboratory air

Equation [9] is based on the assumption that the failure of the [90] ply is determined by the failure of the matrix in the net section based on the fracture profiles shown in Fig- ure 2.

The predictions of creep-rupture life of transverse layups obtained from net-section and WC models are shown in Figures 4 through 8 for all the TMCs listed in Tables I and II. The WC model predictions were made assuming g = -2 .02 consistent with Wright's I41 predictions. In Figures 4 through 8, the data and the fit for the corresponding un- reinforced matrix are also shown. The data for the unrein- forced matrix and the composite clearly show that the slope of the composite data is indeed identical to that of the un- reinforced matrix. The proposed model predicts the trans- verse t, satisfactorily for all the TMCs over a wide range of stress levels and temperatures. The WC predictions over- estimate t r for TMC with Vj > 0.30. The predictions using the proposed model are close to the WC predictions for V f < 0.30.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, OCTOBER 1996--3077

1000 1000 Vf = 0.35

Data: matrix ~ /

. Fit: matrix co : ......... " ' " ~ " . . ' ~ / Prediction: o~ ~ " ' " " - . . . ~ W/C, g=-2.02 �9 ~ 100

~ Data: [90]8 . ~ �9 ...~. .......... < - ,_ .., P r e d i c f ~ o n : / / ~ , ~

This study t0 I I ! t I , I , I I

17 18 19 20 21 22 23 24 25 26

P = T [ 2 0 + l o g t r ] , (K, hr) (x 1/1000)

Fig. 5~omparison of measured and predicted creep-rupture lives of the [90]8 layup of foil/fiber/foil SCS-6/T~-24AI-11Nb

1000 Vf : 0.28

"~" Data: matrix

" " " ' * . . , Fit: matrix

m ~ ...... " ' " - . , . Prediction: 100 . . . . w/c, :-2.02

Dat " . - . . . .

~_~ [90]4 P r e d i c t i o n : ~ < This study

10 , I , I , ' ' ' i i i 17 18 19 20 21 22 23 24 25 26

P = T [ 2 0 + l o g t r ] , (K,hr) (x 1/1000)

Fig. 6~Companson of measured and predicted creep-rupture lives of the [90]4 layup of plasma-sprayed SCS-6/Ti-24AI-11Nb.

1000 Vf = 0.33 Data: matrix

" ' ~ . Fit: matrix '%, ~o,, / ffl �9

Of) ~ - ' * * * * *'~~ ~ ~,

�9 (~ 1 0 0 ~ . o /~ ~ . . . P r e d i c t i o n :

" ~ , . . "~. W/C, g=-2.02 -o D a t a : ~ . . ~ . . ~ . zx A. / .~_

< redicti s tudy ' - This

10 , I , i , i , i . I ~ ? " - I . ~ , I

17 18 19 20 21 22 23 24 25 26

P = T [ 2 0 + l o g t r ] , (K, hr) (x 1/1000)

Fig. 7--Comparison of measured and predicted creep-rapture lives of the [90]4 layup of foil/fiber/foil SCS-6/Timetal21S

V. D I S C U S S I O N

The predictive capabil i ty o f the WC and proposed mod- els are shown in Figures 9 through 11, in which the pre- dicted t~ is plotted vs the measured t~ for all the TMCs. The solid line corresponds to the exact correlation between the predicted and measured t~. The dashed lines correspond to a variation o f factor o f 2 in predicted life. Figure 9 shows

13_ v

r

m

-o

Q. <

100

10 17 26

I ~ Data: matrix Vf = 0.23

' "~' ~7 ~. "-. . Fit: matrix

' _ ~a~.,," ~. Prediction: w/c, g=-2.02

t~u]4 ~ ""',, /

18 19 20 21 22 23 24 25

P = T [ 2 0 + l o g t r ] , (K, hr) ( x l / 1 0 0 0 )

Fig. 8--Comparison of measured and predicted creep-rupture lives of the [90]4 layup of foil/fiber/foil SCS-9/Timetal21S.

..~ 104 11r s " S

103 * . - ' " . - "

~- 102

rr ~r A,IY

~ . . . , "~" ,,

1 01 .T'" . ' "

I - 0 ~ "~ 1

"o 04 .e 1 13_

10 "1 100 101 102 103 104

M e a s u r e d T i m e to Rupture, tr (hr)

Fig. 9--Plot highlighting the predictive capability of the WC model with geometric factor g = -2.02.

,_ 104 v . - ~ *

103

~. 102 ~-" . " n-

10 ~ . ~ ~ ' " .E_ I--- .'7. -o 100 . ~ " . . " �9

m 104 ..; ...... , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . 0..

I 10 0 101 10 2 10 ~ 10 4

Measured T ime to Rupture, t r (hr)

Fig. 10~Plot highhghting the pre&ctive capability of the WC model with geometric factor g = -2.28.

that most o f the predictions obtained using the WC model with fully separated fibers (i .e. , g = 2.02) lie above the solid line, implying that the predictions tend to overestimate the time to rupture. Some possible reasons for the overes- t imation by the WC model were discussed earlier. Based on the results of our analysis, we used the WC model with g = - 2 . 2 8 , i.e., assuming zero modulus fibers consistent

3078--VOLUME 27A, OCTOBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A

L - , , . . ,

r r

.E_ t-'- "0 0

10 4 _

lOlL ,..-"

101

1o 0

10 "I ..*. ...... i ........ i ........ i ..... ~,,I , , ,,,,,,

I 0 q I 0 ~ 101 10 2 10 3 10 4

Measured Time to Rupture, tr (hr)

Fig. 11--Plot highlighting the predictive capabihty of the proposed net- section model.

o 0 t~ LL. t- O

r r

N e-

0.0 1.0

0.8

0.6

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8

Fiber Diameter/Ply Thickness, 2Rf/Bp

Fig. 12~omparison of proposed net-section and WC models.

Volume Fraction of Fibers, Vf

0.1 0.2 0.3 0.4 0.5

\ ""~-'~- ~ - W - C '

\ \ ~ ' ~ . . ~ . . g = -2.28 "',. I " "" ".'.A

vv-c: .... .[ g =-4.00 "'....... ~ .......... .t.'.?.:.:..:

, i , IPr~ 7 ~

1.0

0.8

O "5 LI. c- O "6 -'1

n" r

r

0.6

0.4

0.2 2.0 3.5

2Rf /Bp = 0 .5 W-C mode l

g=-2.02 W-C model S

..... S';" f'~';'7";'[';';'7" ....................... ~"Proposed model

' ~ ' ' ' ~ , . , . . . . .

. . . . | . . . . I . . . .

2.5 3.0

Fiber spacing/Fiber radius, s/R f

Fig. 13--Predicted effect of fiber spacing (s) on the strength reduction factor.

with the proposed net-section model. The predictions ob- tained using the modified WC model are compared with the data in Figure 10. Comparing Figures 9 and 10, we see that the modified WC model with zero modulus fiber predicts the rupture life better than the WC model with fully sepa- rated fibers. Figure 11 highlights the predictive capabilities

of the proposed net-section model which also assumes zero modulus fiber. Most of the predictions obtained using the net section model are within a factor of 2 compared to the data. Figure 11 also shows that the proposed model tends to underestimate the rupture life in some cases, consistent with the assumption that the failure is controlled by the net- section stress in the matrix ligament neglecting the fibers completely. Figures 10 and 11 show that for the WC and net-section models, the same number of data points (~68 to 70 pct) are within the dashed lines. But most of the predictions by the net-section model lie below the solid line in contrast to the predictions by the W-C model. Hence, the proposed net-section model predicts a better lower bound solution and should be a useful conservative design tool.

Using the schematic shown in Figure 3, the volume frac- tion of the fiber can be derived as

~-a~ [11]

Substituting Eq. [11] in Eq. [8], the strength reduction of the WC model can be written as

F~.wc = exp [ g -~sY ~ ] [12]

Using Eqs. [10] and [12], the strength reduction factor pre- dicted using net-section and WC models is plotted vs the ratio of fiber diameter to ply thickness in Figure 12. For WC model predictions, the ratio s/R r was assumed equal to 2.78 based on the data in Table I. The proposed model predicts that F~ decreases linearly from 1.0 to 0.0 when the ratio 2R/Bp increases from 0.0 to 1.0. The WC model pre- dictions are close to that of the proposed model for 2R/Bp < 0.50, which corresponds to Vj < 0.30. For 2R/Bp > 0.50, the WC model predictions are greater than that of the pro- posed model. For the limiting case corresponding to 2R/Bp = 1.0, i.e., when all the fibers in a row are touching each other, the proposed model correctly predicts zero strength but the WC model predicts a finite strength. The data shown in Figure 12 appear to support the trend predicted by the proposed model for //1 > 0.30.

When IgV~ < < 1.0, Eq. [12] can be written as

~R r 2R s F ..... = 1 + g [13]

2s /~

Substituting the values of RI, s, and g, Eq. [13] can be shown to be close to the net-section model given by Eq. [10]. Hence, the net-section and WC models give close pre- dictions for V s < 0.30.

Comparing Eqs. [10] and [12], we see that the proposed model predicts that F~ is independent of s for a given ratio of 2R/Bp, while the WC model depends explicitly on V I. Figure 13 compares F~ predicted by the models with in- creasing s/R I for a constant 2R/Bp = 0.5. Note that the minimum s/R i = 2.0 corresponds to two rows of fibers touching each other. Figure 13 shows that for a given ratio of 2R/Bp, increasing s/R r corresponds to decreasing V~ as given by Eq. [11]. The net-section model predicts that the strength reduction is independent of s, while the WC model predicts an increase in transverse strength with increasing ratio of s/R r In a near-square arrangement, the failure oc-

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, OCTOBER 199~3079

curs at the weakest link, i.e., at a row of fibers. Hence, for the same 2R/Bp, the rupture life of a composite with one row can be expected to be the same as that of a composite with multiple rows. Consequently, the net-section model predicts that F~ is independent of s. Note that while the rupture life is independent of fiber spacing, the creep de- formation will depend on fiber spacing, s. Additional data are required for verifying the dependence of the rupture life on the fiber spacing, s.

In the composites considered during this study, the fibers were arranged in near-square arrays. The WC model with g = -2.02 and -2.28 and the proposed model correspond to the square array arrangement. Crossman et al. tg~ also sug- gested using g = -4 .0 for a composite with zero-modulus fibers arranged in diamond arrays. The WC prediction with g = -4 .0 is also shown in Figure 12, along with other predictions and the data. The figure shows that the WC model with g = -4 .0 does not predict the overall trend of the data for all V/. The prediction is close to one data point at V I ~ 0.35 (2R/Bp ~ 0.6) and close to the proposed model for V s > 0.35. As discussed earlier, data are required for V s > 0.35 (2R/Bp > 0.65) to verify the predictive capabilities of the various models for high V i.

The proposed net-section model is expected to be appli- cable for all MMCs which satisfy the following conditions: (1) weak fiber/matrix interfacial bond such that the fiber does not contribute to the creep resistance in the transverse direction; and (2) the fibers are arranged in near-square ar- rays such that a row of fibers forms a weak link, as shown in Figure 2.

In this study, we had neglected the effect of processing- induced residual stresses on the creep-rupture life. Many investigations (for example, Reference 17) have shown that the residual stresses become negligible at elevated temper- atures. Hence, the assumption of zero residual stresses should be valid.

VI. CONCLUSIONS

A new model to predict the transverse creep-rupture life of a continuously reinforced titanium matrix composite was developed and verified during this study. The proposed model is based on the net-section stress in the matrix lig- ament assuming that the performance of the composite of the transverse layup is identical to the matrix with holes. The net-section model predicted the transverse creep-rup- ture life satisfactorily over a wide range of stress levels, temperatures, and volume fraction of fibers. The predictions tended to be conservative, which is ideal for application in the industry. The results confirm that the failure depends on the ratio of fiber diameter to ply thickness, and the fail- ure is independent of fiber spacing parallel to the loading direction. The proposed model correctly predicts that the composite has zero transverse strength when the ply thick- ness is equal to the fiber diameter. The proposed model predicts a reasonable lower bound solution and should be a useful conservative design tool.

The WC model was found to be nonconservative, and better predictions were obtained assuming zero modulus fi- ber. The WC model is based on the assumption that the

failure depends on the volume fraction of fibers; i.e., failure depends on the ratio of fiber diameter to ply thickness and fiber spacing along the length. Contrary to the net-section model, the WC model predicts finite transverse strength when the ply thickness is equal to the fiber diameter.

Studies are in progress to extend the net-section model to predict the primary creep response and the minimum creep strain rate of unidirectionally reinforced TMCs sub- jected to sustained transverse loading.

ACKNOWLEDGMENTS

This research was conducted at the Materials Directorate, Wright Laboratory (Wright-Patterson Air Force Base, OH). Drs. Reji John and M. Khobaib were supported under Con- tract No. F33615-94-C-5200 (Project Monitor: J.R. Jira). Mr. Paul R. Smith was supported under internal Air Force funding. The authors acknowledge discussions with Dr. N.E. Ashbaugh and thank Dr. K. Wright for providing plasma-sprayed SCS-6/Ti-24Al-llNb data. The authors also gratefully acknowledge the help of Ms. D. Garner and Mr. D. Knapke in conducting the tests and assisting with data reduction.

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