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Modeling and simulation of the effect of fiber breakage on creep behavior of fiber-reinforced metal matrix composites
MECHANICS OF MATEBALS ELSEVIER
Mechanics of Materials 21 (1995) 303-312
Modeling and simulation of the effect of fiber breakage on creep behavior of fiber-reinforced metal matrix composites Sabing Lee, S.M. Jeng, J.-M. Yang Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024-1595, USA
Received 8 December 1994; revised version received 20 April 1995
Abstract A theoretical model and computer simulation methodology was developed to predict the effect of fiber fracture on creep behavior of continuous fiber-reinforced metal matrix composites. Initially, a single fiber model was developed based upon the fiber statistical characteristics and a shear-lag analysis to establish the computation simulation route. Then, the methodology was extended to predict the creep behavior of a multiple fiber composite. A failure criterion was also incorporated in the model to predict the rupture life of the composite. A parametric study was also conducted to investigate the effects of properties of the constituents on the longitudinal creep behavior of the SCS-6/Ti composite. Keywords: Creep; Metal matrix composite;Computer simulation
I. Introduction Continuous fiber-reinforced titanium matrix composites (TMC) are attractive materials for applications in which high specific strength, high specific stiffness and good creep resistance are required, such as gas turbine engines. However, before these composite can be safely implemented into aerospace structures, it is essential that the fracture, fatigue and creep properties of these materials are well characterized. In the past few years, a significant amount of works has been conducted to study the mechanical behavior and damage mechanisms of the composites under static and cyclic loading (Yang et al., 1991; Jeng et ai. 1991a-c, 1992a,b). The results showed that the damage mechanisms and flaw evolution are strongly dependent on the properties of the constituents (fiber strength distribution, matrix toughness, interfacial shear strength and frictional stress) and loading conditions. Corre-
sponding micromechanical models have also been developed to predict the mechanical behavior and service life of the composites (Halford et al., 1993, Bartolotta et al., 1991; Nicolas and Russ, 1994). However, the creep behavior and damage mechanisms of the composites are not yet well understood. Recently, constant loading creep tests have been conducted to study the longitudinal creep behavior of the SCS-6/Ti-6242 composite under both tensile and bending condition (Jeng and Yang, 1993; Jeng et al., 1993). The results indicated that the composite exhibited a transient creep stage followed by a quasi-steady state creep. At high applied stress levels, a short tertiary creep stage leading to final failure of composite was also observed. A typical tensile creep deformation curve and the corresponding microstructural damage mechanisms of the titanium matrix composite in each stage are shown in Fig. 1. The key mechanism is random fiber breakage and microvoids forma-
0167-6636/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0167-6636(95)00013-5
S. Lee et aL/Mechanics of Materials 21 (1995) 303-312
304
to predict the effect of fiber fracture and interfacial debonding on the longitudinal creep behavior of the fiber-reinforced titanium matrix composites. Initially, a single fiber composite model based upon random fiber fragmentation and the shear-lag model is developed to establish the computer simulation route. Then, the technique is extended to multiple fiber composite for a complete representation of the real composite. 1. random fiber brea 2. microvoid format
2. Modeling creep behavior of fiber-reinforced metal matrix composite without damage 0.8
0.6
"4 0.4
0.2
0 0
Fig.
1. T y p i c a l
100
21)0
300 Time (hour)
400
500
tensile creep curve and corresponding
600
damage
mechanisms in SCS-6/Ti alloy matrix composites. tion along the reaction layer/matrix interface during the early stages of creep deformation. Matrix cracking that extended from the broken fiber ends were also observed during the later stages of creep deformation. The loss of the load carrying capability of the broken fiber will essentially increase the stress state of the matrix which, in turn, would reduce the creep resistance of the composite. Microvoid formation along the reaction layer/matrix interface will degrade the interfacial shear strength. As a result, the load transfer efficiency between the broken fiber and matrix will be degraded as a function of creep loading history. Meanwhile, the matrix cracking induced by the stress concentration near the broken fiber ends will gradually decrease the load carrying area of the matrix and further reduces the creep resistance of the composite. A theoretical prediction for the creep deformation and rupture life of the composite that incorporates these damage mechanisms has not been developed yet. The objective of this paper is to develop a theoretical model and a computer simulation methodology
The expected service temperatures of the CVD SiC fiber-reinforced titanium matrix composite range from 500 to 800°C. Under these conditions, the SiC fiber is known to possess excellent creep resistance and will deform elastically. However, the titanium alloys are susceptible to creep deformation. McLean (1985) proposed a model to describe the longitudinal creep deformation of an unidirectional fiber-reinforced metal matrix composite. The model assumed that the composite contained a power-law creep matrix, an elastically deformed fiber and a perfect fiber/matrix interface bonding. The overall strain rate of the composite can be described as = ~lO~2dm(
l -- Sl )m
(
1)
where eel =
[ 1-k-Ef~ ] -1 Era(1 - ~ ) ' ,
ce2 = (1 - ~ ) - m ,
where ~m represents the creep rate of the matrix carrying all of the applied stress o'0, Ef and Em are Young's modulus of fiber and matrix, respectively, Vf is the volume fraction of the fibers, m is the exponent constant of the matrix, and $1 represents a normalized value for the stresses carried by the fiber: Sl -
o-fVf or0
(2)
Meanwhile, $1 can be described by
$1 = Hl~, where H1-
~Ef or0
(3)
305
S, Lee et al./Mechanics of Materials 21 (1995) 303-312
For the composite under constant applied stress o'0, the initial conditions for Eqs. (1) and (3) are e(0) =
- ~ - Virgin' SCS~6, SPOOI~ l Oo=5142 MPa, m=16.2 ---<)-- Virgin SCS-6, SPOOL 2 <~o=3646 MPa, m=15.7 ----O--- As-Consolidated SCS-6 =3943 MPa, m=6.2
O"0
EfV~ + Em( 1 - V~)'
S1 (0) = Hie ( 0 ) .
(4)
The creep deformation of fiber-reinforced metal matrix composites without microstructural damage can be predicted by the governing equations, and initial conditions (Eq. ( 4 ) ) . Physically, S1 can be treated as a dimensionless internal stress (stress carried by the fibers) that modifies the matrix creep behavior. The constant H is a strain-hardening coefficient that control increments of the internal stress. If the fibers do not fail, the internal stress SI would gradually increase as the deformation accumulates, which, in turn, will reduce the stress carried by the matrix. When the fibers fully support the applied stress (S1 = 1 ), the composite will not deform as a function of loading time.
3. Modeling creep behavior with microstructural damage When damages are induced during processing and/or creep loading, the equations developed by McLean are inadequate to describe the creep behavior of the composites. Since the majority of the fibers remains intact, the creep deformation of the composite is basically controlled by the noncreeping fibers. In order to accurately predict the creep behavior, the following two damage-induced characteristics need to be taken into consideration. The first consideration is to change the internal stress Sl and its increase rate Sl resuiting from fiber breakage and microvoids formation between the reaction layer/matrix interface. To determine the effects of these two damage mechanisms on Sl, several aspects must be considered. These include the interfacial shear strength between the fiber and matrix, stress recovery at the broken fibers, fiber modulus degradation, and the statistical distribution of fiber strength. The second consideration comes from the modification of the matrix creep characteristics, dm, resulting from the stress concentration and matrix cracking near the broken fiber ends. In this paper, a major effort is made to study the effects of the fiber breakage and other constituent properties on the variations of internal stress (SI and Sl) and the
=,
-3
V
4 []
--~ 6/ 7.4
<> 7.6
7.8
[] 8
8.2
8.4
8.6
8.8
Ln (Our s)
Fig. 2. Strength distribution of SCS-6 fiber at as-fabricated and after compositeprocessing. creep strain of composite. A detailed discussion of each aspect is given in the following sections.
3.1. Fiber fragmentation and stress redistribution during loading The strength of a brittle fiber is limited primarily by the defects found along the length of the fiber and can generally be characterized by the Weibull distribution. The failure probability as a function of stress for a fiber with gauge length L is given by (Weibull, 1951 ) Pf
=
1 -exp
--~R \ o ' m /
J '
where o-m is a reference stress, LR is the reference length for o-m, and p is the Weibull modulus, which describes the strength variation. This equation provides the percentage of fibers in a bundle which are broken as a function of the stress. Fig. 2 shows a typical fiber strength characteristics of the SCS-6 fiber before and after composite processing (Gambone and Wawner, 1994). It clearly indicates that the Weibuli modulus of the SCS-6 fiber degraded from 16 to 6 before and after composite processing, respectively. The population of low strength fibers also increases significantly. These weak fibers will fail prematurely during the early stages of creep loading. When fiber breakage occurs in the composite, the matrix can transfer the stress back into the broken fiber through a recovery length & The stress distribution along the recovery length is strongly dependent
306
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
on the fiber/matrix interfacial condition. If the interfacial shear strength is strong enough to suppress the interfacial debonding, the load transfer will be activated by shear stress only. However, if interfacial debonding occurs, the load transfer in the debonded region will be achieved by frictional stress. The load transfer characteristics have been extensively studied using the shear-lag analysis (Dilandro et al., 1988). For simplicity, the shear and frictional stresses are assumed to be constant, Ty and rf, respectively, in this simulation. The stress in the broken fiber was also assumed to build up linearly in these two regions. Using these two shear components, the fiber stress near the broken end can be described by a bilinear stress recovery model and for a bonded region is given by (Henstenburg and Phoenix, 1989) 4Tfx O'f ( X ) --
d
,
(6a)
X < 6 -- 6b
Off --
d
d
o-f (LT -- nbt~) LT
(7)
where nb is the number of breaks, o-f is the applied fiber stress without breakage, and LT is the length of the fiber. For the debonded case, the effective average stress will be __
O'f [ L T
-- H b ( ~ -- ~ b ) ]
L
-- ~ r f b n b ~
(8)
For the creep deformation simulation, the internal stress in Eq. (2) is represented in a form that is prescribed either by Eq. (7) or Eq. (8).
O-f ( X ) = 4 r i ( 6 -- t~b) + 4"ry [ x -- (t~ -- t3b) ] b
_
O'f =
and for a debonded region by
8--8
tained. A detail discussion of the problem can be found in Curtin ( 1991 ) After fiber breakage, the effective load carrying capability of the composite will be gradually reduced. It is important to know the effective fiber stress for implanting the fiber breakage effect on the creep governing Eqs. ( 1 ) and (2). The effective average stress carried by the fibers, ~f, will depend on the recovery length 6 and whether or not debonding has occurred. For the perfect bonding case, the average stress is
' (6b)
where 6b is the length to which the interface remains inperfect bonding condition. As creep deformation proceeds further, stress in the fiber increases continuously. Fiber breakage can occur in the same fiber or in the adjacent fibers. A complete discussion of the fiber breakage sequence and its effective fiber stress distribution is given in the following section.
3.1.1. Single filament case For solving the single fiber fragment distribution at certain stress level, the model developed by Curtin ( 1991 ) has been used. The original solution was used to examine the fiber fragment distribution for a single fiber under tensile loading. In this model, a uniquestrength fragment distribution function P(x; n, 6) has been used to characterize the fiber fragment distribution as a function of applied stress and interfacial mechanical properties. By solving a set of coupled differential equations with a given fiber statistics (Weibull distribution) and interfacial quantities (interfacial shear strength and frictional stress), the fiber fragment distribution for a known stress can be ob-
3.1.2. Multiple filament case In a multiple fiber composite, the fragment distribution function described for a single filament composite may no longer apply because of interaction among the fibers. In order to account for these effects, a random fiber failure process based upon the Monte Carlo simulation is used. The simulation assumes that the interaction is based only on the load transfer characteristics between the broken fiber and matrix in the composite. Since the stress concentration induced by fiber breakage will be incorporated in the matrix creep characteristics, it was not considered in this fiber fragmentation simulation. In order to determine the fragment distribution for a multiple fiber composite, av. appropriate model must be chosen to represent the statistical strength distribution for each of the fibers. This can be accomplished by dividing the composite into a number of unit cells, where each cell represents a strength value determined by the Weibuil distribution. For a composite containing nf fibers, each fiber was further divided into nc cells, the total number of unit cells for the entire composite will be nf × he. Fiber strength values which followed by the Weibull distribution were assigned to
307
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
each of the nf x nc unit cells by the Monte Carlo technique. Fiber fragmentation in this model assumes that breaks occur in the center of the unit cell, and that the cells are sufficiently long so that the recovery length 6 does not extend into the other cells. If the stress state of the composite is known, the fiber fragment distribution can be determined by comparing the strength data with the fiber stress distribution. When there are n bc fibers break occurring in any given row of cells, the stress will be transferred to the matrix and fibers adjacent to the broken fibers. In the fiber broken plane, the additional stress that is released from the broken fibers is simply equal to the stress that was once carried by the broken fibers, Act0 = nco'fV b ~i ,
(9)
where Vfi (= Vf/nf) is the volume fraction taken up by each individual fiber. However, at any given distance x away from a fiber broken end, the total stress transferred to the other fibers and the matrix is Off (X) = Off -- O'f (X) ,
(10)
where o-f(x) is the stress recovery near the fiber breaks obtained by Eqs. (6a) and (6b). By substituting o-fR(x) for o-f in Eq. (9) and manipulating the parameters, the total stresses in the fibers and matrix adjacent to fiber breakage, as a function of distance x away from the plane of fracture, can be described by Tc
1+
r/c [Off -- O'f(X)]
,
,
(11)
for the fibers and n c [ o f f - O'f(X)] O"To(x) m = O"m +
1 --
where n f is the number of broken cells in the f - t h fiber, and the summation term adds together is the stress transferred from adjacent fiber to each of the k-th unbroken cells in the fiber. In a situation where debonding occurs, the average stress is given by ~fJ = Z
t--n
f (~-
~b)
.c-n
+ v__f[of(~-~b) +~ro, S] ~ (TO
k=1
°'fbnfS L
b
]
nf- ncbk/ (14)
The effective average fiber stress for the multiple fiber composite can be obtained using a rule of mixtures: ~f = v f i Z ~ f =1~ f ( j ) .
(15)
3.2. Effective elastic modulus of composite with fiber breakage The fiber breakage that occurred during creep loading will not only degrade the average fiber load carrying capability but also the composite modulus. In order to account for the degradation of elastic modulus, the model developed by Karandikar and Chou (1993) has been adopted and modified. The original model was used to predict the effect of matrix cracking on the degradation of the elastic modulus for a fiberreinforced brittle matrix composite. When the fibers are under an applied stress o'f the corresponding composite stress can be expressed by
Vf1
~fEc
O'0 J
~° -
1-Vf
Ef
(16)
(12) for the matrix, where the superscript To denotes the row of cells to which the stresses is transferred. Based upon the above equation, the effective average stress for the individual fiber can be obtained. For the perfect bonding situation ~f(j) = ~O"f L - n f ( ~ +
O'f Vf(~ OrO
nc(k)
~
b nf -- nc(k)
(13)
Then, by combining 0-8 and ~f by the rule-of-mixtures, the average static stress carried by the matrix becomes ~ m --
0- 8 - ~ f ~
1-~
,
(17)
where ~f is obtained from the stress redistribution simulation by adapting Eqs. (6a) and (6b). Dividing the average matrix stress by the matrix elastic modulus, the average static strain of the matrix can be obtained. Since there is no damage occurring in the matrix, this
308
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
static strain is equal to the average static strain in the composite,
--s
~m ----~cc"
i ~, ~
Governingequation l ~ = CtlC~2~m(l-Sl)m SI =H1E
(18)
6m = E-~
The reduced composite modulus can be calculated by dividing the applied static stress by the static strain: E,c = cr~Em
Calculatingc ando'f
(19)
--S O"m
fiberfi'agmentation i
~Yes
In order to incorporate the reduced modulus into computer simulation, it can be manipulated into an effective fiber modulus by the rule-of-mixtures. The resulting form for a single fiber composite will be (1 -E~(j) = ( o . . f E c / E f ) _
(20)
~f(j)E m (1 - V0
/~f~.i~ = (o-fEc/Ef)
(21)
butinn
calculatingeffective fiber stressand modulus T _T comparingwith failurecriteria
4NNo[
~f(J)~"'
With the same approach, the effective modulus for each of fibers in a multiple fiber composite will be
I
initail condition / E(O)=Oo/[EfVf +Em(1- Vf)]] Sl(0) = HI~(0) ]
[i
ic°mp°sitef_aiil-- ~_
Fig. 3. A computer simulation route for predicting creep behavior of the TMC.
- ~f(~)Vf
Then the overall effective fiber modulus for the multiple fiber composite is given as
E~-- v ~ i Z nr t f ./=lEf(j).
(22)
3.3. Computer simulation of creep deformation Based upon the above micromechanical models, the computer simulation route shown in Fig. 3 was developed. Before simulating the creep deformation, the fiber fragment distribution and fiber strength data were generated according to the above models. The generated data was used as a database for determining the fiber breakage as the creep loading proceeded. The composite was assumed to be damage free before creep loading. Also, the residual stresses caused by the thermal expansion mismatch between the fiber and matrix were not considered in the simulation since significant stress relaxation occurred near 600°C (E1dridge, 1992). Combining the creep Eqs. ( 1) and (2) and the initial conditions, the creep deformation and fiber stress distribution of the composite can be obtained as a function of loading time. For each time increment, the calculated fiber stress was compared with
the fiber strength data. When the fiber stress is higher than the strength of the weakest fiber, fiber breakage occurs. The stress redistribution in the fiber is evaluated. Meanwhile, the static effective elastic modulus of the composite as function of the number of fiber breakages is also simulated. The resulting "Sl and HI were further used for the simulation of the next step. By repeating the computation routes for each time increment, the creep deformation curve can be obtained for each loading condition. The failure of the single fiber composite was not considered. However, for the case of a multiple fiber composite, composite failure occurs when a certain row of unbroken fibers cannot sustain the applied stress. A fourth-order Runge-Kutta technique with a adaptive time scale was used for numerical analysis. The current model was developed to simulate the creep behavior of a titanium matrix composite reinforced with SiC fibers. Because the specific properties of the composite depend on the matrix composition and processing conditions, the simulation was conducted to study the effect of the various parameters of the materials on the creep deformation curves. For this reason, the parameters used in this simulation do not necessarily represent a specific material,
309
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
Table 1 Mechanical characteristics of the SCS-6 fiber Parameter
Value
Young's modulus (El) Weibull modulus (p) Fiber diameter (df) Reference strength (O'fR) Volume fraction (Vf) Fiber length (ld)
400 GPa 2-6 140/zm 4000 MPa 0.35 0.5 m
Table 2 Properties of interface and matrix used for computer simulation Parameter
Value
Matrix Young's Modulus (Em) Stress exponent (m) Strain rate (Em)
100 GPa 3-5 a 1.59 × 10 - 6 to 8.21 × 10-3 (hr- l ) b 20-160 MPa 20-100 MPa
Ty ~'f
Depending on the temperature. b Depending on applied stress and temperature.
a
but rather they encompass the entire range of properties pertaining to S i C / T i composites. For the SiC fiber, the characteristics o f a C V D grown SCS-6 fiber was used, which are listed in Table 1. For the titanium matrix, the mechanical properties and creep characteristics were obtained from available materials that have been used for composites such as Ti6242 ( T i - 6 w t % A1-2wt% S n - 4 w t % Z r - 2 w t % M o ) , Ti-15-3 ( T i - 1 5 w t % V - 3 w t % A1-3wt% C r - 3 w t % S n ) and Ti-6-4 ( T i - 6 w t % A1-4wt% V) alloys. Because the creep data for these materials were obtained at applied stresses below the simulation conditions, the strain rates were extrapolated to the applied stress values based on the linear relationship between In e and In o-0. The room temperature interfacial shear strength for the S C S - 6 / T i composites have been characterized to be between 120 to 160 MPa (Yang et al., 1991 ). However, due to the reduction o f the residual clamping at elevated temperatures, the interfacial shear strength of the composites was reduced to between 4 0 - 7 5 MPa at temperatures between 600 to 800°C (Eldridge, 1992). The post-debonded interfacial frictional stress o f the S C S - 6 / T i composite has also been measured to be between 60 to 100 MPa (Yang et al., 1991). Due to the relaxation o f residual clamping stress at elevated tern-
perature, the frictional stress was found to be 2 0 - 3 0 MPa (Eldridge, 1992). The properties o f the interface and matrix used in the computer simulation are listed in Table 2 (Bania and Hall, 1983; Bania et al., 1985; Rosenberg, 1980)
4. Results and discussion The numerical simulation described in Fig. 3 was performed on a composite with different properties o f the constituents. In order to study fiber breakage as function o f fiber stress in the SCS-6 fiber, a fiber with Weibull modulus p = 2 and a composite with perfect bonding was analyzed. The results are plotted in Fig. 4. It clearly seen that the first fiber breakage occurs around 1000 MPa. This implies that fiber breakage will occur immediately when the initial load is applied, and the broken segments will continuously fracture as the creep loading time is increased. The effect o f the fiber breakage on the creep behavior o f the composite is shown in Fig. 5. The results show that a significant amount o f additional strain will be induced by the fiber breakage. Furthermore, the effect o f fiber breakage and interfacial debonding on creep behavior was also simulated and plotted in Fig. 5. It becomes apparent that debonding at fiber broken ends will further increase the overall strain. The effects o f applied stresses on creep deformation of the composites were also simulated and compared with the prediction using the model without any fiber breakage. The result8
-
7 ~ 6
/i
ii
/
5 /
"~ Z
3
•
2
~
1
*
0
~:::::
/
0
• i
+
+
1000
2000
3000
4.000
Fiber stress (MPa)
Fig. 4. Fiber fragmentation distribution as a function of stress for single fiber composite.
310
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
0.0075
1000 MPa
Breakage and debonding ,}
0.007 ~
Breakage and perfect bonding
,t
0.0065
i
= No breakage 0.006
i,%*= = = =
I
, ,
,>
~
'
*:
>
/,
800 MPa
0.005 ~ , ~ * * 600 MPa 0.004
0.0055 i
0.003
I 0.0045
i 0
0.002 1000
2000
3000
4000
400 MPa ~- - ~
5000
~-
I
1000
2000
Time(hr)
3000
4000
5000
Time (hr)
Fig. 5. Creep curves for single fiber composite for (1) no fiber breakage, (2) fiber breakage without debonding, and (3) fiber breakage with debonding (o0 = 1000 MPa, 6m = 8.21 × 10-3 hr-t,m = 5, p = 2,~'y = 130 MPa, rf = 80 MPa).
Fig. 6. Effect of applied stress on creep of single fiber composite, with dashed curves representing no fiber breakage (m = 5, p = 2, Zy = 130 MPa, ~'f = 80 MPa)
ing creep curves were plotted in Fig. 6. For an applied stress below 600 MPa, the composites exhibited a similar creep response as predicted by these two models. This is because the applied stress is not high enough to cause fiber breakage to occur. This suggests that a critical threshold stress exists below which the fiber breakage mechanisms will not play any role in creep deformation. The effect o f the fiber strength distribution was also investigated in this simulation and the results are shown in Fig. 7. The creep strain o f the composite reinforced with an SCS-6 fiber having a low Weibull modulus ( p = 2) was approximately 30% higher than that o f composites with an high Weibull modulus fiber ( p > 4). This suggests that the increase of the low fiber strength population resulting from high temperature consolidation will significantly reduce the creep resistance o f the composite. The effect of the interfacial shear strength and post-debonded frictional stress on the creep deformation is shown in Figs. 8 ( a ) and 8 ( b ) . This clearly indicates that a composite with a low interfacial shear strength (20 MPa) exhibits a significant increase in creep strain (more than 4 0 % ) as compared with the one with a high interfacial shear strength (160 MPa). The post-debonded frictional stress also influences the overall strain o f the composite. With a frictional stress of 20 MPa, the creep strain in the composite is substantially higher
than in a composite with higher frictional stresses. Therefore, maintaining the integrity of the interface during creep loading is essential to facilitate load transfer from the creeping matrix to the noncreeping fiber. 4.1. Multiple fiber composite
When simulating the creep deformation of the multiple fiber composite, the selection of a suitable size of the computation cells is important for adjusting the computation time and the accuracy of the predicted creep behavior. A composite with constant length and 0.0075 -
0.007 ~
I
.~
ir
0.0065
0.006
•
2
~
4
¸
0.0055
0.005 1000
2000
3000
4000
5000
Time (hr)
Fig. 7. Effect of the Weibull modulus on creep of single fiber composite ( ~m 8.21 × 10-3 hr - l ). =
S. Lee et al./Mechanics o f Materials 21 (1995) 303-312
311 ~u
0.01
t
0.0053
i
•
1000 M P a
0.0048
800MPa
0.0043 0.008 .~ 0.0038 600
0.0033 0.006
0.0028 4
20
•
0.0023 i
100 0.004 0
i
i
i
I
,
1000
2000
3000
4000
5000
Time
0.0075
b. 0.0065
•
20
*
80
60 0.0055 I00 I 1000
2000
MPa
0.0018 0
1000
2000
3000
4000
5000
Time (hr)
Fig. 9. Effect of cell size and number on the creep of multiple fiber composite (rim = 5.51 X 1 0 - 4 hr-1, p = 2, ry = 130 MPa, "rf = 8 0 MPa).
0.0085 i i
0
400
(hr)
(a)
0.0045 t
MPa
L
I
r
3000
4000
5000
(hr) (b)
Time
Fig. 8. Effect of interfacial mechanical properties on creep of single fiber composite, (a) interfacial shear strength, (b) frictional stress (em = 8.21 × 10-3 hr- l , p = 2). with different cell lengths was used to investigate the cell size effect. The results are plotted in Fig. 9. The simulation results indicate that the differences in the creep strain for three cell sizes were within the 5% range, especially, the results for the 15 × 15 and 20 × 20 cases were almost identical (within 2%). This suggests that the predicted creep behavior for the multiple fiber composite is not sensitive to the cell size. Meanwhile, the creep behavior for multiple fiber composites with different interfacial properties and fiber characteristics was also studied. One of the results was compared with the single fiber composite, as shown in Fig. 10. This result shows that the creep strain predicted by the multiple fiber model is slightly higher than that by the single fiber model. However, the difference in the predicted creep strain from these two models was
minimal. This may be due to the stress concentration and matrix cracking effects which have not been considered in this simulation yet. Furthermore, the important distinction of the multiple fiber composite model is its ability to determine the rupture of the composite as shown in Fig. 11. When compared with Fig. 6, it clearly shows that the creep rupture life of a multiple fiber composite is predicted to be approximately 2500 hours at an applied stress o f 1000 MPa. 5. Concluding r e m a r k s The model and simulation presented in this work represent a methodology for predicting the effect of
0.0027
0.0025
/
"I
•
lOxl0
0.0023 !
~
I ~
0.0021 f
0.0019
[.
, 1000
--~ - 2000 Time
3000
15x15 *
I 4000
20x20
5000
(hr)
Fig. 10. Comparison between predicted creep curves for single fiber composite and 20 × 20 multiple fiber composite (~:m = 1.82 × 10-3 hr- l , p = 2,ry = 130 MPa, ~'f = 80 MPa).
312
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
Acknowledgements
0.0041 i
0.0039
This work was supported by the National Science Foundation (DDM 905730) and Allied-Signal Corporation.
m m ,
0.0037
w~ •
0.0035
MFC
References
SFC 0.0033 "~
I ~" o.oo3J
+
0.0029 ~ 0
1000
2000
3000
4000
5000
Time (hr)
Fig. 11. Effect of applied stress on creep of multiple fiber composite (m = 5, p= 2, ,'ry 130 MPa, 7f = 80 MPa). =
fiber breakage on the creep behavior of a continuous fiber-reinforced metal matrix composite. However, other factors involved in the actual physical deformation of the material may need to be considered in further analysis. First, because the approach was from a load transfer standpoint, the effect of stress concentration and matrix cracking near the broken fiber ends was not considered yet. In fact, stress concentration and matrix cracking will make a significant contribution towards creep by causing local deformation in the matrix as well as further fiber fracture. Therefore, the creep equations described in this study need to be modified to include stress concentration and matrix cracking effects. Second, experimental data must be obtained to determine the validity of the model. Although some creep data does exist for S i C / T i composites, the amount of available information is limited. Also, because of the numerous parameters that are input into the simulation, experiments need to be designed specifically to test the limitations of this model. It should prove especially useful to obtain creep and fragmentation data for S i C / T i single filament composites, of which there is very little experimental data. Another factor to be considered is the formation of interfacial layers. It is well-known that materials such as titanium and SiC form a brittle interface zone, so that the deformation characteristics of this third constituent must be examined separately in the creep equations.
Bania, P.J., G.A. Lenning,and J.A. Hall (1983), in: Beta Titanium Alloys in the 80"s, eds. R.R. Boyer and H.W. Rosenberg,AIME, Atlanta, Georgia, pp. 209-229. Bania, EJ., and J.A. Hall (1985), in: Titanium Science and Technology, Vol. 4, eds. G. Liitjering,U. Zwicker, and W. Bunk, Deutsche Gesellschaft fur Metallkundee.V., Federal Republic of Germany,pp. 2371-2378. Bartolotta, PA. ( 1991), NASA TM 104494, Lewis Research Center, Cleveland, OH. Curtin, W.A. (1991), J. Mater. Sci. 26, 5239-5253. Dilandro, L., A.T. DiBenedetto, and J. Groeger (1988), Polym. Comp. 9, 209-221. Eldridge, J.I. (1992), in: lntermetallic Matrix Composites 11, eds. D.B. Miracle, D.L. Anton and J.A. Graves, MRS, Pittsburgh, Pennsylvania,pp. 325-330. Gambone, M.L. and EE. Wawner (1994), unpublishedwork. Halford, G.R., B.A. Lerch, LE Saltsman and V.K. Arya (1993), in: Thermomechanical Fatigue Behavior of Materials, ASTM STP 1186, eds. H. Sehitoglu, Philadelphia, 176-194. Henstenburg, R.B., and S.L. Phoenix (1989), Polym. Comp. 10, 389-408. Jeng, S.M., and J.-M. Yang (1993), Mater. Sci. & Eng. A 171, 65-75. Jeng, S.M., J.-M. Yang and C.J. Yang (1991a), Mater. Sci. & Eng. A 138, 169-180. Jeng, S.M., J.-M. Yang and C.J. Yang (1991b), Mater. Sci. & Eng. A 138, 181-190. Jeng, S.M., E Alassoeur,J.-M. Yangand S. Aksoy ( 1991c),Mater. Sci. & Eng. A 148, 67-77. Jeng, S.M., P Alassoeurand J.-M. Yang (1992a), Mater. Sci. & Eng. A 154, 11-19. Jeng, S.M., J.-M. Yangand S. Aksoy (1992b), Mater. Sci. & Eng. A 156, 117-124. Jeng, S.M., E-C. Wangand J.-M. Yang(1993), in: Proc. American Society for Composite, 8th Technical Conf. on Composite Materials, pp. 451-460. Karandikar, P., and T-W. Chou (1993), Comp. Sci. Technol. 46,
253-263. Mclean, M. (1994), Comp. Sci. Technol. 23, 37-52. Nicholas, T. and S.M. Russ (1994), in: Proc. American Society for Composite, 9th Technical Conf on Composite Materials.
Rosenberg, H.W. (1980), in: Titanium "80: Science and Technology, Vol. 2, eds. H. Kimura and O. Izumi, AIME, New York, pp. 1071-1080. Weibull, W. (1994), J. Appl. Mech. 18, 293-297. Yang, J.-M., S.M. Jeng and C.J. Yang (1991),Mater. Sci. & Eng. A 138, 155-167.
Mechanics of Materials 21 (1995) 303-312
Modeling and simulation of the effect of fiber breakage on creep behavior of fiber-reinforced metal matrix composites Sabing Lee, S.M. Jeng, J.-M. Yang Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024-1595, USA
Received 8 December 1994; revised version received 20 April 1995
Abstract A theoretical model and computer simulation methodology was developed to predict the effect of fiber fracture on creep behavior of continuous fiber-reinforced metal matrix composites. Initially, a single fiber model was developed based upon the fiber statistical characteristics and a shear-lag analysis to establish the computation simulation route. Then, the methodology was extended to predict the creep behavior of a multiple fiber composite. A failure criterion was also incorporated in the model to predict the rupture life of the composite. A parametric study was also conducted to investigate the effects of properties of the constituents on the longitudinal creep behavior of the SCS-6/Ti composite. Keywords: Creep; Metal matrix composite;Computer simulation
I. Introduction Continuous fiber-reinforced titanium matrix composites (TMC) are attractive materials for applications in which high specific strength, high specific stiffness and good creep resistance are required, such as gas turbine engines. However, before these composite can be safely implemented into aerospace structures, it is essential that the fracture, fatigue and creep properties of these materials are well characterized. In the past few years, a significant amount of works has been conducted to study the mechanical behavior and damage mechanisms of the composites under static and cyclic loading (Yang et al., 1991; Jeng et ai. 1991a-c, 1992a,b). The results showed that the damage mechanisms and flaw evolution are strongly dependent on the properties of the constituents (fiber strength distribution, matrix toughness, interfacial shear strength and frictional stress) and loading conditions. Corre-
sponding micromechanical models have also been developed to predict the mechanical behavior and service life of the composites (Halford et al., 1993, Bartolotta et al., 1991; Nicolas and Russ, 1994). However, the creep behavior and damage mechanisms of the composites are not yet well understood. Recently, constant loading creep tests have been conducted to study the longitudinal creep behavior of the SCS-6/Ti-6242 composite under both tensile and bending condition (Jeng and Yang, 1993; Jeng et al., 1993). The results indicated that the composite exhibited a transient creep stage followed by a quasi-steady state creep. At high applied stress levels, a short tertiary creep stage leading to final failure of composite was also observed. A typical tensile creep deformation curve and the corresponding microstructural damage mechanisms of the titanium matrix composite in each stage are shown in Fig. 1. The key mechanism is random fiber breakage and microvoids forma-
0167-6636/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0167-6636(95)00013-5
S. Lee et aL/Mechanics of Materials 21 (1995) 303-312
304
to predict the effect of fiber fracture and interfacial debonding on the longitudinal creep behavior of the fiber-reinforced titanium matrix composites. Initially, a single fiber composite model based upon random fiber fragmentation and the shear-lag model is developed to establish the computer simulation route. Then, the technique is extended to multiple fiber composite for a complete representation of the real composite. 1. random fiber brea 2. microvoid format
2. Modeling creep behavior of fiber-reinforced metal matrix composite without damage 0.8
0.6
"4 0.4
0.2
0 0
Fig.
1. T y p i c a l
100
21)0
300 Time (hour)
400
500
tensile creep curve and corresponding
600
damage
mechanisms in SCS-6/Ti alloy matrix composites. tion along the reaction layer/matrix interface during the early stages of creep deformation. Matrix cracking that extended from the broken fiber ends were also observed during the later stages of creep deformation. The loss of the load carrying capability of the broken fiber will essentially increase the stress state of the matrix which, in turn, would reduce the creep resistance of the composite. Microvoid formation along the reaction layer/matrix interface will degrade the interfacial shear strength. As a result, the load transfer efficiency between the broken fiber and matrix will be degraded as a function of creep loading history. Meanwhile, the matrix cracking induced by the stress concentration near the broken fiber ends will gradually decrease the load carrying area of the matrix and further reduces the creep resistance of the composite. A theoretical prediction for the creep deformation and rupture life of the composite that incorporates these damage mechanisms has not been developed yet. The objective of this paper is to develop a theoretical model and a computer simulation methodology
The expected service temperatures of the CVD SiC fiber-reinforced titanium matrix composite range from 500 to 800°C. Under these conditions, the SiC fiber is known to possess excellent creep resistance and will deform elastically. However, the titanium alloys are susceptible to creep deformation. McLean (1985) proposed a model to describe the longitudinal creep deformation of an unidirectional fiber-reinforced metal matrix composite. The model assumed that the composite contained a power-law creep matrix, an elastically deformed fiber and a perfect fiber/matrix interface bonding. The overall strain rate of the composite can be described as = ~lO~2dm(
l -- Sl )m
(
1)
where eel =
[ 1-k-Ef~ ] -1 Era(1 - ~ ) ' ,
ce2 = (1 - ~ ) - m ,
where ~m represents the creep rate of the matrix carrying all of the applied stress o'0, Ef and Em are Young's modulus of fiber and matrix, respectively, Vf is the volume fraction of the fibers, m is the exponent constant of the matrix, and $1 represents a normalized value for the stresses carried by the fiber: Sl -
o-fVf or0
(2)
Meanwhile, $1 can be described by
$1 = Hl~, where H1-
~Ef or0
(3)
305
S, Lee et al./Mechanics of Materials 21 (1995) 303-312
For the composite under constant applied stress o'0, the initial conditions for Eqs. (1) and (3) are e(0) =
- ~ - Virgin' SCS~6, SPOOI~ l Oo=5142 MPa, m=16.2 ---<)-- Virgin SCS-6, SPOOL 2 <~o=3646 MPa, m=15.7 ----O--- As-Consolidated SCS-6 =3943 MPa, m=6.2
O"0
EfV~ + Em( 1 - V~)'
S1 (0) = Hie ( 0 ) .
(4)
The creep deformation of fiber-reinforced metal matrix composites without microstructural damage can be predicted by the governing equations, and initial conditions (Eq. ( 4 ) ) . Physically, S1 can be treated as a dimensionless internal stress (stress carried by the fibers) that modifies the matrix creep behavior. The constant H is a strain-hardening coefficient that control increments of the internal stress. If the fibers do not fail, the internal stress SI would gradually increase as the deformation accumulates, which, in turn, will reduce the stress carried by the matrix. When the fibers fully support the applied stress (S1 = 1 ), the composite will not deform as a function of loading time.
3. Modeling creep behavior with microstructural damage When damages are induced during processing and/or creep loading, the equations developed by McLean are inadequate to describe the creep behavior of the composites. Since the majority of the fibers remains intact, the creep deformation of the composite is basically controlled by the noncreeping fibers. In order to accurately predict the creep behavior, the following two damage-induced characteristics need to be taken into consideration. The first consideration is to change the internal stress Sl and its increase rate Sl resuiting from fiber breakage and microvoids formation between the reaction layer/matrix interface. To determine the effects of these two damage mechanisms on Sl, several aspects must be considered. These include the interfacial shear strength between the fiber and matrix, stress recovery at the broken fibers, fiber modulus degradation, and the statistical distribution of fiber strength. The second consideration comes from the modification of the matrix creep characteristics, dm, resulting from the stress concentration and matrix cracking near the broken fiber ends. In this paper, a major effort is made to study the effects of the fiber breakage and other constituent properties on the variations of internal stress (SI and Sl) and the
=,
-3
V
4 []
--~ 6/ 7.4
<> 7.6
7.8
[] 8
8.2
8.4
8.6
8.8
Ln (Our s)
Fig. 2. Strength distribution of SCS-6 fiber at as-fabricated and after compositeprocessing. creep strain of composite. A detailed discussion of each aspect is given in the following sections.
3.1. Fiber fragmentation and stress redistribution during loading The strength of a brittle fiber is limited primarily by the defects found along the length of the fiber and can generally be characterized by the Weibull distribution. The failure probability as a function of stress for a fiber with gauge length L is given by (Weibull, 1951 ) Pf
=
1 -exp
--~R \ o ' m /
J '
where o-m is a reference stress, LR is the reference length for o-m, and p is the Weibull modulus, which describes the strength variation. This equation provides the percentage of fibers in a bundle which are broken as a function of the stress. Fig. 2 shows a typical fiber strength characteristics of the SCS-6 fiber before and after composite processing (Gambone and Wawner, 1994). It clearly indicates that the Weibuli modulus of the SCS-6 fiber degraded from 16 to 6 before and after composite processing, respectively. The population of low strength fibers also increases significantly. These weak fibers will fail prematurely during the early stages of creep loading. When fiber breakage occurs in the composite, the matrix can transfer the stress back into the broken fiber through a recovery length & The stress distribution along the recovery length is strongly dependent
306
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
on the fiber/matrix interfacial condition. If the interfacial shear strength is strong enough to suppress the interfacial debonding, the load transfer will be activated by shear stress only. However, if interfacial debonding occurs, the load transfer in the debonded region will be achieved by frictional stress. The load transfer characteristics have been extensively studied using the shear-lag analysis (Dilandro et al., 1988). For simplicity, the shear and frictional stresses are assumed to be constant, Ty and rf, respectively, in this simulation. The stress in the broken fiber was also assumed to build up linearly in these two regions. Using these two shear components, the fiber stress near the broken end can be described by a bilinear stress recovery model and for a bonded region is given by (Henstenburg and Phoenix, 1989) 4Tfx O'f ( X ) --
d
,
(6a)
X < 6 -- 6b
Off --
d
d
o-f (LT -- nbt~) LT
(7)
where nb is the number of breaks, o-f is the applied fiber stress without breakage, and LT is the length of the fiber. For the debonded case, the effective average stress will be __
O'f [ L T
-- H b ( ~ -- ~ b ) ]
L
-- ~ r f b n b ~
(8)
For the creep deformation simulation, the internal stress in Eq. (2) is represented in a form that is prescribed either by Eq. (7) or Eq. (8).
O-f ( X ) = 4 r i ( 6 -- t~b) + 4"ry [ x -- (t~ -- t3b) ] b
_
O'f =
and for a debonded region by
8--8
tained. A detail discussion of the problem can be found in Curtin ( 1991 ) After fiber breakage, the effective load carrying capability of the composite will be gradually reduced. It is important to know the effective fiber stress for implanting the fiber breakage effect on the creep governing Eqs. ( 1 ) and (2). The effective average stress carried by the fibers, ~f, will depend on the recovery length 6 and whether or not debonding has occurred. For the perfect bonding case, the average stress is
' (6b)
where 6b is the length to which the interface remains inperfect bonding condition. As creep deformation proceeds further, stress in the fiber increases continuously. Fiber breakage can occur in the same fiber or in the adjacent fibers. A complete discussion of the fiber breakage sequence and its effective fiber stress distribution is given in the following section.
3.1.1. Single filament case For solving the single fiber fragment distribution at certain stress level, the model developed by Curtin ( 1991 ) has been used. The original solution was used to examine the fiber fragment distribution for a single fiber under tensile loading. In this model, a uniquestrength fragment distribution function P(x; n, 6) has been used to characterize the fiber fragment distribution as a function of applied stress and interfacial mechanical properties. By solving a set of coupled differential equations with a given fiber statistics (Weibull distribution) and interfacial quantities (interfacial shear strength and frictional stress), the fiber fragment distribution for a known stress can be ob-
3.1.2. Multiple filament case In a multiple fiber composite, the fragment distribution function described for a single filament composite may no longer apply because of interaction among the fibers. In order to account for these effects, a random fiber failure process based upon the Monte Carlo simulation is used. The simulation assumes that the interaction is based only on the load transfer characteristics between the broken fiber and matrix in the composite. Since the stress concentration induced by fiber breakage will be incorporated in the matrix creep characteristics, it was not considered in this fiber fragmentation simulation. In order to determine the fragment distribution for a multiple fiber composite, av. appropriate model must be chosen to represent the statistical strength distribution for each of the fibers. This can be accomplished by dividing the composite into a number of unit cells, where each cell represents a strength value determined by the Weibuil distribution. For a composite containing nf fibers, each fiber was further divided into nc cells, the total number of unit cells for the entire composite will be nf × he. Fiber strength values which followed by the Weibull distribution were assigned to
307
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
each of the nf x nc unit cells by the Monte Carlo technique. Fiber fragmentation in this model assumes that breaks occur in the center of the unit cell, and that the cells are sufficiently long so that the recovery length 6 does not extend into the other cells. If the stress state of the composite is known, the fiber fragment distribution can be determined by comparing the strength data with the fiber stress distribution. When there are n bc fibers break occurring in any given row of cells, the stress will be transferred to the matrix and fibers adjacent to the broken fibers. In the fiber broken plane, the additional stress that is released from the broken fibers is simply equal to the stress that was once carried by the broken fibers, Act0 = nco'fV b ~i ,
(9)
where Vfi (= Vf/nf) is the volume fraction taken up by each individual fiber. However, at any given distance x away from a fiber broken end, the total stress transferred to the other fibers and the matrix is Off (X) = Off -- O'f (X) ,
(10)
where o-f(x) is the stress recovery near the fiber breaks obtained by Eqs. (6a) and (6b). By substituting o-fR(x) for o-f in Eq. (9) and manipulating the parameters, the total stresses in the fibers and matrix adjacent to fiber breakage, as a function of distance x away from the plane of fracture, can be described by Tc
1+
r/c [Off -- O'f(X)]
,
,
(11)
for the fibers and n c [ o f f - O'f(X)] O"To(x) m = O"m +
1 --
where n f is the number of broken cells in the f - t h fiber, and the summation term adds together is the stress transferred from adjacent fiber to each of the k-th unbroken cells in the fiber. In a situation where debonding occurs, the average stress is given by ~fJ = Z
t--n
f (~-
~b)
.c-n
+ v__f[of(~-~b) +~ro, S] ~ (TO
k=1
°'fbnfS L
b
]
nf- ncbk/ (14)
The effective average fiber stress for the multiple fiber composite can be obtained using a rule of mixtures: ~f = v f i Z ~ f =1~ f ( j ) .
(15)
3.2. Effective elastic modulus of composite with fiber breakage The fiber breakage that occurred during creep loading will not only degrade the average fiber load carrying capability but also the composite modulus. In order to account for the degradation of elastic modulus, the model developed by Karandikar and Chou (1993) has been adopted and modified. The original model was used to predict the effect of matrix cracking on the degradation of the elastic modulus for a fiberreinforced brittle matrix composite. When the fibers are under an applied stress o'f the corresponding composite stress can be expressed by
Vf1
~fEc
O'0 J
~° -
1-Vf
Ef
(16)
(12) for the matrix, where the superscript To denotes the row of cells to which the stresses is transferred. Based upon the above equation, the effective average stress for the individual fiber can be obtained. For the perfect bonding situation ~f(j) = ~O"f L - n f ( ~ +
O'f Vf(~ OrO
nc(k)
~
b nf -- nc(k)
(13)
Then, by combining 0-8 and ~f by the rule-of-mixtures, the average static stress carried by the matrix becomes ~ m --
0- 8 - ~ f ~
1-~
,
(17)
where ~f is obtained from the stress redistribution simulation by adapting Eqs. (6a) and (6b). Dividing the average matrix stress by the matrix elastic modulus, the average static strain of the matrix can be obtained. Since there is no damage occurring in the matrix, this
308
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
static strain is equal to the average static strain in the composite,
--s
~m ----~cc"
i ~, ~
Governingequation l ~ = CtlC~2~m(l-Sl)m SI =H1E
(18)
6m = E-~
The reduced composite modulus can be calculated by dividing the applied static stress by the static strain: E,c = cr~Em
Calculatingc ando'f
(19)
--S O"m
fiberfi'agmentation i
~Yes
In order to incorporate the reduced modulus into computer simulation, it can be manipulated into an effective fiber modulus by the rule-of-mixtures. The resulting form for a single fiber composite will be (1 -E~(j) = ( o . . f E c / E f ) _
(20)
~f(j)E m (1 - V0
/~f~.i~ = (o-fEc/Ef)
(21)
butinn
calculatingeffective fiber stressand modulus T _T comparingwith failurecriteria
4NNo[
~f(J)~"'
With the same approach, the effective modulus for each of fibers in a multiple fiber composite will be
I
initail condition / E(O)=Oo/[EfVf +Em(1- Vf)]] Sl(0) = HI~(0) ]
[i
ic°mp°sitef_aiil-- ~_
Fig. 3. A computer simulation route for predicting creep behavior of the TMC.
- ~f(~)Vf
Then the overall effective fiber modulus for the multiple fiber composite is given as
E~-- v ~ i Z nr t f ./=lEf(j).
(22)
3.3. Computer simulation of creep deformation Based upon the above micromechanical models, the computer simulation route shown in Fig. 3 was developed. Before simulating the creep deformation, the fiber fragment distribution and fiber strength data were generated according to the above models. The generated data was used as a database for determining the fiber breakage as the creep loading proceeded. The composite was assumed to be damage free before creep loading. Also, the residual stresses caused by the thermal expansion mismatch between the fiber and matrix were not considered in the simulation since significant stress relaxation occurred near 600°C (E1dridge, 1992). Combining the creep Eqs. ( 1) and (2) and the initial conditions, the creep deformation and fiber stress distribution of the composite can be obtained as a function of loading time. For each time increment, the calculated fiber stress was compared with
the fiber strength data. When the fiber stress is higher than the strength of the weakest fiber, fiber breakage occurs. The stress redistribution in the fiber is evaluated. Meanwhile, the static effective elastic modulus of the composite as function of the number of fiber breakages is also simulated. The resulting "Sl and HI were further used for the simulation of the next step. By repeating the computation routes for each time increment, the creep deformation curve can be obtained for each loading condition. The failure of the single fiber composite was not considered. However, for the case of a multiple fiber composite, composite failure occurs when a certain row of unbroken fibers cannot sustain the applied stress. A fourth-order Runge-Kutta technique with a adaptive time scale was used for numerical analysis. The current model was developed to simulate the creep behavior of a titanium matrix composite reinforced with SiC fibers. Because the specific properties of the composite depend on the matrix composition and processing conditions, the simulation was conducted to study the effect of the various parameters of the materials on the creep deformation curves. For this reason, the parameters used in this simulation do not necessarily represent a specific material,
309
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
Table 1 Mechanical characteristics of the SCS-6 fiber Parameter
Value
Young's modulus (El) Weibull modulus (p) Fiber diameter (df) Reference strength (O'fR) Volume fraction (Vf) Fiber length (ld)
400 GPa 2-6 140/zm 4000 MPa 0.35 0.5 m
Table 2 Properties of interface and matrix used for computer simulation Parameter
Value
Matrix Young's Modulus (Em) Stress exponent (m) Strain rate (Em)
100 GPa 3-5 a 1.59 × 10 - 6 to 8.21 × 10-3 (hr- l ) b 20-160 MPa 20-100 MPa
Ty ~'f
Depending on the temperature. b Depending on applied stress and temperature.
a
but rather they encompass the entire range of properties pertaining to S i C / T i composites. For the SiC fiber, the characteristics o f a C V D grown SCS-6 fiber was used, which are listed in Table 1. For the titanium matrix, the mechanical properties and creep characteristics were obtained from available materials that have been used for composites such as Ti6242 ( T i - 6 w t % A1-2wt% S n - 4 w t % Z r - 2 w t % M o ) , Ti-15-3 ( T i - 1 5 w t % V - 3 w t % A1-3wt% C r - 3 w t % S n ) and Ti-6-4 ( T i - 6 w t % A1-4wt% V) alloys. Because the creep data for these materials were obtained at applied stresses below the simulation conditions, the strain rates were extrapolated to the applied stress values based on the linear relationship between In e and In o-0. The room temperature interfacial shear strength for the S C S - 6 / T i composites have been characterized to be between 120 to 160 MPa (Yang et al., 1991 ). However, due to the reduction o f the residual clamping at elevated temperatures, the interfacial shear strength of the composites was reduced to between 4 0 - 7 5 MPa at temperatures between 600 to 800°C (Eldridge, 1992). The post-debonded interfacial frictional stress o f the S C S - 6 / T i composite has also been measured to be between 60 to 100 MPa (Yang et al., 1991). Due to the relaxation o f residual clamping stress at elevated tern-
perature, the frictional stress was found to be 2 0 - 3 0 MPa (Eldridge, 1992). The properties o f the interface and matrix used in the computer simulation are listed in Table 2 (Bania and Hall, 1983; Bania et al., 1985; Rosenberg, 1980)
4. Results and discussion The numerical simulation described in Fig. 3 was performed on a composite with different properties o f the constituents. In order to study fiber breakage as function o f fiber stress in the SCS-6 fiber, a fiber with Weibull modulus p = 2 and a composite with perfect bonding was analyzed. The results are plotted in Fig. 4. It clearly seen that the first fiber breakage occurs around 1000 MPa. This implies that fiber breakage will occur immediately when the initial load is applied, and the broken segments will continuously fracture as the creep loading time is increased. The effect o f the fiber breakage on the creep behavior o f the composite is shown in Fig. 5. The results show that a significant amount o f additional strain will be induced by the fiber breakage. Furthermore, the effect o f fiber breakage and interfacial debonding on creep behavior was also simulated and plotted in Fig. 5. It becomes apparent that debonding at fiber broken ends will further increase the overall strain. The effects o f applied stresses on creep deformation of the composites were also simulated and compared with the prediction using the model without any fiber breakage. The result8
-
7 ~ 6
/i
ii
/
5 /
"~ Z
3
•
2
~
1
*
0
~:::::
/
0
• i
+
+
1000
2000
3000
4.000
Fiber stress (MPa)
Fig. 4. Fiber fragmentation distribution as a function of stress for single fiber composite.
310
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
0.0075
1000 MPa
Breakage and debonding ,}
0.007 ~
Breakage and perfect bonding
,t
0.0065
i
= No breakage 0.006
i,%*= = = =
I
, ,
,>
~
'
*:
>
/,
800 MPa
0.005 ~ , ~ * * 600 MPa 0.004
0.0055 i
0.003
I 0.0045
i 0
0.002 1000
2000
3000
4000
400 MPa ~- - ~
5000
~-
I
1000
2000
Time(hr)
3000
4000
5000
Time (hr)
Fig. 5. Creep curves for single fiber composite for (1) no fiber breakage, (2) fiber breakage without debonding, and (3) fiber breakage with debonding (o0 = 1000 MPa, 6m = 8.21 × 10-3 hr-t,m = 5, p = 2,~'y = 130 MPa, rf = 80 MPa).
Fig. 6. Effect of applied stress on creep of single fiber composite, with dashed curves representing no fiber breakage (m = 5, p = 2, Zy = 130 MPa, ~'f = 80 MPa)
ing creep curves were plotted in Fig. 6. For an applied stress below 600 MPa, the composites exhibited a similar creep response as predicted by these two models. This is because the applied stress is not high enough to cause fiber breakage to occur. This suggests that a critical threshold stress exists below which the fiber breakage mechanisms will not play any role in creep deformation. The effect o f the fiber strength distribution was also investigated in this simulation and the results are shown in Fig. 7. The creep strain o f the composite reinforced with an SCS-6 fiber having a low Weibull modulus ( p = 2) was approximately 30% higher than that o f composites with an high Weibull modulus fiber ( p > 4). This suggests that the increase of the low fiber strength population resulting from high temperature consolidation will significantly reduce the creep resistance o f the composite. The effect of the interfacial shear strength and post-debonded frictional stress on the creep deformation is shown in Figs. 8 ( a ) and 8 ( b ) . This clearly indicates that a composite with a low interfacial shear strength (20 MPa) exhibits a significant increase in creep strain (more than 4 0 % ) as compared with the one with a high interfacial shear strength (160 MPa). The post-debonded frictional stress also influences the overall strain o f the composite. With a frictional stress of 20 MPa, the creep strain in the composite is substantially higher
than in a composite with higher frictional stresses. Therefore, maintaining the integrity of the interface during creep loading is essential to facilitate load transfer from the creeping matrix to the noncreeping fiber. 4.1. Multiple fiber composite
When simulating the creep deformation of the multiple fiber composite, the selection of a suitable size of the computation cells is important for adjusting the computation time and the accuracy of the predicted creep behavior. A composite with constant length and 0.0075 -
0.007 ~
I
.~
ir
0.0065
0.006
•
2
~
4
¸
0.0055
0.005 1000
2000
3000
4000
5000
Time (hr)
Fig. 7. Effect of the Weibull modulus on creep of single fiber composite ( ~m 8.21 × 10-3 hr - l ). =
S. Lee et al./Mechanics o f Materials 21 (1995) 303-312
311 ~u
0.01
t
0.0053
i
•
1000 M P a
0.0048
800MPa
0.0043 0.008 .~ 0.0038 600
0.0033 0.006
0.0028 4
20
•
0.0023 i
100 0.004 0
i
i
i
I
,
1000
2000
3000
4000
5000
Time
0.0075
b. 0.0065
•
20
*
80
60 0.0055 I00 I 1000
2000
MPa
0.0018 0
1000
2000
3000
4000
5000
Time (hr)
Fig. 9. Effect of cell size and number on the creep of multiple fiber composite (rim = 5.51 X 1 0 - 4 hr-1, p = 2, ry = 130 MPa, "rf = 8 0 MPa).
0.0085 i i
0
400
(hr)
(a)
0.0045 t
MPa
L
I
r
3000
4000
5000
(hr) (b)
Time
Fig. 8. Effect of interfacial mechanical properties on creep of single fiber composite, (a) interfacial shear strength, (b) frictional stress (em = 8.21 × 10-3 hr- l , p = 2). with different cell lengths was used to investigate the cell size effect. The results are plotted in Fig. 9. The simulation results indicate that the differences in the creep strain for three cell sizes were within the 5% range, especially, the results for the 15 × 15 and 20 × 20 cases were almost identical (within 2%). This suggests that the predicted creep behavior for the multiple fiber composite is not sensitive to the cell size. Meanwhile, the creep behavior for multiple fiber composites with different interfacial properties and fiber characteristics was also studied. One of the results was compared with the single fiber composite, as shown in Fig. 10. This result shows that the creep strain predicted by the multiple fiber model is slightly higher than that by the single fiber model. However, the difference in the predicted creep strain from these two models was
minimal. This may be due to the stress concentration and matrix cracking effects which have not been considered in this simulation yet. Furthermore, the important distinction of the multiple fiber composite model is its ability to determine the rupture of the composite as shown in Fig. 11. When compared with Fig. 6, it clearly shows that the creep rupture life of a multiple fiber composite is predicted to be approximately 2500 hours at an applied stress o f 1000 MPa. 5. Concluding r e m a r k s The model and simulation presented in this work represent a methodology for predicting the effect of
0.0027
0.0025
/
"I
•
lOxl0
0.0023 !
~
I ~
0.0021 f
0.0019
[.
, 1000
--~ - 2000 Time
3000
15x15 *
I 4000
20x20
5000
(hr)
Fig. 10. Comparison between predicted creep curves for single fiber composite and 20 × 20 multiple fiber composite (~:m = 1.82 × 10-3 hr- l , p = 2,ry = 130 MPa, ~'f = 80 MPa).
312
S. Lee et al./Mechanics of Materials 21 (1995) 303-312
Acknowledgements
0.0041 i
0.0039
This work was supported by the National Science Foundation (DDM 905730) and Allied-Signal Corporation.
m m ,
0.0037
w~ •
0.0035
MFC
References
SFC 0.0033 "~
I ~" o.oo3J
+
0.0029 ~ 0
1000
2000
3000
4000
5000
Time (hr)
Fig. 11. Effect of applied stress on creep of multiple fiber composite (m = 5, p= 2, ,'ry 130 MPa, 7f = 80 MPa). =
fiber breakage on the creep behavior of a continuous fiber-reinforced metal matrix composite. However, other factors involved in the actual physical deformation of the material may need to be considered in further analysis. First, because the approach was from a load transfer standpoint, the effect of stress concentration and matrix cracking near the broken fiber ends was not considered yet. In fact, stress concentration and matrix cracking will make a significant contribution towards creep by causing local deformation in the matrix as well as further fiber fracture. Therefore, the creep equations described in this study need to be modified to include stress concentration and matrix cracking effects. Second, experimental data must be obtained to determine the validity of the model. Although some creep data does exist for S i C / T i composites, the amount of available information is limited. Also, because of the numerous parameters that are input into the simulation, experiments need to be designed specifically to test the limitations of this model. It should prove especially useful to obtain creep and fragmentation data for S i C / T i single filament composites, of which there is very little experimental data. Another factor to be considered is the formation of interfacial layers. It is well-known that materials such as titanium and SiC form a brittle interface zone, so that the deformation characteristics of this third constituent must be examined separately in the creep equations.
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