Roy M. Sullivan

Glenn Research Center, Cleveland, Ohio

Time-Dependent Stress Rupture Strength Degradation 

of Hi-Nicalon Fiber-Reinforced Silicon Carbide

Composites at Intermediate Temperatures

NASA/TM—2016-218939

January 2016
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Roy M. Sullivan

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Time-Dependent Stress Rupture Strength Degradation 

of Hi-Nicalon Fiber-Reinforced Silicon Carbide

Composites at Intermediate Temperatures

NASA/TM—2016-218939

January 2016

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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NASA/TM—2016-218939 1 

Time-Dependent Stress Rupture Strength Degradation of 

Hi-Nicalon Fiber-Reinforced Silicon Carbide Composites 

at Intermediate Temperatures 

Roy M. Sullivan 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135

Abstract 

The stress rupture strength of silicon carbide fiber-reinforced silicon carbide (SiC/SiC) composites with 

a boron nitride (BN) fiber coating decreases with time within the intermediate temperature range of 700 to 

950 °C. Various theories have been proposed to explain the cause of the time-dependent stress rupture 

strength. The objective of this paper is to investigate the relative significance of the various theories for the 

time-dependent strength of SiC/SiC composites. This is achieved through the development of a numericallybased progressive failure analysis routine and through the application of the routine to simulate the 

composite stress rupture tests. The progressive failure routine is a time-marching routine with an iterative 

loop between a probability of fiber survival equation and a force equilibrium equation within each time step. 

Failure of the composite is assumed to initiate near a matrix crack and the progression of fiber failures 

occurs by global load sharing. The probability of survival equation is derived from consideration of the 

strength of ceramic fibers with randomly occurring and slow growing flaws as well as the mechanical 

interaction between the fibers and matrix near a matrix crack. The force equilibrium equation follows from 

the global load sharing presumption. The results of progressive failure analyses of the composite tests 

suggest that the relationship between time and stress-rupture strength is attributed almost entirely to the slow 

flaw growth within the fibers. Although other mechanisms may be present, they appear to have only a minor 

influence on the observed time-dependent behavior. 

Introduction 

The stress rupture strength of silicon carbide fiber-reinforced silicon carbide (SiC/SiC) composites 

with a boron nitride (BN) fiber coating is time dependent within the intermediate temperature range of 

700 to 950 °C. The strength of these composites in the fiber direction under constant stress loading 

conditions decreases with time. Alternatively, one may view this behavior as a time delayed failure, 

where the time-to-failure is inversely proportional to the applied stress magnitude. 

This time-dependent strength behavior is illustrated by the experimental data plotted in Figure 1, 

which was originally published in Morscher, Hurst and Brewer (Ref. 1). Figure 1 contains fast fracture 

strength data at 815 °C and stress rupture strength data at 815 and 880 °C. The data was obtained in tests 

performed on Hi-Nicalon fiber-reinforced, melt infiltrated silicon carbide composites. The material was 

reinforced with a 5HS woven fabric, with a tow spacing of 17 ends per inch (epi). Each tow contained 

500 fibers, with an average fiber radius of 7 m. The tests were performed on flat dog-bone specimens 

which were 8 plies thick. The arrow on one of the stress rupture data points in Figure 1 signifies that this 

specimen did not fail. This test was interrupted at the time indicated before failure occurred. 

The time-dependent strength of SiC/SiC composites in the intermediate temperature range has been 

the topic of many previous studies. Various theories have been proposed to explain the cause of the 

time-dependent stress rupture strength (Refs. 1 to 11). A common theme among all proposed strengthlimiting mechanisms is that they are all enabled by an oxidation reaction that is expected to occur at a 

sufficient rate within the intermediate temperature range of 700 to 950 °C. 
NASA/TM—2016-218939 2 

Figure 1.—Stress versus time-to-failure of Hi-Nicalon composite specimens at intermediate 

temperatures. Data from Morscher, Hurst and Brewer (Ref. 1). 

A number of researchers (Refs. 1 to 7) attribute a time-dependent loss of strength to the oxidation of 

the BN fiber coating. The oxidation of the BN coating and the formation of silica at the interface causes 

the fibers to fuse to the matrix and to one another. They argue that failure of any of the fused fibers will 

result in a stress concentration in the nearest intact fiber, making it more likely that the nearest neighbor 

will also fail. This results in a cascading of fiber failures and failure of the composite cross-section at a 

stress value significantly lower than the fast fracture strength. This proposition is consistent with a timedependent strength degradation, since the process of oxidation and fiber fusing takes time and the extent 

to which the cross-section is embrittled will increase with time. As such, one would expect the stress 

rupture strength of the composite to decrease with time. 

More recently, Xu, Zok and McMeeking (Ref. 8) have suggested two other possible mechanisms 

which could be responsible for the loss of composite strength over time. The first involves oxidation of 

the SiC fibers and the second involves oxidation of the fibers or matrix within a matrix crack. Regarding 

the first, they have suggested that the growth of a silica scale on the fiber surface induces a tensile stress 

in the fibers. The tensile stress is generated by the difference in the molar volumes of SiC and SiO2. 

Under the second scenario, oxide growth within a matrix crack can act as a wedge which pries the matrix 

crack open. This also induces a tensile stress in the crack bridging fibers. In both scenarios, the tensile 

stress in the fibers increases with time since the magnitude of the tensile stress is a function of the amount 

of oxide growth. The difference between the inherent strength of the SiC fibers and the oxide growthinduced tensile stress results in the apparent strength reduction over time. 

A third group of authors (Refs. 9 to 11) propose that the observed composite strength behavior is due to 

an inherent time-dependent strength degradation of the fibers and that the cause of the fiber strength 

degradation is the slow growth of flaws within the fibers. They contend that slow flaw growth within the 

fibers is facilitated by oxidation of either SiC or the free carbon at the grain boundaries. These authors were 

influenced by the previous work of Davidge, et al. (Ref. 12), who recognized the effect of flaws on the time-
NASA/TM—2016-218939 3 

dependent and stochastic strength in ceramic materials and who established a three-way relationship 

between stress, time and the probability of failure. Based upon this previous work, Gauthier and Lamon 

(Ref. 11) developed a similar three-way relationship for Hi-Nicalon, Hi-Nicalon Type S, and Nicalon fibers. 

The objective of this paper is to investigate the relative significance of the various theories for the 

time-dependent strength degradation of SiC/SiC composites. This is achieved through the development of 

a numerically-based progressive failure analysis routine and through the application of the routine to 

simulate the composite stress rupture tests which produced the results in Figure 1. The progressive failure 

routine assumes that the progression of fiber failures occur by global load sharing. Failure of the 

composite is assumed to initiate near a matrix crack, as this is the location where the fibers carry the 

highest stress and likely have the highest probability of failure. 

The progressive failure analysis solution is based on an iterative solution between an equation for the 

probability of survival of the reinforcing fibers and an equation of force equilibrium within each time 

step. Both equations were derived from consideration of the mechanical interaction between the fibers 

and the matrix near a matrix crack (Refs. 13 to 19). The equation of force equilibrium is obtained from 

the work of Thouless and Evans (Ref. 15) and Curtin, et al. (Ref. 18). The progressive failure analysis of 

the composite is similar to the analysis described in Halverson and Curtin (Ref. 19), except herein the 

analysis is performed numerically and the formulation of the probability of fiber survival equation follows 

from the triadic relation between stress, time and probability of failure for ceramic fibers provided by 

Gauthier and Lamon (Ref. 11). 

The progressive failure analyses presented in this paper neglect the effects of BN fiber coating 

oxidation and the resulting fiber fusing and local load sharing. They neglect to include the effect of oxide 

scale growth either within the matrix cracks or on the fiber surfaces and the resulting stress that develops 

in the bridging fibers. The only time-dependent strength-limiting mechanism included in the progressive 

failure analyses is that proposed by Lamon and his coworkers (Refs. 9 to 11), which results from the timedependent growth of flaws within the fibers. Comparison of the results obtained by the progressive failure 

analysis to the measured behavior shown in Figure 1 reveals the significance of the slow flaw growth 

effect. By this comparison, one may also judge the significance of the other strength-limiting mechanisms 

proposed by the previous authors. 

Fiber Failure Model (Relationship Between Pf –  – t) 

The strength of ceramic fibers is limited by the presence of flaws either within the fibers or on the 

surface of the fibers. Fiber failure occurs when the stress intensity factor equals the fracture toughness KIc. 

The stress intensity factor is given by KI   f Y C , where f is the average stress in the fiber, C is the 

size of the critical flaw and Y is a unit-less geometrical parameter, which has a specific value for each 

flaw shape. Thus, the fast fracture strength of the fiber, denoted as s  f , is related to the initial flaw size Ci

according to 

2

















  Y

K C s

f

Ic

i (1) 

In stress rupture tests, the combination of the applied stress magnitude and initial flaw size are 

typically not large enough to result in immediate fiber failure. However, over time the flaws will slowly 

propagate under the fiber stress f. At some point, one of the flaws will reach a sufficient size to satisfy 

the failure criteria, KI = KIc. The time-to-failure is the time it takes a flaw to grow from its initial size Ci to 

the critical size under the fiber stress f. Previous authors (Refs. 9 to 11) claim that the slow growth of 

flaws in SiC fibers is enabled by oxidation at the fiber grain boundaries. 
NASA/TM—2016-218939 4 

Davidge, et al. (Ref. 12) developed an expression for the time-to-failure of ceramic materials, 

subjected to an applied stress . They assumed that the flaw growth velocity is a function of the stress 

intensity factor according to n KI v dC dt 1  /   , where 1 and n are flaw growth parameters, and 

obtained the following expression for the relation between the applied stress , the initial flaw size Ci and 

time-to-failure t: 

( 2)

2

1

(2 )/2

   



Y n

C

t n

n n i (2) 

Equation (2) may also be applied to ceramic fibers. A plot of t versus f obtained from stress rupture tests 

performed on a group of single fibers, all with an identical initial flaw size and shape, will lie along a 

curve defined by t A n  f  , where A is a constant. 

Both the fast fracture fiber strength and the product t n  f are stochastic quantities since they depend 

on the initial size of randomly occurring flaws on or within the fibers. It is not practical to obtain the flaw 

size distribution in a population of fibers. However, the nature of the statistical distribution of initial flaw 

sizes can be obtained from Equation (1) and the statistical treatment of the fast fracture strength. To this 

end, one may substitute Equation (1) into (2) and denote the applied stress  as f to obtain 

2

2

1 ( 2)

2 















 

   

n

Ic

s

n f

f Y n K

t (3) 

The statistical nature of the fast fracture strength of ceramic fibers is usually modeled using the 

two-parameter Weibull strength model (Refs. 12, 14, 17, 19, and 20). In the Weibull strength theory, the 

cumulative probability of failure of a population of ceramic fibers is given by 































    

m

o

f

o

f L S

L P 1 exp (4) 

where Lo is the reference length, m is the Weibull modulus and So is the characteristic strength of a fiber 

of length Lo (Ref. 17). The values of So and m are obtained from tensile strength tests performed on a 

population of individual fibers. Lo is the gage length used in the strength tests. The ratio L/Lo accounts for 

the length effect, i.e., the number of flaws in a length of fiber L are L/Lo times the number of flaws that 

occur in a length Lo. 

By rearranging Equation (4), the fiber fast fracture strength may be written as a function of the 

cumulative probability of failure and the statistical parameters according to 

m

f

o o s

f L P

L S

1/

1

1 ln 















   (5) 

Substituting Equations (5) into (3) leads to the following relationship between stress, time and probability 

of failure: 

m

n

f

o

n

Ic

n o f L P

L

K

S

Y n

t

2

2

2

1 1

1 ln

( 2)

2

 

















 















    (6) 
NASA/TM—2016-218939 5 

Figure 2.—Time-to-failure versus applied stress of Hi-Nicalon single fibers (gold circles) and tows 

(black squares) at 800 °C. Data from Gauthier and Lamon (Ref. 11). 

Equation (6) includes the flaw growth parameters, the statistical fast fracture strength parameters as 

well as the length effect. It is very similar to the equation derived in Gauthier and Lamon (Ref. 11) except 

that the equation derived by the previous authors contains an additional term (Note: The parameter 1 is 

denoted as A1 in Reference 11). Gauthier and Lamon (Ref. 11) state that the additional term is 

insignificant under most conditions of interest and may be neglected. A form of Equation (6) will be used 

in progressive failure analyses of the Hi-Nicalon composite stress rupture tests in the following sections 

of this paper. 

The time-to-failure versus applied stress reported by Gauthier and Lamon (Ref. 11) for Hi-Nicalon 

fibers and tows at 800 °C is plotted in Figure 2. The single fiber results are plotted as gold circles and the 

tow results are plotted as black squares. Figure 2 reveals the scatter in the initial flaw size and shape in 

Hi-Nicalon fibers, as it reveals the scatter in the product t n f . One can imagine a series of curves of the 

form t A n  f  , each one with a different value of A and each one passing through one or more of the 

single fiber data points in Figure 2. 

The values for the constants in Equation (6), which will be used in the progressive failure analyses, 

are listed in Table 1. Some of the values listed were the values listed in Table III of Gauthier and Lamon 

(Ref. 11) for Hi-Nicalon fibers. The rest were obtained by fitting Equation (6) to match the plots of timeto-failure versus fiber stress at specific values of the probability of failure. This is illustrated by the plots 

of t versus f at Pf = 99.8% and Pf = 0.2% shown in Figure 2. The dashed lines are the lines from 

Figure 7 in Gauthier and Lamon (Ref. 11) and the solid lines are a plot of Equation (6) using the constants 

listed in Table 1. The plots of t versus f obtained from Equation (6) and the constants in Table 1 provides 

a very good match to those from the previous authors. Both sets of plots encapsulate almost all of the 
NASA/TM—2016-218939 6 

TABLE 1.—TABLE OF CONSTANTS FOR THE Pf –  – t MODEL FOR Hi-NICALON 

FIBERS AT INTERMEDIATE TEMPERATURES 

n m K (MPa m ) Ic  (MPa) o S L (m) o (MPa ) 2 1 (2 ) / 2 1 n n Y s m       

9 4.3 1.4 61 0.025 3.0410–20

single fiber data points from Gauthier and Lamon (Ref. 11). Note that in Table 1 the quantities Y and 1

are not treated separately, but are accounted for by the product 2 1Y . This eliminates the need to identify 

the shape of the flaws, since the value of the product 2 1Y is determined directly from the 

aforementioned treatment of the test data. 

The only significant difference between the values listed in Table 1 and those suggested by Gauthier 

and Lamon (Ref. 11) is the value of 2 1Y . The source of this discrepancy lies in the method by which 

the size effect is treated by the authors. The previous authors use a volume ratio V/Vo to account for the 

size effect and assigned the reference volume a value of Vo = 1 m3

. Given the length (25 mm) and radius 

(7.5 m) of the single fiber tensile specimens, the volume ratio is V/Vo = 4.410–12. The present treatment 

of the experimental data assumes L = Lo = 25 mm. The result is a significant adjustment to the value of 

2 1Y . Further validation of the constants in Table 1 and the accuracy of the resulting fiber failure model 

will be obtained by the results of the tow tensile tests simulations discussed in the following section. 

Progressive Failure Routine 

A progressive failure routine was developed to simulate the progressive failure of a collection of 

fibers. The fibers are aligned in parallel with one another and collectively carry a total tensile load F. The 

progressive failure routine is performed under the assumption of global load sharing (GLS). When a fiber 

fails, the load it previously carried is distributed evenly to all remaining unbroken fibers. Under GLS, 

force equilibrium requires that the stress carried by each unbroken fiber is f = F/(N Af), where N is the 

number of unbroken fibers and Af is the area of each fiber. 

For a large number of fibers, the probability of survival is approximately equal to the ratio N/No, 

where No is the initial number of unbroken fibers. Rearranging Equation (6) to obtain an expression for Pf

in terms of f and t, and recognizing Ps = 1 – Pf, the probability of survival may be expressed as 





















 





























      2 2 2

1

2

( 2) exp n

mn

f

m

o

Ic n

m

o o s S

K

t

Y n

L

L

N

N P (7) 

The progressive failure routine is a time-marching routine with an iterative loop between the 

probability of survival and force equilibrium within each time step. The number of unbroken fibers N is 

calculated for the current time and current fiber stress using the probability of survival equation. An 

update to the fiber stress is calculated using the new value of N and the equilibrium equation. The 

iteration between the equilibrium and probability of survival equations occurs until convergence is 

obtained. When convergence is reached, time is marched forward and the process is repeated. Time-tofailure is determined numerically as the time step where solution convergence is not obtained within a 

large number of iterations. 

A FORTRAN source code was written to implement the progressive failure routine. The solution was 

performed on a desktop computer. 

 
NASA/TM—2016-218939 7 

Analysis of Tow Tests 

The progressive failure routine was applied to simulate the time-dependent strength behavior in the 

tow tensile tests reported in Gauthier and Lamon (Ref. 11). In these tests, the time-to-failure of 

Hi-Nicalon and Hi-Nicalon Type S tows was measured at various constant applied stress levels and test 

temperatures. The analyses reported here are focused on the Hi-Nicalon tests performed at 800 °C only. 

The Hi-Nicalon tows contained five hundred fibers (No = 500). The constants listed in Table 1 were used 

as input to Equation (7). 

The results of four progressive failure simulations are shown in Figure 3. The results are shown for 

simulations which used an initial tow tensile stress of 1100, 800, 600, and 400 MPa. The stress in the 

fibers is plotted in Figure 3(a) and the number of unbroken fibers is plotted in Figure 3(b). Both sets of 

results are plotted as a function of time. The time at which failure of the tow occurred in each of the 

simulations is obvious from the results in Figure 3. 

(a) 

(b) 

Figure 3.—Numerical results from progressive failure analysis of HiNicalon tow tensile tests at 800 °C. Results are shown for numerical 

simulations using an initial fiber stress loading of 1100, 800, 600, and 

400 MPa. (a) Fiber stress versus time. (b) Number of unbroken fibers 

versus time. 
NASA/TM—2016-218939 8 

Figure 4.—Initial applied stress versus time-to-failure from tensile tests on Hi-Nicalon tows at 800 °C. 

Measured data from Gauthier and Lamon (Ref. 11) are shown as black squares. Dashed line is a curvefit to the experimental data from Gauthier and Lamon (Ref. 11). Numerical results from progressive 

failure analysis are shown as the solid line with hollow circular markers. 

Figure 4 is a comparison of the measured and predicted results from the tow tensile tests. Here the 

applied stress is plotted as a function of time-to-failure. The experimental measurements from Gauthier 

and Lamon (Ref. 11) are shown as square markers. The dashed line is a curve-fit of the experimental data 

using the equation t A n   . The curve-fit values are A = 3.361026 and n = 8.34. These were obtained by 

Gauthier and Lamon (Ref. 11) using regression analysis. The progressive failure analysis results are 

shown as the solid black line with hollow circular markers. The results of the progressive failure 

simulations are in good agreement with the curve fit to the measured results provided by Gauthier and 

Lamon (Ref. 11). This provides a validation of the constants listed in Table 1, the resulting fiber failure 

model and the progressive failure analysis routine. 

Analysis of Composite Failure 

Mechanics at a Crack Plane 

In this section, a progressive failure analysis is performed for the fibers within a composite in an 

attempt to predict the time-to-failure versus applied stress behavior of the composite. It is assumed that 

failure of the composite initiates near a matrix crack, since the fiber stresses are considerably higher near 

a matrix crack than in an uncracked zone. The mechanics in the vicinity of a matrix crack have been 

described previously by many authors (Refs. 13 to 19). Some of the basic concepts that are necessary for 

the current discussion will be reviewed. 
NASA/TM—2016-218939 9 

Consider a collection of fibers in a composite specimen, aligned in the direction of loading and 

bridging a single matrix crack. Herein, the loading direction is denoted as the z direction. Over a short 

distance on either side of the crack plane, the fibers and matrix become de-bonded and may slip with 

respect to one another. Slipping is resisted by a shear stress τ acting at the fiber/matrix interface over a 

slip length l on either side of the crack plane. The shear stress is assumed to be constant along the slip 

length. At distances greater than the slip length away from the crack plane, the composite response is 

linear elastic with a modulus given by the rule of mixtures relation, E  f Ef  (1 f )Em . The symbols 

Ef and Em represent the Young’s modulus of the fiber and matrix, respectively, and f is the fiber volume 

fraction in the direction of loading. The slip length is given by l  (R/ 2)(1 f )Em / f E , where  is 

the far field composite stress and R is the fiber radius. 

The fiber stress is a maximum at the matrix crack plane having a value of T = /f. The fiber stress 

decreases linearly from its maximum value at the crack plane with a slope of 2τ/R reaching a value of 

Ef / E at, and beyond, the slip length. The variation of the fiber stress within the slip length is given by 

z T R z f  ( )   (2/ ) where z = 0 at the crack plane. The matrix stress in the loading direction at the 

crack plane is zero. The matrix stress increases linearly with distance from the crack plane with a slope of 

(2 / R)( f /1 f ) , reaching a maximum value of Em/ E at, and beyond, the slip length. 

Matrix cracking will occur at a composite stress value of mc. At this composite stress, the stress in 

the matrix is Em mc E m

mc    / . Consider now the possibility that multiple matrix cracks may form 

perpendicular to the direction of loading. Since the matrix stress at a crack plane is zero and increases 

linearly with distance from the crack plane with a slope of ) (2 / R)( f /1 f , there’s a length lm on both 

sides of a crack, given by 

fE

R f E l mc m m

(1 )

2





  , (8) 

where no additional cracks will form because the matrix stress in this length cannot reach the matrix 

cracking stress mmc . As loading is increased, cracks will continue to form until crack saturation is reached 

at which point the crack spacing x must satisfy lm  x  2lm (Refs. 13 and 14). The average distance 

between cracks at saturation is x  1.337lm (Ref. 21). 

Probability of Survival 

It is assumed that the failure of the fibers will only occur within the slip length near the crack plane so 

the contribution of the fiber stress outside the slip length is ignored. As previously discussed, the fiber 

stress varies linearly near a matrix crack and has a maximum value of T = /f at the crack plane. The 

probability of survival of a fiber with a varying stress along its length may be obtained by dividing each 

fiber into Q fiber segments each of length i  , such that 



 

Q

i

i l

1

2 . Each fiber segment carries a different 

stress value i depending on its distance from the crack plane. The probability of survival of a fiber is 

then the product of the probability of survival of all fiber segments 



Q

i

Ps Ps i

1

. 

 
NASA/TM—2016-218939 10 

The time-dependent probability of survival of a single fiber of length L under stress  is given by 

Equation (7). Thus, the time-dependent probability of survival of the chain of Q fiber segments each of 

length i  carrying a stress i would be 



 





















 





























     

Q

i

n

mn

i

m

o

Ic n

m

o

i s S

K

t

Y n

L

P

1

2

2 2

1

2

( 2) exp (9) 

The right hand side of Equation (9) may be replaced with the mathematically equivalent form 





















 





























     



  Q

i

n

mn

i

m

o

Ic n

m

o

i s S

K

t

Y n

L

P

1

2

2 2

1

2

( 2) exp (10) 

The expression for the probability of survival is obtained by replacing the summation sign in 

Equation (10) with an integral and the length segments i  with the differential length dz and by 

performing the integration of the fiber stress function f (z) over the slip length. It is more convenient to 

approximate the fiber stress variation near the crack plane with f (z) = T(1–z/l) and to approximate the 

slip length as l = TR/2τ (Ref. 15). Using these approximations and upon performing the integration one 

obtains 

 

































 







  





























       

2

2 1 2

2 ( 2) exp

2 2 2 1

n

n m

T

S

K

t

Y n

L

l

N

N P n

mn m

o

Ic n

m

o o

s (11) 

Force Equilibrium 

Fibers that break a finite distance from the crack plane are capable of carrying a load at the crack 

plane. The stress that a broken fiber can carry at the crack plane is referred to as the pullout stress and is a 

function of the shear stress transferred between the fiber and matrix and the distance between the fiber 

fracture location and the crack plane. This distance is referred to as the pull-out length. For a single matrix 

crack in the gage section, force equilibrium at the crack plane requires 

     F TNAf No N Af p ( ) (12) 

The first term in Equation (12) is the load carried by the unbroken fibers and the second term is the load 

carried by the broken fibers via the pullout stress. The quantity No – N is the number of broken fibers 

within the slip length of the crack and   p is the average pull-out stress, given by p   2 h / R , 

where h is the average pull-out length. The analyses discussed in the next section utilize an equation for 

the average pull-out length in terms of τ, R and the statistical parameters, So and m, which was given in 

Thouless, et al. (Ref. 16). Rearranging Equation (12) leads to the following equation for the fiber stress at 

the crack plane under the assumption of a single matrix crack: 
NASA/TM—2016-218939 11 

















  













    p o f o

o

N

N

N A

F

N

N T 1 (13) 

If there are multiple cracks within the specimen gage length, it is possible that the slip lengths 

associated with adjacent cracks may overlap. Following Curtin, et al. (Ref. 18), let us consider a crack in 

the matrix and designate this crack the central crack. Within the slip lengths on either side of the central 

crack plane there are 2l/x other cracks. The probability of survival (or failure) of a fiber is the same in 

each of the crack planes and may be calculated using Equation (11). The number of fibers that are not 

failed in any crack plane within the shear transfer length 2l of the central plane is given by 

N    N N l x o o   1 2 / . These fibers carry a stress T at the central plane. The other fibers that have 

broken in any one of the other crack planes cannot carry a stress of T at the central plane, since there is 

not sufficient distance between their fracture location and the central plane. It is assumed that, on average, 

they carry a pull-out stress of T/2 at the central crack plane (Ref. 18). 

Force equilibrium at the central crack plane now requires 

o f o o f o

N A T

x

l

N

N N A

x

l

N

N F T

2

2 1 2 1 1 1 















  













 



 



  













    (14) 

The first term is the load carried by the fibers which have not failed in any crack within the slip length of 

the central crack plane. The second term is the load carried by the pull-out stress of the broken fibers in 

the vicinity of all cracks within the slip lengths of the central crack plane. Rearranging Equation (14), one 

obtains the following equation for the fiber stress at the crack plane if multiple matrix cracks are present: 





 



 



 



  













  



x

l

N

N A N

F T

o

o f 1 1 1

1 (15) 

Analysis Results 

Progressive failure analyses were performed to simulate the progression of failure in the tensile tests 

performed on the Hi-Nicalon composites, reported in Morscher, Hurst and Brewer (Ref. 1). The tensile 

tests were performed on flat dog-bone specimens with a gage section width of 12.6 mm. The specimens 

were 8 plies thick, with a tow spacing of 17 epi and 500 fibers per tow. Thus, the total number of tows in 

the specimen gage section, in the direction of loading, was (12.6 mm/25.4)(17 epi)(8 plies)  67.5 tows 

and the number of fibers which are initially unbroken is No = 67.5(500) = 33,750. The thickness of 

8 plies is 2.1 mm and the average fiber radius is R = 7 μm. Thus, the initial fiber volume fraction in the 

direction of loading is f = 0.1965. 

Morscher and Cawley (Ref. 7) measured the crack density as a function of the applied composite 

stress for Hi-Nicalon-reinforced silicon carbide composite specimens. The crack density study was 

performed on a Hi-Nicalon composite similar to the test material that produced the results shown in 

Figure 1. The results of the crack density study by Morscher and Cawley (Ref. 7) are shown in Figure 5; 

the crack spacing, which is the inverse of the crack density, is plotted as a function of the composite 

stress. These results indicate that the matrix cracking stress is mc130 MPa and that the matrix crack 

spacing at saturation is x = 0.4 mm. Given these results, and using x = 1.337lm and Equation (8), an 

approximate shear stress value of 5 MPa is obtained. 

 
NASA/TM—2016-218939 12 

Figure 5.—Composite plot of crack spacing and ratio of slip length-to-crack spacing versus composite stress. 

Crack spacing versus composite stress data was obtained from crack density versus composite stress data 

on Hi-Nicalon composites reported in Morscher and Cawley (Ref. 7). 

The choice of whether the single or multiple matrix crack equilibrium equation is more appropriate 

depends on the value of the ratio l/x. Equation (13) applies as l / x  0 . Otherwise, Equation (15) is the 

more appropriate choice. The value of l/x was calculated for the range of composite stresses used in the 

tensile tests that produced the results shown in Figure 1. The slip length was calculated at each composite 

stress value from l = TR/2τ using τ = 5 MPa and R = 7 μm. The crack spacing results obtained from 

Morscher and Cawley (Ref. 7) were used to calculate the ratio l/x as a function of composite stress. Note 

that the crack spacing data was extended to higher composite stress values since the highest stress in the 

crack density study by Morscher and Cawley (Ref. 7) was only 227 MPa. This is indicated with a dashed 

line and hollow markers in Figure 5. It was extended under the assumption that crack saturation was 

reached at 180 MPa. The slip length-to-crack spacing ratio is also plotted as a function of composite stress 

in Figure 5. Given these results, it appears that the multiple matrix crack equation of force equilibrium 

should be applicable for the range of composite stress values which are of interest here. For example, one 

would expect to have at least two other cracks within the slip length of each matrix crack at composite 

stress values above 160 MPa, where l/x1. 

The progressive failure analysis simulations of the Hi-Nicalon composite tests were performed using 

Equation (11) for the probability of survival and either Equations (13) or (15) for the equation of force 

equilibrium. The time-marching scheme with iterations between the equilibrium and probability of 

survival equations within each time step were performed in a similar manner to that described previously. 

Within each time step, Equation (11) is used to obtain an update to the number of unbroken fibers N using 

the current time and the previous value of T. The slip length is calculated with l = TR/2τ. The equilibrium 

equation then provides an update to the value of T. The process is repeated until convergence is reached. 

Time is then marched forward to the next time step. 
NASA/TM—2016-218939 13 

Figure 6.—Comparison of progressive failure analysis results with the experimental data from Morscher, 

Hurst and Brewer (Ref. 1) on Hi-Nicalon fiber-reinforced composites at intermediate temperatures. 

Progressive failure analysis results are shown for three series of analysis solutions: a single crack 

formulation (using Eq. (13) as equilibrium equation) with = 5 MPa, a multiple crack formulation (using 

Eq. (15) as equilibrium equation) with = 5 MPa and a multiple crack formulation with = 3.5 MPa. 

A series of progressive failure analyses were performed for each set of solution parameters. Within 

each series, progressive failure analyses were performed for a variety of applied composite stress 

magnitudes and the time-to-failure was calculated for each stress magnitude. The results from three series 

of progressive failure analysis solutions are shown as the solid black lines in Figure 6. The experimental 

results from Morscher, Hurst and Brewer (Ref. 1) are included for comparison. One series of solutions 

assumed a single matrix crack and used Equation (13) as the equilibrium equation. A shear stress value of 

5 MPa was used in these solutions. Another series of solutions assumed multiple matrix cracks within the 

gage section and used Equation (15) as the equation of force equilibrium. The ratio of l/x as a function of 

composite stress shown in Figure 5 was used as input to Equation (15). These solutions also used a shear 

stress of 5 MPa. This series of solutions will be referred to as the baseline solutions. As one would expect 

from the discussion of the ratio l/x in Figure 5, the results obtained using the multiple matrix crack 

formulation are closer to the measured results than those obtained using a single matrix crack formulation. 

The baseline solutions did not produce results that were sufficiently close to the measured fast 

fracture strength. The progressive failure analysis method must be able to predict the fast fracture strength 

in order to have any expectation of predicting the composite strengths at longer times. There are a handful 

of possible reasons for the discrepancy between the baseline progressive failure analysis predictions and 

the measured fast fracture strength. One reason may be that the true shear stress between the fibers and 

matrix may not be exactly 5 MPa, which is the estimated shear stress value from the calculations using 

the crack density measurements. An additional series of progressive failure analysis solutions were 
NASA/TM—2016-218939 14 

performed with the multiple matrix crack formulation and a slightly lower shear stress value. Using a 

lower shear stress value shifts the results either to a lower composite strength or a shorter time-to-failure, 

since using a lower shear stress value will result in a longer slip length and thus a lower probability of 

survival for the same composite stress magnitude. The results in Figure 6 show that by reducing the shear 

stress to a value of 3.5 MPa, there is good agreement between the predicted results and the measured fast 

fracture strength. Note that the value of 3.5 MPa is very close to the value obtained by Marshall and 

Evans (Ref. 22) for other SiC fiber-reinforced ceramic composites. They consistently obtained values in 

the range of 2 to 2.5 MPa using multiple techniques. Note also that, for the multiple matrix crack 

solutions using a shear stress value of 3.5 MPa, there is a small difference between the measured and 

predicted composite strengths at later times. 

The matrix cracking stress has been identified with a dashed horizontal line in Figure 6. Note that the 

data point representing the specimen that did not fail in the tensile tests lies slightly below the matrix 

cracking stress value. It’s not surprising that this specimen did not fail at the applied stress value, since 

the stress is much lower, and the probability of survival is much higher, for fibers in an un-cracked matrix 

than for fibers bridging a matrix crack. 

Another possible reason for the discrepancy between the baseline progressive failure analysis 

predictions and the measured fast fracture strength may be that some of the fibers were damaged during 

processing. Progressive failure analyses were performed to investigate the effect of fiber damage. This was 

achieved by starting the progressive failure analysis simulations with a reduced initial fiber volume fraction. 

Progressive failure analysis results for an assumed fiber damage of 9 percent are shown in Figure 7. The 

experimental results from Morscher, Hurst and Brewer (Ref. 1) and the baseline solution results are included 

for comparison. Both sets of progressive failure simulation results shown in Figure 7 used the multiple 

Figure 7.—Comparison of progressive failure analysis results with the experimental data from Morscher, Hurst 

and Brewer (Ref. 1) on Hi-Nicalon fiber-reinforced composites at intermediate temperatures. Progressive 

failure analysis results are shown for two series of progressive failure analysis solutions: a multiple matrix 

crack formulation with no fiber damage and a multiple matrix crack formulation with 9 percent fiber damage. 

Both sets of progressive failure analysis solutions used a shear stress value of 5 MPa. 
NASA/TM—2016-218939 15 

matrix crack equilibrium equation and a shear stress value of 5 MPa. Notice that the progressive failure 

analysis solutions using a shear stress of 5 MPa and 9 percent fiber damage produced a very similar set of 

results as the solutions shown in Figure 6 that used a shear stress of 3.5 MPa and no fiber damage. In both 

cases, the predicted strengths are very close to the measured fast fracture strength and there is only a small 

difference between the measured and predicted composite strengths at later times. 

A third possible reason for the discrepancy between the baseline progressive failure analysis solution 

results and the measured fast fracture strength may be due to the fiber architecture in the composite. Keep 

in mind that the progressive failure model was developed from data on straight fibers. In the composite 

system which is the subject of the current discussion, the reinforcement is in the form of a woven fabric. 

Thus, the tows are crimped (or curved) at regular intervals along the length. Applying a tensile load to 

curved tows results in a higher tensile stress, within the tow cross-section, than a tensile load on straight 

tows due to the additional stress caused by bending. If one were to account for the additional stress caused 

by crimped tows in the baseline progressive failure analysis, the baseline results plot would be shifted 

closer to the measured results. The magnitude of this shift has not yet been determined. The effect of fiber 

crimping on the progressive failure analysis will be left as a topic of future study. 

Discussion of Results 

The progressive failure analysis solutions were successful in simulating the measured relationship 

between time-to-failure and composite stress for a Hi-Nicalon composite tested within the intermediate 

temperature range. The progressive failure analysis solutions were developed under the assumption that 

composite failure initiated at a matrix crack and that the progression of fiber failure occurred by global 

load sharing. Failure of a fiber is stochastic and time-dependent due to flaws within the fiber. A model for 

the statistically-based and time-dependent fiber strength proposed by Gauthier and Lamon (Ref. 11) was 

utilized in the progressive failure analyses. No other strength-limiting mechanism, such as BN fiber 

coating oxidation or silica formation in a matrix crack or on the fibers, was included in the formulation. 

The success of the progressive failure analysis in simulating the measured relationship between time-tofailure and composite stress indicates that the time-dependent strength degradation in SiC/SiC composites 

at intermediate temperatures is primarily due to the intrinsic time-dependent strength of the fibers, which 

is the result of slow growth of flaws in the fibers. 

Calculations for the ratio l/x in the Hi-Nicalon composite tests indicate that the selection of the 

multiple crack equilibrium equation is appropriate for these simulations. Furthermore, calculations for the 

shear stress using the measured crack spacing at saturation and the first matrix cracking stress reveal that 

the shear stress between the fibers and matrix is approximately 5 MPa. A very good match between the 

experimental data from Hi-Nicalon composite testing and the progressive failure analysis results was 

obtained using the multiple matrix crack equilibrium equation. Two sets of analysis solutions provided 

the best correlation with the measurements: one that used a shear stress value of 5 MPa and assumed 

9 percent fiber damage and one that used a shear stress value of 3.5 MPa and assumed no fiber damage. 

Other combinations of shear stress and fiber damage values would provide equally favorable results. 

These simulations would necessarily require a shear stress value somewhere between 3.5 and 5 MPa and 

an assumed fiber damage less than 9 percent. These findings provide validation of the progressive failure 

analysis solutions, since all of these solutions require a shear stress value close to the estimated value of 

5 MPa and a reasonable estimate of fiber damage. 

The difference between the measured and predicted strengths at longer times may be due to the other 

strength-limiting mechanisms which were discussed in the introduction. However, in view of the results 

in Figures 6 and 7, it appears that these effects are secondary. Although the conclusions drawn here 

regarding the mechanisms responsible for the time-dependent strength behavior in SiC/SiC composites 

were based on testing and analysis of a specific composite material in a specific series of stress rupture 

tests, it is believed that these same conclusions apply to a wide range of SiC/SiC composites. 
NASA/TM—2016-218939 16 

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