July 2010 

NASA/TM–2010-216721 

A Critique of a Phenomenological Fiber 

Breakage Model for Stress Rupture of 

Composite Materials 

James R. Reeder 

Langley Research Center, Hampton, Virginia 
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July 2010 

NASA/TM–2010-216721 

A Critique of a Phenomenological Fiber 

Breakage Model for Stress Rupture of 

Composite Materials 

James R. Reeder 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Abstract

Stress rupture is not a critical failure mode for most 

composite structures, but there are a few applications where it 

can be critical. One application where stress rupture can be a 

critical design issue is in Composite Overwrapped Pressure 

Vessels (COPV’s), where the composite material is highly and 

uniformly loaded for long periods of time and where very high 

reliability is required. COPV’s are normally required to be 

proof loaded before being put into service to insure strength, but 

it is feared that the proof load may cause damage that reduces 

the stress rupture reliability. Recently, a fiber breakage model 

was proposed specifically to estimate a reduced reliability due to 

proof loading. The fiber breakage model attempts to model 

physics believed to occur at the microscopic scale, but validation 

of the model has not occurred. In this paper, the fiber breakage 

model is re-derived while highlighting assumptions that were 

made during the derivation. Some of the assumptions are 

examined to assess their effect on the final predicted reliability. 

By improving some assumptions at the micro-mechanics level, 

the predicted reliability was shown to improve with proof 

loading instead of being drastically reduced, as predicted by the 

original model. The fiber breakage model may be useful for 

understanding the underlying mechanism of stress rupture 

failures. However, until validation of the model at both the 

micro and macro scales can be accomplished, it should not be 

used for programmatic reliability estimates. Even in the 

improved form, the fiber breakage model is undependable 

because it still relies on some assumptions at the micromechanics level that have not been validated and to which the 

predicted reliability is quite sensitive. Therefore, the effect of 

proof loading on reliability is still unknown, and testing should 

be performed to understand the true effect of proof loading.

Nomenclature

α – the Weibull shape parameter for composite (strand or vessel) strength distribution

β – the Weibull shape parameter for a stress rupture distribution

γ – shear strain in composite matrix

δ – overload region in fiber breakage model

ς – shape factor for the strength distribution of individual fibers

μ – matix shear modulus

σ – stress (composite stress unless otherwise noted)

σf– nominal stress in fiber

!

" vessel

o – composite stress in the vessel when the vessel is pressurized to its burst scale parameter 

pressure

!

"# e

o – scale strength level for fibers with length δe

θ – power law exponent in a visco-elastic creep model

ρ – classic model parameter describing stress rupture scale parameter changes with stress ratio
2

ζ – the Weibull shape parameter for individual fiber strength

τ – shear stress in composite matrix

D – ratio of composite stress to fiber stress 

j – cluster size

jc– critical cluster size leading to vessel or strand failure

jp – critical cluster size needed for continued growth after proof

J(t) – visco-elastic compliance of matrix material

Je – elastic compliance of matrix material

K – stress concentration around a cluster of broken fibers

l – characteristic length used to describe individual fiber strength

n – number of fibers immediately adjacent to a cluster of broken fibers

N – number of characteristic lengths of fiber in a structure (strand or vessel)

P – Probability of Failure

Pj– Probability of a cluster growing from size j-1 to j

!

P1"j

– Probability of a cluster initiating and growing to size j

R – Reliability (Probability of Survival)

s – stress ratio defined as nominal stress divided by the strength scale parameter (stress ratio can 

be defined at the fiber or composite level)

t – time 

tc – time constant for matrix creep

tref – scale parameter for stress rupture failure time at s=1

tsafe – time required for a danger area to initiate along a fiber neighboring a broken fiber cluster

!

small

probability – indicates a small probability approximation (1-e

-x

≈x when x<0.1)

!

t>>tc

t>>0

 – simplification of equations at long time periods

Superscript

cp – conditional on proof survival (conditional reliability given survival of proof loading)

ip – including the chance of failure during proof loading

o – characteristic value (Weibull scale parameter value which corresponds to a 63.2% chance of 

survival)

p – related to a proof load

Subscript

e – related to elastic response

f – related to an individual fiber

j– related to a cluster of fibers of size j

o – nominal value, e.g. stress in fibers away from a fiber break.

p – related to a proof load

vessel – related to vessel reliability or probability of failure

1→j – combined probability of cluster growing from cluster size 1 to j

Acronyms

SR - Stress Ratio

WSTF -White Sands Test Facitilty

LLNL -Lawrence Livermore National Laboratory

NESC –NASA Engineering and Safety Center
3

Introduction

Stress rupture of composite material is a critical failure mode in a few composite applications 

such as Composite Overwrapped Pressure Vessels (COPV’s) where the composite material is 

highly and uniformly loaded for long periods of time. Although time delayed failures under 

constant load are quite rare, applications that demand high reliabilities must address the stress 

rupture mechanism. Traditionally, the probability of stress rupture failure is addressed by 

empirical models where the probability of failure measured at very high stress levels is 

extrapolated to lower stress levels where the chance of failure is too low to be adequately 

measured. 

There are a number of empirical models that are generally quite similar which can be used to 

estimate reliability as a function of time and stress level. Equation 1 is an example of an 

empirical model that will be called the “classic” model[1]. This model is in the mathematical 

form and notation proposed by Phoenix and predicts the life of a COPV as a Weibull distribution 

where the scale parameter is a powerlaw function of the stress ratio. 

!

Pvessel(t) =1" e

" t

s

"# tref

$

%

&

&

&

'

(

)

)

)

*

(1) 

 where stress ratio, 

!

s = nominalstress

strength scale parameter = ""o

This classic model has three curve fitting parameters (ρ, β and tref) which are fit to experimental 

data at high stress ratios. The model then allows predictions at much lower stress levels where 

the probability of failure is too low to be adequately measured. COPV are often designed around 

stress ratios of 0.5. An example of estimates from the classic model are shown in Figure 1. The 

results in Figure 1 are plotted as the number of 9’s of reliability. Reliability is the probability of 

not having a failure, and since one wants to design for high reliability (low chance of failures), 

the reliability value gets very close to 1, e.g. 0.999 or 0.99999. Here three 9’s of reliability 

implies that 1 out of 1000 articles are predicted to fail or five 9’s is 1 out of 100,000. By using 

Figure 1. Weibull plot of reliability including proof loading effects. 
4

the scale of 9’s of reliability, predictions in the high reliability region can be more easily 

distinguished. Precisely, the scale is

!

Weibull Scale = "(Log["Ln(R)]) = "(Log["Ln(1" P)]) (2) 

 where R is the reliability and P is the probability of failure.

This scale is a Weibull scale because a Weibull distribution will appear linear. However, it is a 

specific Weibull scale where above 1, it can be directly interpreted as the number of 9’s of 

reliability as indicated on the figure and at zero it is the Weibull scale parameter. From the 

figure, one can see that at a stress ratio of 0.5 the predicted reliability after 10 yrs (104.9 hrs) is 

approximately six 9’s or 1 in a million chance of failure. These predictions were made with the 

curve fitting parameters given in Table 1, which have previously been used for T1000/Epoxy 

composites [2].

In 2004, the NASA Engineering and Safety Center (NESC) sponsored an engineering review of 

how the risk of stress rupture failures has been addressed [1]. During that study, a critical issue 

was identified that COPV’s in service are exposed to proof loading prior to use. It is feared that a 

proof load could cause damage in the COPV that would actually reduce its reliability. This effect 

cannot be modeled with traditional models of carbon vessels. One can account for proof loading 

with the classic model, but because of the low beta found when fitting carbon data (β=0.22 used 

here), the model predicts an ever decreasing rate of failure with time. A reliability after proof can 

be calculated with the classic model, but because stress at a higher level behaves as a lower stress 

for a much longer period of time, the decreasing failure rate of carbon composites will cause the 

predicted reliabilities after proof to be much higher. In the example shown in Figure 1, a 5 min. 

proof load at 0.75 stress ratio causes the predicted reliability to increase from six 9’s to greater 

than fifteen 9’s. This is a nine order of magnitude increase in reliability. Other traditional 

models, like the classic model, will never be able to predict a decreased reliability due to a proof 

load. Because the effect of proof loading in traditional models is to drastically increase the 

reliability estimate, and this increase cannot be verified, the effect of proof loading is generally 

omitted when making reliability estimates. So in the current example, the estimated reliability 

would be six 9’s for a COPV in service for 10 yrs.

To address the fear that a proof load could damage a COPV and result in lower predictions of 

reliability, a fiber breakage model was proposed [2]. The fiber breakage model attempts to model 

physics thought to occur at the microscopic scale, but validation of the model has not occurred. 

In the example presented in Figure 1, the predicted reliabilities from the fiber breakage model, 

dropped from six 9’s to two 9’s due to the proof load. (Fiber breakage model equations and 

Table 1. T1000 composite parameter input values for the classic and fiber breakage models 

Classic Model Fiber Breakage Model

Model Parameters

tref=0.001 hr

β=0.22

ρ=114

tc=0.01 time constant for matrix power law creep

θ=0.11 exponent for matrix power law creep

jc=5 critical cluster size

ζ=5 shape parameter for individual fiber strength

α=25 shape parameter for vessel strength (which is jcζ

so not independent in this case)
5

parameters used in the example problem will be presented later in the paper.) This dramatic 

decrease in the reliability estimate (four orders of magnitude), of course caused concern among 

project managers responsible for managing risk in programs that use COPV’s. In this paper, the 

fiber breakage model is re-derived while highlighting assumptions that were made during the 

derivation. Some of the assumptions are examined to assess their effect on the final predicted 

reliability. 

General description of the fiber breakage model

The fiber breakage model [2] assumes 

that the stress rupture response is due 

to individual fiber breaks at the 

microscopic level as shown in Figure 

2. When an individual fiber breaks, 

the load that had been carried by that 

fiber must be transferred through the 

matrix to the neighboring fibers thus 

causing a stress increase in the 

neighboring fibers over some distance 

close to the fiber break. If all the 

fibers had essentially the same 

strength, the failure of one fiber 

would cause an overload in the neighboring fibers. This would instantaneously cause a cascade of 

fiber failures leading to the failure of the composite. The strength of carbon fibers has been 

shown to exhibit a high degree of scatter [3]. Therefore, when the weakest spot in a group of 

fibers fails, there is a significant probability that the surrounding fibers will all have significantly 

higher strengths and be able to carry the elevated stress level caused by that first failure. When 

the composite stress is raised high enough, there will be many individual failed fibers and some 

of these failed fibers will have created an overload in neighboring fibers that is large enough to 

fail a neighboring fiber, thus causing an even higher overload in the remaining fibers. This 

results in groups of failed fibers in the composite (also called clusters). Finally, the stress 

concentrations created by the larger groups of failed fibers becomes large enough that it is likely 

that the remaining fibers will not be able to withstand the elevated stress level, resulting in an 

unstoppable cascade of fiber failures and failure of the composite. The strength of the composite 

is therefore described by the stress level required to create a broken fiber cluster of adequate size 

to initiate an unstoppable cascade of failures. This micro-mechanical description of progressive 

composite failure has been proposed by many researchers and is the prevailing micro-mechanical 

theory regarding composite strength [4]. 

The fiber breakage stress rupture model then introduces a time dependent response due to the 

visco-elastic response of the matrix. Because the matrix transfers the load from a broken fiber to 

its neighbors through shear and because polymeric matrix materials will creep over time, the 

displacement will increase with time under constant stress. For a broken fiber in a composite, this 

means that the load that was carried by the broken fiber gets transferred to the neighboring fibers, 

and over time, the length of the neighboring fibers that are overloaded will increase from 2δe to 

2δ as shown in Figure 3. The amount of stress increase in the neighboring fiber can be 

characterized by a stress concentration (K) that is a function of the number of neighboring fibers 

that share the load and the number of fibers in a broken fiber cluster. With the increasing length, 

there is a chance that a newly overloaded portion of the neighboring fiber will have a weak spot 

and fail. This newly overloaded region will be referred to as the danger area, the area where 

additional damage could occur with time. So, without any increase in global stress, the increasing 

Figure 2. Fiber break geometry. 
6

danger area creates the 

potential for clusters of 

broken fibers to grow 

and eventually creates 

a critical cluster size 

that propagates to 

failure.

The fiber breakage 

model predictions of 

reliability after proof 

are quite different 

from the classic model 

predictions. With the 

fiber breakage model, 

a detrimental effect of 

proof loading can be 

predicted because the proof loading will create many more broken fibers and larger cluster sizes, 

all of which cause elevated stresses in neighboring fibers. The danger area associated with each 

break then grows with time. Therefore, the proof load causes an increased total danger area along 

neighboring fibers because of the additional number of breaks. However, the elevated stress 

region creating a danger area around a fiber break after proof is more complicated because all of 

the fibers endured an elevated load during proof. For the neighboring fiber to still be unbroken, 

the fiber strength must be higher than the overload that occurred during proof as illustrated in 

Figure 4. For small cluster 

sizes, the stress 

concentration caused by 

the broken cluster may not 

be larger than the stress 

endured by the 

neighboring fiber during 

proof (i.e. Kσf <

!

" f

p

), and 

therefore these clusters of 

broken fibers are predicted 

to never grow with time. 

In contrast, larger fiber 

clusters, where Kσf <

!

" f

p , 

may grow with time and 

have the potential to 

eventually cause a stress 

rupture failure. 

Mathematical derivation of the fiber breakage model. 

In this section, the equations used to calculate the fiber breakage model reliabilities will be rederived in order to highlight the assumptions that are made during the derivation. The next 

section will look at the effect of some of these assumptions.

 Figure 3. Change in fiber stress near fiber break with time.

 

Figure 4. Change in neighboring fiber danger area after proof loading.
7

Visco-elastic response of matrix 

The fiber breakage model assumes the matrix obeys a power law creep compliance given by

!

J(t) = Je 1+

t

tc

"

#

$

%

&

'

) (

*

+

+

,

-

.

. (3)

where J(t) is the matrix compliance as a function of time

Je is the elastic compliance of the matrix

tc is a time constant for the matrix creep

θ is a power law exponent

In a simple elastic shear lag model [5], a characteristic distance δe over which the broken fiber 

stress is shared with neighboring fibers is related to the effective shear modulus of the matrix μ

by: 

!

"e # 1

μ

= Je (4) 

The fiber breakage model assumes that the time dependent length of the overload region can be 

described by simply substituting the time dependent compliance into the elastic equation for the 

characteristic distance.

!

"(t)

"e

= J(t)

Je

=

Je 1+

t

tc

#

$

% &

'

(

)

Je

*"(t) = "e 1+

t

tc

#

$

%

&

'

(

)

t>>tc

t>>0

"e

t

tc

#

$

%

&

'

(

)

2

(5) 

where δe is the overload distance due to the elastic response of the matrix

δ(t) is the overload distance as a function of time

!

t>>tc

t>>0

 simplification of equations at long times periods

One critical assumption inherent in these equations is that:

• the visco-elastic response of the matrix can be represented by an elastic model with a time 

dependent compliance.

The implications of critical assumptions will be examined later in the paper. 

Probability of broken fiber cluster growth

By using the time dependent deformation in the overload region, the stress rupture response can 

be related to the following parameters:

1. the increased overload distance as a function of time, δ(t)

2. the likelihood of having an additional break in the overload distance

3. the size of the broken fiber cluster, j

4. the total number of broken fiber clusters that could grow

5. the probability of cluster growth under constant load 
8

Assuming the strength of a fiber obeys a Weibull distribution, the probability of a failure in a 

fiber of length, l, exposed to a fiber stress level σf is

 

!

Pfib =1" e

"

# f

# !

o

$

%

&

&

'

(

)

)

*

small

probability

Pfib = # f

# !

o

$

%

&

'

(

)

*

(6) 

where 

!

" !

o

 is the strength scale parameter for fibers of length l

ζ is the fiber strength shape parameter

 

!

small

probability indicates a small probability assumption which is used extensively in this 

derivation and is valid whenever the exponent in the probability equation is 

small (1-e

-x

≈x when x<0.1). This approximation greatly simplifies the 

probability equations. 

Note that the probability of the fiber surviving (i.e. the reliability) is just the converse of the 

probability of failure.

 

!

Rfib =1" Pfib = e

"

# f

# !

o

$

%

&

&

'

(

)

)

*

(7) 

Because the strength of a fiber would obey a weakest link behavior, the strength distribution of a 

fiber would change with the fiber length. The probability of fiber failure with length δ is

 

!

Pfib =1" e

" #

!

$ f

$ !

o

%

&

'

'

(

)

*

*

+

(8) 

Eq. 8 is equivalent to Eq. 6 when the strength scale parameter is defined in terms of length δ

fibers as follows 

 

!

"#

o = !

#

$

%

& '

(

)

1

*

" !

o (9) 

To begin the fiber breakage model derivation, it is convenient to calculate the probability of an 

initial failure in a fiber of length 

!

"e . (

!

"e is the characteristic overload distance due to elastic 

deformation along a neighboring fiber which will enter the derivation later).

!

P" e =1# e

#

$ f

$" e

o

%

&

'

'

'

(

)

*

*

*

+

small

probability

P" e = $ f

$" e

o

%

&

'

'

(

)

*

*

+

(10) 
9

Once an initial break occurs, the neighboring fibers will see elevated stresses. The probability of 

a second break occurring in a neighboring fiber due to the initial break is: 

!

P2 =1" e

"n1

2#(t)

#e

K1$ f

$# e

o

%

&

'

'

'

(

)

*

*

*

+

small

probability

P2 = n1

2#(t)

#e

K1$ f

$# e

o

%

&

'

'

(

)

*

*

+

 (11) 

where K1 is the stress concentration in a neighboring fiber caused by a cluster size j=1

n1 is the number of neighboring fibers for a cluster size j=1

2δ(t) is the length along a neighboring fiber that is overloaded

The combined probability of having an initial break followed by the second break is then 

!

P1"2 = P# e P2

small

probability

P1"2 = 2n1

#(t)

#e

K1$ f

$# e

o

%

&

'

'

(

)

*

*

+

$ f

$# e

o

%

&

'

'

(

)

*

*

+

(12) 

which can be generalized to the probability of a cluster of size j forming in time t as

 

!

P1" j = P# e P2P3...Pj

small

probability

P1" j = (n1n2..n j$1) 2#(t)

#e

%

&

'

(

)

*

j$1

K1K2 ( ...K j$1)

+ , f

,# e

o

%

&

'

'

(

)

*

*

j+

 (13) 

A conservative assumption is made in Eq. 13 that the entire time is available at each new cluster 

size for the overload zone to grow. 

K and n are related and somewhat compensating parameters. If one assumes fewer neighboring 

fibers, then the elevation in stress in the neighboring fibers increases. To simplify equations, a 

planar array of fibers was assumed. Thus, two intact fibers on either side of a line of broken 

fibers carry the load of the broken fiber cluster. The two neighbors are assumed to carry most of 

load of the broken cluster, but not all the load as would be expected from a fracture mechanics 

type decaying stress state around a sharp notch. Therefore, n and K for any cluster of size, j, are 

assumed to be: 

!

n j = 2 and K j = 1+

" j

4 (2D assumption) (14) 

Although a planar array of fibers is generally assumed in the fiber breakage model, a 3D array of 

fibers was also been postulated in the original derivation [6]. By assuming a circular cluster of 

broken fibers, the following equations were derived,

!

n j = "

4 j

"

+1

#

$

%

&

'

( and K j = 1+

4 j

"3 (3D assumption) (15) 
10

The values of the number of surrounding fibers (n) and stress concentration values (K) for the 2D 

and 3D assumptions are compared in Table 2. With the 3D assumption, the number of 

neighboring fibers is higher which decreases the stress concentration for a given cluster size.

Critical assumptions in this section include:

• Stress state in the overload region assumed constant at a level Kσ.

• A 2D arrangement of fibers adequately represents the damage state.

• Each cluster grows for the entire time.

The implications of critical assumptions will be examined later in the paper.

Probability of strand or vessel failure due to clusters of broken fibers

The final step is realizing that there is a large number (N) of δe length fibers in a strand or vessel 

that could originate the cluster. This derivation will assume a scale up to a vessel, but the same 

equations would apply to strands by substituting the scale parameter of a strand (

!

" strand

o ) for the 

scale parameter of the vessel (

!

" vessel

o ). The reliability of the vessel failing is the probability that 

none of the fibers initiate a cluster that grows to failure. The probability that none of the δe 

lengths initiate a critical cluster is:

!

Rvessel = (R1" j)

N = (1# P1" j)N small

probability

Rvessel = 1# 1# e

#P1" j $

%

& '

(

) $

%

& '

(

)

N

Rvessel = e

#N P1" j

(16) 

The probability of a vessel failure is just the converse of reliability.

 

!

Pvessel =1" e

"N P1# j (17) 

Table 2. Cluster size influence on n, K and δsafe/δe.
11

Note that in Eq. 16, 

!

P1" j is assumed small therefore 

!

P1" j can be expressed as 

!

P1" j =1# e

#P1" j and substituting Eq. 13 into Eq. 17 

!

Pvessel =1" e

" N 2 j"1(n1n2... n j"1)(K1#K2#K3#... K j"1)

$ % (t)

% e

&

'

(

(

)

*

+

+

( j"1) , f

, % e

o

&

'

(

(

(

(

)

*

+

+

+

+

j$ -

.

/

/

/

/

/

/

0

1

2

2

2

2

2

2

(18) 

The assumption in this step is that the probability of any one δe length initiating a critical cluster 

(

!

P1" j ) is small. However, because N will be large, it is possible for Pvessel to be large. 

Assuming that: 

1. the strength of the vessel obeys a Weibull distribution 

!

Pvessel =1" e

" (s) $ # %& '

()

where 

!

s " #

# vessel

o

2. the strength is measured when t is sufficiently small so 

!

"(t) = "e

3. the composite and fiber nominal stresses are proportional 

!

" = D " f

4. the vessel fails when a critical cluster of size jc is formed 

5. the critical cluster size is not a function of 

!

"

Eq. 18 can be shown to reduce to a Weibull distribution by making the following substitutions:

!

" vessel

o = D"# e

o

N

1

jc$ 2

jc %1

jc$ (n1n2... n jc %1)

1

jc$ (K1 &K2 &K3 & ...K jc %1)

1

jc

' = jc$

(19) 

The time dependent probability of failure and reliability become

!

Pvessel(t) =1" e

" # (t)

# e

$

%

&

'

(

)

j ( c "1)

(s) jc*

+

,

-

-

.

/

0

0

(20) 

!

Rvessel(t) = e

" # (t)

# e

$

%

& '

(

)

j ( c "1)

(s) jc*

+

,

-

-

.

/

0

0

(21) 

Substituting the visco-elastic model of Eq. 5 in Eq. 21 produces

!

Rvessel(t) = e

" 1+ t

tc

#

$

% &

'

(

) j ( c "1)

(s) jc*

+

,

-

-

-

.

/

0

0

0 t>>t

c

t>>0 Rvessel(t) = e

" t

tc

#

$

%

&

'

(

) j ( c "1)

2

(s) jc*

+

,

-

-

-

.

/

0

0

0

(22) 
12

Eq. 22 can be used to make reliability predictions with the fiber breakage model when no proof 

loading is applied, and has modeling parameters tc, θ, jc, ζ, and the inputs are s and t. σ and

!

" vessel

o are also needed to calculate s. Common parameters used for the T1000 fiber based 

composite are presented in Table 1. 

At large t, the fiber breakage model can be shown to be equivalent to the classic model given in 

Eq. 1 by making the following substitution into Eq. 22:

!

" = #

2

( jc $1)

% = 2 jc&

( jc $1)#

 

!

tref = tc

s = "

" vessel

o

(23) 

Therefore, the fiber breakage model predictions can be comparable to the classic model 

predictions at long times, when a proof load is not applied. However, the fiber breakage model is 

based on micro-mechanics that requires numerous assumptions and more modeling parameter 

inputs than required in the classic model. The two models are only comparable in their 

predictions if specific values of the fiber breakage parameters are chosen, and these values may or 

may not be representative of the physical properties that they are meant to represent.

Figure 1 shows the fiber breakage model and the classic model reliability predictions for a T1000 

composite before proof, and it is clear that the models agree at longer times. The models use the 

input parameters listed in Table 1, that have been suggested for this composite [2]. However, the 

models agree because the fiber breakage model parameters were chosen to fit the classic model 

predictions instead of being measured directly from the physical phenomenon that the parameters 

are intended to represent.

Critical assumptions in this section are that:

• Input parameters back calculated from the classic model adequately represent the actual 

micro-mechanical response.

• The critical cluster size, jc, remains constant over a wide range of load levels.

The implications of critical assumptions will be examined later in the paper. 

Strand or vessel reliability after proof loading

The predictions of the classic model and the fiber breakage model diverge once proof loading is 

applied. From Eq. 13, the probability of a critical cluster of size j forming during proof loading 

from a 

!

"e section of fiber is 

!

P1" jc

p small

probabilityP1" jc

p = P# epP2pP3p ...Pjcp

= (n1n2..n jc $1) 2#(t)

#e

%

&

' (

)

*

jc $1

K1K2...K j ( c $1)

+ , f

,# e

%

&

'

'

(

)

*

*

jc+ (24) 
13

After the proof load, only clusters creating an overload stress larger than the proof loading 

(

!

" fK j#1 $ " f

p

) can grow to size j. As seen in Figure 4, smaller clusters with smaller K’s will 

never develop a danger area. Assuming the 2D load sharing of Eq. 14 leads to

!

" f

p

" f

< K j#1 = 1+

$ (j #1)

4 (25) 

Solving for the minimum cluster size that would be required to form during proof that could 

eventually lead to a stress rupture failure under normal load produces the following equation

!

j p = 4

"

# f

p

# f

$

%

&

&

'

(

)

)

2

*1

$

%

&

&

&

'

(

)

)

)

+1 (26) 

Assuming the nominal stress in the composite is proportional to the nominal fiber stress (σ=Dσf), 

jp can also be expressed in terms of the composite stress ratio, s, as

!

j p = 4

"

sp

s

#

$

%

%

&

'

(

(

2

)1

#

$

%

%

&

'

(

( +1 (27) 

where jp is not the cluster size created under proof, but is instead the cluster size that would be 

needed for further growth under nominal stress. Thus, jp is a function of both s and sp. 

Furthermore, since the vessel did not burst, jp<jc .

The probability of a flaw in a 

!

"e long section of fiber growing to jp during proof, and then on to jc

during use pressure, is 

!

P1" j p

p P( j p +1)" jc = P# epP2p ...Pj pp $ Pj p +1Pj p +2...Pjc (28) 

Note that 

!

"(t) in 

!

Pj p must be adjusted for the growth that already occurred in the proof test 

cycle

!

Pj p +1 = 2n j p

"(t) #"(t p )

"e

K j p$ f

p

$" e

o

%

&

'

'

(

)

*

*

+

= 2n j p

"(t p )

"e

$ f

p

$" e

o

%

&

'

'

(

)

*

*

+

K j p

, "(t)

"(t p )

#1

%

&

'

'

(

)

*

*

$ f

$ f

p

%

&

'

'

(

)

*

*

- +

.

/

/

/

0

1

2

2

2

(29) 

When calculating reliability after a proof loading, two numbers can be calculated. One can 

calculate the reliability based on all failures, Rip (including the chance of failures during proof 

loading). Alternatively, a conditional reliability, Rcp, can be calculated (conditional on survival of 

proof loading) where only the failures that occur due to post proof loading are counted. The 

conditional reliability that ignores the risk of proof loading failures is the more relevant number 

for long-term reliability estimates because proof tests are conducted so that proof failures do not 

cause significant damage to surrounding equipment or personnel. Although the Rcp is the more 
14

relevant number, calculating Rip and the associated total probability of failure Pip are helpful in 

calculating, Rcp..

Pip is the probability of a cluster of size jp being created during proof and then either growing to 

failure under proof loading or later under nominal stress and is given by

!

P1" jc

ip small

probabilityP1" jc

p + P1" j pp P( j p +1)" jc = P1" j pp (P( j p +1)" jcp + P( j p +1)" jc

) (30) 

!

P1" jc

ip small

probability 2 jc #1 (n1n2... n jc #1)(K1 $K2 $K3 $ ...K jc #1)

% &(t p )

&e

'

(

)

*

+

,

j p #1 - fp

-& e

o

'

(

)

)

*

+

,

,

% j p

.

&(t p )

&e

'

(

)

*

+

,

- f

p

-& e

'

(

)

)

*

+

,

,

+

&(t p )

&e

'

(

)

*

+

,

- p

-& e

'

(

)

)

*

+

,

,

&(t)

&(t p )

#1

'

(

)

)

*

+

,

,

- f

- f

p

'

(

)

)

*

+

,

,

0 /

1

2

2

2

3

4

5

5

5

'

(

)

)

)

*

+

,

,

,

( jc # j p ) (31) 

!

P1" jc

ip small

probability 2 jc #1

(n1n2... n jc #1)(K1 $K2 $K3 $ ...K jc #1)

% &(t p )

&e

'

(

)

*

+

,

jc #1 - f

p

-& e

o

'

(

)

)

*

+

,

,

% j ' c

(

)

)

)

*

+

,

,

,

.

1+

&(t)

&(t p )

#1

'

(

)

)

*

+

,

,

- f

- f

p

'

(

)

)

*

+

,

,

0 /

1

2

2

2

3

4

5

5

5

( jc # j p ) (32) 

!

P1" jc

ip small

probabilityP1" jc

p 1+

# f

# f

p

$

%

&

&

'

(

)

)

*

+(t)

+(t p )

,1

$

%

&

&

'

(

)

)

-

.

/

/

/

0

1

2

2

2

jc , j p

(33) 

The conditional proof (cp) reliability (probability of failing during service after surviving proof 

loading) is:

!

R1" jc

cp =

R1" jc

ip

R1" jc

p =

1# P1" jc

ip

1# P1" jc

p

small

probability

1# (1# e

#P1" jc

ip

)

1# (1# e

#P1" jc

p

)

= e

#P1" jc

ip

e

#P1" jc

p = e

#(P1" jc

ip #P1" jcp ) (34)

The conditional proof (cp) probability is then just the converse of the reliability.

!

P1" jc

cp =1# R1" jccp =1# e

#(P1" jc

ip #P1" jcp ) small

probability

P1" jc

ip # P1" jcp = P1" jcp 1+

$ f

$ f

p

%

&

'

'

(

)

*

*

+

,(t)

,(t p )

#1

%

&

'

'

(

)

*

*

-

.

/

/

/

0

1

2

2

2

jc # j p

#1

%

&

'

'

'

(

)

*

*

*

(35) 
15

Scaling to the vessel level as was done in Eq. 16-22, the vessel reliability including proof 

becomes

!

Rvessel

ip = e

"

# (t p )

# e

$

%

&

&

'

(

)

)

j

c ( "1) s( p ) j

c*

+

,

-

-

-

.

/

0

0

0

1+ s

sp

$

%

&

&

'

(

)

)

* # (t)

# (t p )

"1

$

%

&

&

'

(

)

)

+

,

-

-

-

.

/

0

0

0

j

c" j p $

%

&

&

&

&

'

(

)

)

)

)

(36) 

Conditional reliability given proof survival 

!

Rvessel

cp = e

"

# (t p )

# e

$

%

&

&

'

(

)

)

j

c ( "1) s( p )

j

c*

+

,

-

-

-

.

/

0

0

0

1+ s

s p

$

%

&

&

'

(

)

)

*

# (t)

# (t p )

"1

$

%

&

&

'

(

)

)

+

,

-

-

-

.

/

0

0

0

j

c" j p

"1

$

%

&

&

&

&

'

(

)

)

)

)

(37) 

Substituting the visco-elastic model of Eq. 5 and writing the equation in terms of stress ratio

!

Rvessel

cp = e

" 1+

t p

t

c

#

$

%

%

&

'

(

(

) #

$

%

%

%

&

'

(

(

(

j

c ( "1)

s( p )

j

c*

+

,

-

-

-

-

-

.

/

0

0

0

0

0

1+ s

s p

#

$

%

%

&

'

(

(

* 1+ t

t

c

#

$

%

%

&

'

(

(

)

1+

t p

t

c

#

$

%

%

&

'

(

(

)

"1

#

$

%

%

%

%

%

%

%

&

'

(

(

(

(

(

(

(

+

,

-

-

-

-

-

-

-

.

/

0

0

0

0

0

0

0

j

c" j p

"1

#

$

%

%

%

%

%

%

%

%

&

'

(

(

(

(

(

(

(

(

(38) 

which is equivalent to equation 46 in the ITAI report [6] given for the conditional vessel 

reliability. Note that the original equation was expressed with α the scale parameter on 

composite strength which was assumed equal to jcζ.

Eq. 38 like Eq. 22 has modeling parameters tc, θ, jc, ζ, and the inputs are s, t, but has the 

additional input parameters, sp and tp, to describe be proof loading. Note that jp is not an 

independent parameter because it is a combination of the input parameters as given by Eq. 27. 

Note also that if α is used in the equation, it is an additional modeling parameter if 

!

" # $ jc since 

ζ and jc are used elsewhere in the equation. The fiber breakage model therefore has four or five 

independent parameters as opposed to the classic model which has three.

Using the fiber breakage model and parameters in Table 1, the reduction in predicted reliability 

due to proof loading can be seen to be dramatic as shown in Figure 1 and, therefore, reason for 

concern. For the example of a 5 min. proof at stress ratio 0.75, the predicted reliability is only 

two 9’s or one in 100 being predicted to fail in 10 yrs. This is a reduction in reliability of four 

orders of magnitude due to the proof loading. However, there are numerous assumptions that 

were made during this derivation that have not been validated. The effect that these assumptions 

have on the reliability predictions will be addressed in the next section.

Effect of Simplifying Assumptions

As pointed out in the last section, numerous assumptions were made in the derivation of the fiber 

breakage model equations to make the equations tractable. In this section the effect of these 
16

simplifying assumptions will be discussed. The more critical assumptions highlighted in the 

previous sections are listed here in the order in which they will be discussed:

• Stress state in the overload region assumed constant.

• Input parameters back calculated from classic model adequately represent the actual 

micro-mechanical response.

• A 2D arrangement of fibers adequately represents the damage state.

• The visco-elastic response of the matrix can be represented by an elastic model with a time 

dependent compliance.

• Critical cluster size, jc , assumed independent of nominal stress.

• Each cluster grows for entire time.

Assumption: Stress state in the overload region assumed constant.

A very important part of the fiber breakage model relates to the way load is shared by a 

neighboring fiber in the area of a broken fiber. The mathematical derivation assumes a 

characteristic distance δ over which the load is transferred from the broken fiber to the 

neighboring fiber. The δ is assumed to grow over time due to the visco-elastic deformation of the 

matrix material. In the mathematical derivation, δ is related to the creep modulus through Eq. 5 

which comes from an elastic shear lag model. 

However, to make the statistical model tractable, the load in the neighboring fibers is assumed to 

be constant over the δ distance. This is in contrast to the results of a shear lag model where the 

stress in the first unbroken fiber is predicted to decay rapidly as one moves along the fiber but 

away from the location of a nearby fiber break. Assuming that the stress jumps from the nominal 

level to the constant elevated level at a δ boundary is actually quite unrealistic because a change 

in fiber load must occur due to matrix shear stress which is compliant compared to the fiber. 

Therefore the change in fiber load should be more gradual. This simplification has been 

recognized and debated in the literature but was used in the fiber breakage model to simplify the 

equations because it was believed to cause a negligible effect [4, 7, 8]. This section will look at 

the merit of this assumption and compare it to other possible simplifying assumptions. 

The fiber breakage model is primarily used to make predictions of reduced reliability after proof 

loading. The reduced reliability is due to “danger areas” along the neighboring fiber where stress 

increases over time to a higher level than was experienced during the short exposure to proof 

loading. These danger areas are therefore the areas where there is some chance of additional fiber 

damage that can eventually lead to a stress rupture failure. Although the probability of cluster 

growth from a given cluster after proof may be small, the overall reliability after proof may be 

reduced because the proof load may have created a larger number of clusters. The assumption of 

a constant load in the δ region leads to a simple characterization of the “danger area” because all 

of the fibers in the region are elevated by a constant amount, and the growth of the danger area is 

then completely characterized by how δ grows with time.

A number of other possible assumptions could be made as shown in Figure 5. One could assume 

that plastic deformation occurs in the matrix. If one assumes that because of plasticity, shear 

stress in the matrix remains constant near the break, then the change in fiber stress should be 

linear. This leads to a triangular elevated stress distribution in the danger area with the region 

expanding over time as shown in Figure 5. After proof loading the elevated stress region δ must 

grow significantly before a danger area is created. Using the geometry shown in Figure 6, the 

following expression for when the danger area is created can be derived.
17

!

K " (K "1)

# #e

(1) = $ P

$

(39) 

Solving for δ/δe and assuming the δ growth law of Eq. 5 produces

!

"(tsafe )

"e

= 1+

tsafe

tc

#

$

%

&

'

(

)

= K *1

K * + P

+

(40)

where tsafe is the time required for a danger area to form.

 Solving Eq. 40 for tsafe produces

!

tsafe = tc

K "1

K " # P

#

$

%

&

&

&

'

(

)

)

)

2

"1

*

+

,

,

,

,

-

.

/

/

/

/

1

0

 (41) 

Until this time, the model would predict no chance of continued damage growth and 100% 

conditional reliability after surviving proof loading. This safe time, tsafe , is clearly related to the 

level of proof and the cluster size (through K). After tsafe, the danger area would grow much more 

slowly than predicted by assuming the elevated stress was constant. 

Figure 5. Change in fiber stress near fiber break with time with and without proof loading 

(K=2 and sp/s=1.5 assumed).
18

The tsafe is smaller for higher stress 

concentrations. The worst case stress 

concentration occurs at the last break before a 

cluster becomes critical (jc-1). As shown in 

Table 2 for a 2D cluster, the critical stress 

concentration would be K=2.04 assuming jc=5, 

and the effective overload zone would need to 

grow to 1.32 or 1.93, for proof loads of 

!

sp

s

= 

1.25 and 1.5, respectively. Therefore, the 

effective overload region would have to grow 

by nearly 1/3 before there is any chance of 

failure. The effect of the assumed visco-elastic 

model and input parameters on tsafe will be 

discussed in the next section.

Assumption: Input parameters back calculated from classic model adequately 

represent the actual micro-mechanical response.

The current model assumes a tc=.01 hrs and θ=.11 in order to match the classic Weibull model 

predictions before proof. The predicted growth of the overload region using these parameters is 

shown in Figure 7. Overload lengths of 1.32 and 1.93 would lead to tsafe times of .0006 hrs (2 

sec) and 90 hrs, given 

!

sp

s = 1.25 and 1.5 proof loads, respectively. However, the predictions that 

the deformations would grow by 1/3 due to creep in 1 sec does not seem credible for the cured 

epoxy matrix. More realistic values might be tc=98,000 hr, θ=0.247 [9], which is the measured 

creep compliance of a unidirectional composite (T300/5208) with fibers 10° to the loading 

direction. This type of test primarily loads the matrix in shear. With these parameters, the tsafe

becomes 30,000 (3.4 yrs) and 6,000,000 hrs (680 yrs) for proof levels of 

!

sp

s = 1.25 and 1.5, 

respectively. With more reasonable visco-elastic constants, the model predicts no chance of 

failure for significant periods and reverses the prediction that higher loads are detrimental. These 

predictions are much closer to the predictions of the classic model that predicts extremely high 

reliability following proof and better results with higher proof. But with these updated 

parameters, the fiber breakage model would not replicate the classic Weibull model predictions 

without a proof load.

The long safe periods also 

rely on the relatively small 

jc=5. As shown in Table 2, 

the value of K can 

significantly increase with 

cluster size. With larger jc

values, there could be larger 

clusters that would not have 

the long safe periods that 

were predicted here.

The parameters in reference 1 

are tied to the original matrix 

creep parameters to match the 

classic Weibull model without 

Figure 6. Geometry governing initiation of 

danger zone assuming plastically 

deforming matrix. 

Figure 7. Effect of matrix creep parameters on predicted response.
19

accounting for proof loading. Fitting the parameters to the classic model allowed a direct 

comparison between the two models. However, fitting the parameters to the global response 

invalidates the claim of a physics based model if the resulting values are not realistic in 

representing the response of the phenomenon they are supposed to represent. In this case, it also 

indicates that some of the many assumptions made during the derivation may not have been valid. 

Even basic questions, such as “Does the fiber breakage model represent the right physical 

phenomenon?” may be questioned. Although the mechanism that is being modeled by the fiber 

breakage model does appear to be a reasonable suggestion for the stress rupture phenomenon, 

there is no proof that it is, in fact, the controlling phenomenon. 

Assuming a constant plastic stress in the matrix is just one possible assumption that could have 

been made. Figure 5 shows other possible assumptions. An elastic model would create a more 

gradual decay in stress at the end of the elevated stress region. This would require even more 

growth in δ before the proof and after proof curves would cross and result in longer values of tsafe. 

A debond along the fiber could also be assumed. This would tend to elongate the region of 

elevated stress in the neighboring fiber. Then, one would need to postulate why the elevated 

stress in the neighboring fiber would change with time. There are two possible ways in which the 

debonds could grow with time: 

1) Visco-elastic debond growth. This does not seem likely because the debond stopped growing 

under proof loading. Crack growth (i.e. the debond) is driven by strain energy release rate, which 

is normally proportional to the load squared. The reduced loading under use conditions would 

therefore have a dramatically reduced driving force on the crack. To explore this possibility a 

visco-elastic crack growth model would be required.

2) Debonding under proof followed by visco-elastic matrix deformation. Another possibility is 

that the fiber debonds under proof loading, increasing the length of the overload region, but then 

under nominal loading, the matrix deforms visco-elastically. This would result in values of tsafe as 

produced earlier because the change in stress at the end of the debond would closely resemble the 

change in stress without a debond. An inconsistency still exists in that visco-elastic constants are 

being assumed while the shape of the overload region is assumed to be plastic. Since the plastic 

deformation was assumed to be conservative, and epoxy resins are fairly brittle, large regions of 

plastic deformation are unlikely.

By making small improvements to the micro-mechanics assumed by the fiber breakage model, 

the predicted reliability after proof changes dramatically and the response is more in line with the 

extremely high reliability predicted by the classic model after proof loading.

Assumption: A 2D arrangement of fibers adequately represents the damage state.

A 2D arrangement of broken fibers was assumed. A 3D approximation has also been proposed 

that appears more realistic. The 2D approximation tends to underestimate the number of 

neighboring fibers and then overestimate the stress concentration. In the original formulation of 

the fiber breakage model, these errors would tend to cancel each other. In the shear lag fiber 

breakage model, the safe period is only a function of the stress concentration so the 

overestimation of the stress concentration would tend to significantly underestimate the safe 

period. As seen in Table 2, the required growth in the overload regions required for the formation 

of a damage zone is much larger in a 3D model than in the 2D model for a given size of cluster. 

The larger zone growth would translate into much longer safe periods.
20

Assumption: The visco-elastic response of the matrix can be represented by an elastic 

model with a time dependent compliance.

The stress field around a broken cluster is a complex stress state [10]. Substituting a visco-elastic 

compliance for the elastic compliance assumes that all loading remains constant in the structure. 

With plastic, visco-elastic and elastic deformations occurring in the matrix surrounding the fiber, 

the more heavily loaded parts of the matrix will most certainly shed load into less highly loaded 

regions as the matrix creeps. The substitution of the visco-elastic compliance is therefore invalid. 

Although stress redistribution might tend to slow the time dependent effects, the actual response 

at a given location along a fiber is difficult to measure or predict.

Assumption: Critical cluster size, jc , assumed independent of nominal stress.

The fiber breakage model assumes that the critical cluster size is constant regardless of global 

stress level. This appears to be a poor assumption because as the nominal stress level is reduced 

the critical cluster size would likely increase. Because the fiber breakage model is quite sensitive 

to the cluster size parameters, this would tend to significantly increase the reliability at lower 

stress levels above what is currently predicted by the fiber breakage model. This would be true 

with or without proof.

Assumption: Each cluster grows for entire time.

The fiber breakage model assumes that at each increase in cluster size, the entire time of loading 

is available for increased overload region growth. One exception is that proof time is separated 

from time after proof. The time dependent growth of the overload region would actually be 

divided between the various steps. Allowing all the time at each step certainly yields a 

conservative approximation, but it may not be as conservative as it first appears because growth 

rate decreases after loading. If the majority of time is consumed with growth at one crucial 

cluster size, the approximation would be rather accurate.

Conclusion

The fiber breakage model examined was an attempt at a physics based model for stress rupture of 

composite material that would account for proof loading history. The fiber breakage model 

equation was re-derived while highlighting assumptions required for the derivation. Some of 

these assumptions were then examined. The assumptions, in general, were found to be 

conservative but a few adjustments to the original assumptions completely changed the predicted 

response after proof loading.

The original fiber breakage model postulated a significant decrease in reliability due to damage 

resulting from elevated proof loading. The improved micro-mechanics model does not support 

this postulation but still relies on large numbers of assumptions of the micro-mechanics behavior 

that cannot be validated. Therefore, a damaging effect of proof loading cannot be ruled out by 

analysis. 

The sensitivity of the current fiber breakage model to small changes in assumed behavior at the 

micro-mechanics level make it an inappropriate model for predicting stress rupture. However, it 

may be a useful tool for understanding the mechanics causing the stress rupture phenomenon and 

may eventually be improved and substantiated so that reliability predictions can be made. Until 

validation of a fiber breakage model is accomplished at both a micro- and macro-level, it should 

not be used for reliability estimates. However, the effects of proof loading are still unknown, and 

the present work does not alleviate the general concern. This issue must be addressed by test and 

further analytical work.
21

References

1. NESC, Shelf Life Phenomenon and Stress Rupture Life of Carbon/Epoxy Composite 

Overwrapped Pressure Vessels. Report RP-06-83, 2006, NASA Engineering and Safety 

Center.

2. Phoenix, S.L. and L.N.P. Murthy, "Pro's and Cons of Proof Testing Carbon Composite 

Overwrapped Pressure Vessels: A Comparison of Two Mathematical Models," AIAA Paper 

No. 2007-2325, presented at 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural 

Dynamics, and Materials Conference. 2007. Honolulu, HI.

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of California: Los Angeles.

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Materials," Chapter 19 of Comprehensive Composite Materials. Vol. 1, Elsevier Science Ltd. 

2000. pp. 559-639.

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I: Theory". Journal of Composite Materials, 1988. 22(6): pp. 561-588.

6. NASA Engineering and Safety Center, "Development of the Fiber Breakage Model and 

Implications of Proof Testing on the Reliability of COPVs," in Shelf Life Phenomenon and 

Stress Rupture Life of Carbon/Epoxy Composite Overwrapped Pressure Vessels. Report 

RP-06-83, 2006.

7. Lagoudas, D.C., C.Y. Hui, and S.L. Phoenix, "Time Evolution of Overstress Profiles Near 

Broken Fibers in a Composite With A Viscoelastic Matrix." Int J Solids Structures, 1989. 

25(1): pp. 45-66. 

8. Phoenix, S.L., P. Schwarz, and H.H. Robinson, "Statistics for the Strength and Lifetime in 

Creep-Rupture of Model Carbon/Epoxy Composites," Composites Science and Technology, 

1988. 32: pp. 81-120. 

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in Graphite/Epoxy Composites Under Constant Load at Elevated Temperatures." Journal of 

Material Science, 2003. 38: pp. 877-884.
REPORT DOCUMENTATION PAGE Form Approved

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Technical Memorandum

 4. TITLE AND SUBTITLE

A Critique of a Phenomenological Fiber Breakage Model for Stress 

Rupture of Composite Materials 

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 6. AUTHOR(S)

Reeder, James R.

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NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

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NASA

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14. ABSTRACT

Stress rupture is not a critical failure mode for most composite structures, but there are a few applications where it can be 

critical. One application where stress rupture can be a critical design issue is in Composite Overwrapped Pressure Vessels 

(COPV's), where the composite material is highly and uniformly loaded for long periods of time and where very high 

reliability is required. COPV's are normally required to be proof loaded before being put into service to insure strength, but it 

is feared that the proof load may cause damage that reduces the stress rupture reliability. Recently, a fiber breakage model was 

proposed specifically to estimate a reduced reliability due to proof loading. The fiber breakage model attempts to model 

physics believed to occur at the microscopic scale, but validation of the model has not occurred. In this paper, the fiber 

breakage model is re-derived while highlighting assumptions that were made during the derivation. Some of the assumptions 

are examined to assess their effect on the final predicted reliability. 

15. SUBJECT TERMS

Composite materials; Stress rupture; Reliability

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