April 2012 

NASA/TM–2012-217566 

Composite Stress Rupture: A New Reliability 

Model Based on Strength Decay 

James R. Reeder 

Langley Research Center, Hampton, Virginia 
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April 2012 

NASA/TM–2012-217566 

Composite Stress Rupture: A New Reliability 

Model Based on Strength Decay 

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 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
5 

Abstract 

A model is proposed to estimate reliability for stress rupture of 

composite overwrap pressure vessels (COPVs) and similar composite 

structures. This new reliability model is generated by assuming a 

strength degradation (or decay) over time. The strength decay model 

will be shown to predict a response similar to that predicted by a 

traditional reliability model for stress rupture based on tests at a single 

stress level. In addition, the model predicts that even though there is 

strength decay due to proof loading, a significant overall increase in 

reliability is gained by eliminating any weak vessels, which would fail 

early. The model suggests that strength decays very slowly over most of 

a component’s life. Late in life, as the strength decreases to near the 

applied stress level, the rate of strength decay increases significantly. 

This helps explain why strength measurements following prolonged 

stress rupture testing have not demonstrated a discernable reduction in 

strength. Any reduction in strength that may have occurred is masked by 

the scatter in the data. The strength decay model is also consistent with 

increased levels of acoustic activity late in the life of a COPV. The 

model predicts that there should be significant periods of safe life 

following proof loading, because time is required for the strength to 

decay from the proof stress level to the subsequent loading level. 

Validating reliability models for carbon composite stress rupture is 

difficult because of the high degree of scatter exhibited in observed times 

to failure. Suggestions for testing the strength decay reliability model 

have been made which allows observations with a number of test 

specimens and in a time period, which are reasonable for a test program 

that could actually be performed. If the strength decay reliability model 

predictions are shown through testing to be accurate, COPVs may be 

designed to carry a higher level of stress than is currently allowed, 

which will enable the production of lighter structures. 

Nomenclature 

9’s reliability scale where three 9’s is 1/1000 chance of failure (R=0.999) 

A scaling parameter 

b nonlinearity parameter in strength decay model 

c time scale parameter in strength decay model 

i count of similar specimens that have failed at a given point in stress rupture 

N number of similar specimens tested 

P probability of occurrence 

Pr proof ratio (proof stress/normal operating stress) 

R reliability 

Rc conditional reliability 

s(t) instantaneous strength (strength of a given test specimen at some point in time) 

so initial strength in a given test specimen 

s ˆ 

o scale parameter for Weibull distribution of initial strengths 

S strength distribution of a collection of specimens
6 

SR Stress ratio (σservice / s ˆ 

o ) 

SRP Stress ratio during proof loading (σproof / s ˆ 

o )

t time 

t1 initial loading time for proof load 

tf time to failure 

tref classic model reference time to failure (scale parameter at SR=1) 

tsafe period of safe operation following proof loading 

α Weibull shape parameter for strength 

β Weibull shape parameter for time to failure 

λ Weibull scale parameter 

σ Stress in composite 

σproof Stress in composite at proof load 

σservice Stress in composite during normal operations 

ρ Classic model parameter for sensitivity to stress ratio 

Introduction 

Stress rupture has been observed in composite materials where the material can fail after a period 

of time when no increase in load is applied. One application where stress rupture may be critical 

is in Composite Overwrap Pressure Vessels (COPV). This application may be worse than others 

because the composite material tends to be uniformly loaded to a high stress level for prolonged 

periods of time. As seen in the Figure 1, COPVs come in various sizes and are either cylindrical 

or spherical in shape. Several varieties of composite overwrap have been used including 

Kevlar®/Epoxy, but most COPVs in current use in the aerospace industry are made with carbon 

fibers. In this introduction, a typical data set for creep rupture of polymer epoxy reinforced 

carbon composites will be presented. An existing reliability model will then be described to 

show how reliability predictions are currently established. The concern that proof loading could 

damage a COPV will be described leading to the need for a new model such as the strength 

decay model that will be introduced in this paper. 

These critical structures operate under very 

high pressures, and an unanticipated failure of 

a vessel would likely lead to loss of life and to 

loss of mission. The structure is made of a 

thin metal liner that is then wrapped with 

composite. To avoid stress rupture in the 

composite, the pressure in the COPV is 

normally limited to a relatively low percentage 

of the vessel burst pressure (typically less than 

50%). Reliability models are used to predict 

the likelihood of stress rupture at various 

pressure levels. By keeping the stress level in 

the composite low, the chance of a failure is Figure 1. Composite overwrap pressure vessels 

(COPV). 
7 

kept acceptably small (e.g., 1 in a million). The reliability models are based on data, but there is 

significant extrapolation from the reliability levels where tests can be performed to the 

predictions that are desired. For example, a test of a million vessels to show a one in a million 

chance of failure is not feasible. An example of composite stress rupture data that can be 

collected is presented in Figure 2. The data in Figure 2 are collected from strand test specimens 

which are a single tow of fiber (perhaps 1000 fiber filaments) impregnated with a polymer. 

These strand specimens exhibit the creep rupture phenomenon and are much cheaper test 

specimens than full COPVs. Two things are clear from the stress rupture data presented in 

Figure 2: as the stress level (i.e. load) is decreased, the percentage of surviving specimens grows 

dramatically; and the time to failure of apparently identical test specimens under identical 

loading can be vastly different. 

 

To account for the large variation in failure times for essentially identical test specimens, 

statistical models are used. Many statistical models have been proposed with most assuming a 

Weibull distribution in time to failure, for specimens tested at a given load level. The equation 

for Weibull reliability is given by eq. 1, where β is the Weibull shape parameter and is a measure 

of the scatter, while λ is the scale parameter (the time at which only 36.8% (e-1) would survive). 

R(t) = e

− t

λ

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

β

 (1) 

To account for specimens tested at various load levels, it is often assumed that the scale 

parameter is a power law function of the stress. In eq. 2, the power law exponent is -ρ, and A is a 

proportionality constant. 

λ = A σ−ρ = t ref

σ

s 

ˆ o

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

−ρ

= tref SR−ρ (2) 

 Figure 2. Time to failure data from a stress rupture study on IM6 composite test 

specimens[1]. 
8 

As further shown in eq. 2, it is common to normalize the stress level by the strength. However, 

strengths of a group of specimens also exhibit scatter. The distribution of strengths, S, is often 

characterized with a Weibull distribution similar to that expressed in eq. 1, but where time, t, is 

replaced by strength level so as given by eq. 3. Equation 3 gives the probability of a specimen 

having initial strength greater then so. 

R(so) = P(initial strength > so) = e

− so

s ˆ o

⎛ 

⎝ 

⎜ 

⎞ 

⎠ 

⎟ 

α

 (3) 

It is therefore the scale parameter of initial strength, s ˆ 

o, that is used to normalize the stress level. 

The ratio of applied stress to the strength scale parameter is termed the stress ratio, SR. A 

proportionality constant, tref, is still needed and is chosen to be the scale parameter when SR=1. 

Substituting eq. 2 into eq. 1, a common model for stress rupture is attained. This model is given 

by eq. 4 and has been called the “Classic Model” in previous publications [2, 3]. 

R(t) = e

− t

SR−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

 (4) 

where tref, ρ , β are model parameters, and t and SR are input parameters. 

 

The IM6 data introduced in Figure 2 are replotted in Figure 3 to show how the classic model fits 

the data. The data are now plotted on a “Reliability in 9’s” scale where 1 is a 1 in 10 chance of 

failure (R=0.9) and 3 is a 1/1000 chance (R=0.999). The scale below 1 is not so easily 

interpreted but the time to failure at zero 9’s is the time scale parameter at a given stress level. 

The transformation used to create the scale is given by eq. 5 where R(t) is the reliability given by 

eq. 4 for the classic model. 

Reliability in 9's = −log( ) −ln(R(t) ) (5)

Notice that a combination of natural log and log base 10 is used in this transformation. For the 

test data marked as points joined with solid lines on the right-side plot of Figure 3, the R(t) is 

determined when a specimen fails based on how many similar specimens failed previously and 

how many remain unfailed. When the second specimen out of 10 fails, it would normally have a 

reliability of 0.8 or 1- (2/10). However, it has been shown that this produces a bias in the data 

when sample sizes are small as is generally the case with creep rupture data, so Bernard’s 

Median Rank approximation (eq. 6) [4] is used as a better estimate. 

R(t) = 1− i − 0.3

N + 0.4

 (6) 

where N is the total number of similar specimens tested while i is the number of specimens that 

have failed at a given point in time. So as time progresses more specimens fail, and the 

reliability of specimens at a given stress level decreases. Note that the IM6 stress rupture tests 

were run at four stress levels, but when the data were analyzed, it was found that specimens 

came from two different spools, which had slightly different strengths. When the specimens 

were grouped according to SR (stress normalized by strength), the 7 different groups plotted in 

Figure 3 resulted (the lowest SR from 1 spool resulted in no failures and is not shown). The data 

from the two spools are plotted with different colors. 
9 

Figure 3. IM6/epoxy composite stress rupture data fit with a classic model[3]. 
10 

Reliability predictions from the classic model are shown as dashed lines in the right-side plot of 

Figure 3. While the right-side plot indicates how reliability decreases with time at various stress 

levels, the left-side plot shows how the reliability curves shift with SR. The reliability at 1 hr is 

used to define the shift in the reliability curve with stress ratio as indicated by the arrows linking 

the left and right side plots. The special “Reliability in 9’s” scale allows a Weibull distribution 

as assumed by the classic model to appear linear when plotted against log time. Notice that the 

right-side plot also includes an insert region showing an expanded scale in the range where data 

were collected. 

The parameters of the classic model can then be chosen to fit the model to the data as shown in 

Figure 3. Note that the slopes of the classic model lines in the two plots are defined by the 

model parameters, ρ and β. A number of fitting routines can be used to select model parameters, 

including a simple least squares linear fit to the data. Here, a routine was used which maximizes 

the probability of the model explaining the observed response. This fitting routine is called the 

Maximum Likelihood Estimate (MLE) [4]. The classic model is therefore purely empirical, and 

as shown in Figure 3, there is significant extrapolation from the reliabilities where data are 

collected to the reliability and stress levels (50%) generally used in COPV design. When the 

classic model was fit to the IM6 data, the model parameters were determined to be a β=0.154, 

ρ=154, and tref=0.02 hrs. 

It has been observed that when specimens that have not failed during a stress rupture test 

program are then tested for strength, no discernable reduction in strength is measured[5]. This 

seems inconsistent with the fact that for a stress rupture specimen to fail, the strength must drop 

to the applied stress level. A possible explanation for these inconsistent observations is that a 

small amount of strength reduction may occur early in life, but it is not enough to be observed 

over the normal scatter in strength values. The majority of strength loss then occurs near the end 

of life. 

In addition, acoustic emission data that have been collected on specimens that are tested for 

stress rupture have, at times, measured an increase in activity close to the end of life[6]. 

Assuming that the acoustic activity indicates damage that would reduce the strength of the test 

specimens, these signals would indicate that most of the reduction in strength occurs late in life. 

 

In the following sections, a new model for a stress rupture prediction will be proposed based on 

strength decay. This new strength decay model will help explain why no discernable strength 

loss has been observed in specimens that survive stress rupture testing. For constant stress tests, 

the strength decay model will also predict stress rupture failure rates very similar to the classic 

model. 

COPVs in service are required to survive short periods of elevated loading as a proof test. 

Recently, concerns developed that this short period of high stress may damage COPVs and 

significantly reduce the reliability of these structures in service. The form of the classic model 

does not allow this possible drop in reliability to be modeled. A fiber breakage model was 

proposed to account for damage due to the proof cycle [7] and indicated that significant 

reductions in reliability could be expected. Careful examination of the assumptions underlying 

the model revealed significant flaws which cause the credibility of the model as originally 
11 

presented to be highly questionable [8], but concern that proof loading could damage COPVs 

still exists. 

The strength decay model introduced in this paper models a reduced life of any given vessel due 

to the proof cycle. The model will also predict that the post proof reliability will increase 

significantly by eliminating weak vessels, which would tend to fail early. In fact, the model 

predicts periods of safety when no failures are predicted following a proof test. The strength 

decay model will be shown to predict reliabilities following a proof load that are higher than the 

classic model predictions. If the new model should be validated to be correct through testing, the 

improved understanding of higher reliability following proof loading could lead to certification 

and safe operation of COPVs with higher levels of stress, making the required structure lighter 

and more structurally efficient. The new model also helps explain why so few problems with 

COPVs in service have been reported. In addition, suggestions will be made for how to test to 

determine if the proposed model will predict composite stress rupture reliability after a proof 

test. 

 

 

Strength Decay Model 

Previous reliability models have only used time to failure data and normalized the results by 

original strength test data. No assumption is made about how the material changes with time 

from the initiation of loading until the specimen is failed. In the strength decay model, the 

strength of a given vessel will initially have a strength, so, which is unknown. It is unknown 

because a single vessel cannot be broken to determine a strength and broken at a lower stress 

level to determine a time to failure in stress rupture. This inability to know both the initial 

strength of a given specimen and the time to failure in stress rupture complicates the 

understanding of the stress rupture process. So the initial strength of a vessel under load will be 

unknown, and over time, the strength 

of the vessel, s(t), will decay. 

Eventually the strength may drop to 

the level of the applied stress, σ, at 

which time the vessel fails. 

Instantaneous strength (the strength at 

any point in time) will be treated as 

an internal state variable by the 

strength decay model. Although the 

internal state variable may not be 

directly measurable, it is assumed to 

control the observed failure 

phenomenon. 

 

Figure 4 shows how the strength 

might decay as a function of the time 

before failure (t-tf) assuming a power 

law relation for instantaneous 

strength, s(t), as given by eq. 7. Figure 4. Strength decay to s(tf ) σ =1 
12 

s(t)

σ = 1− t − tfc

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

1

b

 (7) 

so at failure σ = s(tf ) ≡ sf and s(tf )

σ =1. The parameter, b, determines the shape of the 

strength degradation curve with a b of one causing a linear decrease in strength with time. When 

b>1, the strength decay accelerates as the strength approaches the applied stress. The c

parameter scales the rate of the strength decrease for a given value of b. When c is doubled, the 

time to failure from a given initial strength, so, is doubled. For given input parameters b and c, 

the time to failure is only a function of how high the initial strength is compared to the applied 

stress. 

The curves can be shifted in time, as 

shown in Figure 5, so that all curves 

start at the same initial strength which 

shows how the time to failure would 

change as the model parameters b and 

c changed. The strength decay in 

Figure 5 was also normalized by the 

initial strength so . The reduction in 

strength and the time to failure is 

expressed as a function of the initial 

strength normalized by the applied 

stress in eq. 8 and 9.

s(t)

σ = soσ

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

b

− t

c

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

1/b

(8)

 

Failure will occur when s(tf)=σ or 

s(t f ) σ = 1. Therefore, for a 

constant applied stress:

tf = c

so

σ

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

b

−1

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ or σ

so

= 1

1+

tf

c

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

1

b 

or

 so = σ 1+

tf

c

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

1

b

 (9)

Figure 6 shows s(t)/so (eq. 8) for several values of applied stress σ so . Notice all the solid lines 

start near 1 at small values of time and end where s(t) so = σ so . These points can be 

connected by the heavy dashed curve which is defined by eq. 9. In this example, values of b=5 

and c=1 were used, and to include a longer period of time, the results are plotted against a log 

time scale. Notice that the strength is constantly decreasing, but at lower applied stress levels, 

the decrease in strength early in the life is imperceptibly small. The deterministic strength decay 

model suggested in this section was introduced without a clear motivation for precise 

mathematical form chosen. In the next section, this form of strength decay will be shown to 

produce stress rupture reliability response very similar to that shown in Figure 3 once the scatter 

in initial strength is considered. 

Figure 5. Strength decay for various 

parameter values 
13 

Statistical Model 

In the last section, a strength decay 

model was formulated based on the 

initial strength, so. However, when 

applying a stress rupture model, the 

initial strength of a given composite 

part will not be known. Instead, the 

strength distribution, S, of ostensibly 

similar specimens can be found through 

testing. To account for the scatter in 

initial strength, a statistical model is 

required. 

 

Figure 7 shows how the time to failure 

might change assuming a collection of 

specimens that have a distribution of 

strengths, S, described by equation 3. 

Notice that the strength decay is now 

normalized by the scale parameter of 

the distribution of initial strengths, s ˆ o. 

Figure 7 shows that for selected values 

of b and c, a specimen loaded at 80% of 

the nominal value would have a time to 

failure of 2 hrs, if the specimen had the 

nominal strength so s ˆ o =1. However, 

if an individual specimen in the current 

example were 10% weaker than the 

nominal value, the time to failure 

would drop to 0.8 hrs while an 

abnormally strong specimen might fail 

in 6 hrs. 

From eq. 9, the time to failure is 

controlled by the initial strength of a 

given specimen, and eq. 3 describes the 

distribution of initial strengths of a 

collection of specimens, S. The distribution in times to failure can be found by substituting eq. 9 

into eq. 3. More precisely eq. 10 gives the probability of a specimen from the population having 

a time to failure, tf , exceeding some time t. 

R(t) = P(t f > t) = e

− σ

s 

ˆ o

1+ t

c

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

1

b

⎛ 

⎝ 

⎜ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ 

⎟ ⎟ 

α

= e

− SR 1+ t

c

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

1

b

⎛ 

⎝ 

⎜ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ 

⎟ ⎟ 

α

 (10)

 Figure 6. Strength decay with time. 

 Figure 7. Time to failure accounting for 

variation in initial strength. 
14 

This is a reliability prediction similar to that of the classic model given by eq. 4. The ratio of 

applied stress to the scale parameter in strength has already been defined as the stress ratio (SR) 

which leads to the second equality of eq. 10. The strength decay reliability model is plotted with 

the classic model in Figure 8 along with IM6/Epoxy data presented earlier in Figure 3. The 

strength decay model is so similar to the classic model that it can be difficult to discern that they 

are not the same, but the strength decay model does have a small nonlinearity in the predicted 

reliabilities shown in the right side graph at less than 10-1 hr. This agreement was achieved by 

comparing eq. 4 to eq. 10 which leads to setting α = ρβ = 0.154 × 154 = 23.7 and b = ρ = 154 , 

and c=tref =0.02 hr. The models diverge slightly at short time periods, but data reported at very 

short time periods are questionable because the time to failure would be influenced by the time 

spent loading the specimen up to full load. Therefore, by carefully choosing model parameters, 

close agreement between the two models is achieved for all times where the data quality is high. 

The agreement between the models in Figure 8 is artificially high, however, because the strength 

decay model parameters were chosen to fit the classic model. This ignores the fact that the α

parameter has physical meaning as the Weibull shape parameter (scatter) in initial strength. This 

initial strength was measured in the original study [1], and the Weibull strength parameter was 

determined to be 22.3 [3] which is only slightly different from the value that was used in Figure 

8. Using α=22.3 and fitting the strength decay model to the IM6 data, the strength decay model 

shown in Figure 9 is created (b=147, c=0.04 hr). The agreement between the classic model and 

the strength decay model is not quite as precise as it was in Figure 8, but it is still quite close 

considering that the α parameter was determined independent of the plotted data. Note that the 

high value of b indicates that the strength decay is extremely slow until very late in life, when the 

 Figure 8. IM6/epoxy stress rupture data fit with the classic model and the 

strength decay model. 
15 

strength approaches the applied stress. This would explain why no perceptible loss in strength 

has been observed in specimens that survived a stress rupture test. The difference in reliability 

predictions at SR= 0.5 is an indication about how sensitive these extrapolated predictions are to 

small model changes. The two models in Figure 9 are quite close in the region where data are 

collected yet when the models are extrapolated out to SR=0.5, a difference in the models is 

perceived. The strength decay model directly incorporates an influence of initial strength scatter 

into the prediction of stress rupture failures. Because the classic model and the strength decay 

model give similar results for the constant stress load case, there would be little advantage of one 

over the other, if this were the load case of primary interest. However, all COPVs put in service 

receive a proof load, and the reliability predicted by the two models will be shown to be quite 

different after proof loading. Therefore, a validated strength decay model could result in 

improved predictions of reliability for the COPV’s as they are actually used in service. 

 

Classic Model Conditional Reliability 

and Reliability Following Proof Loading 

Conditional reliability in stress rupture is simply the reliability at some point in time given that it 

survived to some shorter time, t1. Equation 11 gives the equation for conditional reliability in 

terms of the reliability that specimen survived to t1 and then to some longer time t, R(t>t1), 

which is then divided by the reliability at time t1, R(t1).

Rc(t) = R(conditional on surviving t1) = R(t > t1)

R(t1) (11) 

 

 

Figure 9. Strength decay model compared to classic model without enforcing α=ρβ. 
16 

So, for the classic model at a constant stress level, the conditional reliability is 

Rc(t > t1) = e

− t

(SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

e

− t1

(SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β = e

t1

(SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

− t

(SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

 (12) 

Figure 10 shows R(t) and Rc(t) for this case when Pr ≡σ pr oof σ service = 1. However, for a proof 

test to be meaningful, the proof ratio should be higher than 1, and to certify COPVs, a proof load 

of between Pr=1.25 and 1.5 is generally required. Eq. 13 and 14 give the classic model 

reliabilities when the proof stress level is different from the normal operating level. The 

influence of short periods (e.g. 4 sec.) of stress overload are also shown in Figure 10. 

R(t1) = e

− t1

(Pr*SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

 and R(t > t1) = e

− t1

(Pr*SR)

−ρ tref

+ t−t1

(SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

 (13) 

Rc(t > t1) = e

t1

(Pr*SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

− t1

(Pr*SR)

−ρ tref

+

t−t1

(SR)

−ρ tref

⎛ 

⎝ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ ⎟ 

β

 (14) 

It is clear from the plot that 

the predicted conditional 

reliability following proof 

loading is much higher than 

the reliability without proof 

loading. The model predicts 

that the larger the proof 

loading, the higher the 

increase in reliability. 

However, this increased 

reliability will eventually 

decrease, and at very long 

times, the reliability with and 

without proof loading will be 

the same. In an effort to be 

conservative, this predicted 

increase in reliability due to 

proof loading is normally not 

acknowledged during 

certification because of a fear 

that it might be artificial. 

 

Figure 10. Increased reliability predicted by classic model 

proof due to loading. 
17 

Strength Decay Model Conditional 

Reliability Following Proof Loading 

In order to predict the reliability following proof loading, it is helpful to first predict how the 

strength is decreased due to the proof load. Using eq. 8, the strength at the end of the proof load 

for a given specimen with initial strength so can be expressed as 

s(t1)

Pr σ = so

Pr σ

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

b

− t1

c

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

1/b

 (15) 

With the strength decay model, the remaining time to failure is only a function of the 

instantaneous strength and the applied stress level. This was shown in Figure 4. Starting with 

the strength at the end of proof loading, the strength decrease under normal loading can then be 

derived by substituting eq. 15 back into eq. 8 as follows 

s(t > t1)

σ = s(t1)σ

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

b

− t − t1

c

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

1

b

=

Pr σ

so

Pr σ

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

b

− t1

c

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

1/b

σ

⎛ 

⎝ 

⎜ 

⎜ 

⎜ 

⎜ 

⎜ 

⎜ 

⎞ 

⎠ 

⎟ 

⎟ 

⎟ 

⎟ 

⎟ 

⎟ 

b

− t − t1

c

⎡ 

⎣ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

⎥ 

⎥ 

⎥ 

⎥ 

⎥ 

1

b

= so

σ

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

b

− t

c

+

t1

c

1− ( ) Pr ⎛ b

⎝ ⎜ ⎞ 

⎠ ⎟ ⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

1

b

 (16) 

 

Figure 11 shows the strength 

decay similar to Figure 6, but 

with the effect of changing the 

stress from Pr σ /so = 0.8 to 

some other ratio after 1 hr. 

Note that again the example 

model parameters of b=5 and 

c=1 are used here. In this 

example, the initial hour of 

stress at σ = 0.8 so, decreased 

the strength to s(t1) = 0.935so . 

If after 1 hr, the applied stress 

is changed, the rate of strength 

degradation will change as 

indicated by the change in 

slope of the strength 

degradation curves. The black 

dashed curve gives the time to 

failure if the specimen was 

held at a constant load for the 

 Figure 11. Strength decay after 1 hr of applied stress at 

s/s o=0.8. 
18 

entire time. The purple 

dashed curve, which 

starts at 1 hr and soon 

drops below the black 

dashed curved, shows 

the time to failure when 

the first hour of loading 

is at Pr σ /so = 0.8. So, 

in this example, a 

specimen with a 

constant stress level of 

0.5 so would fail in 

30 hrs. If the specimen 

received a proof load for 

1 hr at Pr σ /so = 0.8

before being loaded at 

0.5 so, then the total 

time to failure would be 

reduced to 20 hrs. 

Therefore, the strength 

decay model does 

address reduced life due 

to the elevated stress. 

The reliability of a group of specimens surviving proof and going on to survive some additional 

time to tf can be derived by first solving the deterministic strength decay model given by eq. 16 

for so when s(tf)/σ =1 

so = σ 1+

tf

c − t1c

1− Prb ( ) ⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

1/b

 for tf >t1. (17) 

Substituting eq. 17 into eq. 3 translates the variation in initial strengths into a variation in times 

to failure resulting in the reliability equation given by eq. 18. 

R(t) = P(tf > t) = e

− SR 1+ t

c

− t1

c 1−Pr b ( ) ⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

1

b

⎛ 

⎝ 

⎜ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ 

⎟ ⎟ 

α

 (18) 

Figure 12 shows how the instantaneous strength degrades as the initial strength varies about a 

nominal value, s 

ˆ o, similar to Figure 7, but this time with a 1 hr proof loading at Prσ s ˆ 

o = 0.8

followed by nominal loading at σ s ˆ 

o = 0.6. It is clear from the figure that the time to failure is 

significantly influenced by the scatter about the nominal value. If the initial strength is too low, 

the specimen will fail during proof loading. To survive proof loading, the strength at t1 must be 

at least Pr σ. From eq. 9, the original strength must be 

 Figure 12. Strength decay with proof loading. 
19 

so > Pr σ 1+

t1

c

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

1

b (19)

Following proof loading, there is a period where no failures are predicted because time is 

required to decrease the strength from Pr σ to σ. Using eq. 8, the time required for this strength 

decrease can be defined as tsafe , a safe period, where 

 1 = s(t)

σ = Pr σ

σ

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

b

− tsafe

c

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

1/b

 

or tsafe = c Prb ( ) −1 (20) 

Figure 13 shows how tsafe is affected by the model parameters b and c. The c parameter controls 

the approximate time scale when the safe period initiates as the proof ratio is increased above 1. 

This can be seen in the family of curves with c=1 on Figure 13. The b parameter controls how 

quickly the tsafe grows as the proof ratio is further increased as can be seen from the curves where 

b increases from 5 to 250. Changing c would shift the group of curves either left or right, but 

since the time axis is logarithmic, an order of magnitude change in c would be needed to make 

much of a difference. Also shown in Figure 13 are the predicted tsafe periods for the IM6 model 

parameters determined from fitting the data presented in Figure 9. The plots show that to attain a 

safe period of greater than 10 years, a proof level just over Pr=1.1 would be required. 

The reliability for a group of specimens surviving t1 can be found by substituting eq. 19 into eq. 

3 as 

 Figure 13. Safety period, tsafe, predicted for the strength decay model for 

various values of b. 
20 

R(t1) = e

− Pr SR 1+

t1

c

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

1

b

⎛ 

⎝ 

⎜ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ 

⎟ ⎟ 

α

 (21)

and conditional reliability from eq. 11 becomes 

Rc(t) = e

Pr SR 1+

t1

c

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

1

b

⎛ 

⎝ 

⎜ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ 

⎟ ⎟ 

α

− SR 1+

t

c

− t1

c

1−Pr b ( ) ⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

1

b

⎛ 

⎝ 

⎜ 

⎜ ⎜ 

⎞ 

⎠ 

⎟ 

⎟ ⎟ 

α

 (22)

The conditional reliability for IM6/Epoxy is plotted in Figure 14. Notice that the conditional 

reliability from the strength decay model is even higher than that from the classic model. Also, 

notice that for the classic model as time is decreased, the conditional reliability curves become 

linear. In contrast, as time is decreased for the strength decay model, conditional reliability 

curves each appear to approach infinity at some point (the curves become vertical). The point at 

which each curve approaches infinity is t1 + tsafe . At times less than t1 + tsafe, no failures are 

expected, and the reliability is infinite. As shown in Figure 13, the level of proof loading can 

have a dramatic effect on the length of the safe period following proof. Also note, from eq. 20, 

that the safe period is not a function of SR. 

Because of the large b parameter, only a small proof ratio is needed to create very long safe 

periods, and these safe periods are not affected by the operating SR. This suggests that if the 

strength decay model is validated to be correct, COPVs and similar structures can have safe 

operation for very long periods of time with fairly low proof loads, even if an increase in the 

 Figure 14. Conditional reliability from the strength decay model. 
21 

operational stress were allowed from a normal SR=0.5 to perhaps SR=0.7. Although the model 

indicates the SR could be raised to any level without affecting the safe period, setting the SR too 

high would cause problems. For instance, an operating SR=0.8 with a Pr of 1.3 would create a 

proof stress ratio of SRP=1.04 where most of the specimens would fail during the proof cycle. 

Suggested validation study for the strength decay model 

As shown in the last section, if the strength decay model can be validated, COPVs and other 

structures can be operated reliably at much higher stress levels. This will improve the structural 

efficiency of these critical structures. When testing stress rupture theories, it has proven difficult 

to test enough specimens for long enough periods of time to observe sufficient failures. 

Numerous failures are needed to show clear trends in a phenomenon that exhibits significant 

scatter. This is further complicated by the sharply decreasing failure rate, which means that the 

longer a specimen is tested, the less likely a failure is to occur. The problem with creep rupture 

testing is clearly seen by considering an example where predictions are desired. Normal 

operating conditions might be at a SR=0.5 after surviving a proof loading of Pr=1.25. From 

Figure 14, a proof load between 1.2 and 1.3 would have six 9’s of reliability at 108

 hrs. Clearly, 

no one would be willing to test 1 million specimens for 10,000 years to observe just one failure. 

Even if this type of testing could be performed, comparing a model to just one experimentally 

measured failure would not provide much of a model validation because the phenomenon 

exhibits such large scatter. 

Figure 15, shows an “Observable Region” which is the region on the reliability plot where data 

might be collected to validate a model. The region does not extend above 1.25 9’s, which is 

about 1 failure in 20. Because several failures are desired at a condition to show clear trends in a 

 Figure 15. Testing the strength decay model. 
22 

phenomenon that exhibits significant scatter, testing in this range would mean testing between 60 

and 100 specimen to observe 3 to 5 failures, respectively. This is the number of specimens at 

just one condition. Reliability testing must also be completed in a reasonable amount of time. 

For this reason, the observable region does not extend above 104

 hrs or about a year. Testing for 

very short periods of time is also problematic because these very early results will be influenced 

by the time spent loading the specimen to full load. For this reason, the observable region does 

not extend below 0.0015 hrs or about a minute. Although the boundaries of the observable 

region are not absolute, time is plotted on a log scale and reliability is plotted on effectively a log 

scale. Therefore, large changes in the number of specimens tested or in the length of time tested 

would be needed to significantly change the size of the “observable region.” Designing a test 

program to show an effect inside the shaded observable region becomes the objective. To obtain 

failures in a reasonable test time, the proof ratio level must be kept fairly low while keeping the 

operations stress level fairly high as demonstrated later in the example. As the proof load level 

SR·Pr approaches 1, the majority of the specimens will fail in the proofing process, which should 

also be avoided. 

Also shown in Figure 15 are reliability curves for a proposed test plan. This test plan is outlined 

in Table 1 to provide some validation of the strength decay model following proof loading. 

Thirty specimens would be used to determine the initial strength distribution of the specimens. 

Three different proof levels would be tested (Pr= 1, 1.04 and 1.07) with the proof lasting 4 

seconds. A high operating stress ratio of SR=0.88 is chosen to reduce the reliability to a level 

where several failures would be expected within one year. The table shows that with the high 

operating stress ratio and a proof ratio of 1.07, the stress ratio during proof loading is 0.994. At 

this load level, approximately 26 specimens would need to be proof loaded to obtain 20 that 

survived. Fewer initial specimens would be required at the lower proof levels. Starting stress 

rupture testing with 20 specimens surviving proof loading at each level, 5 specimens with Pr =1 

would be expected to fail within 10,000 hours at SR=0.88, similarly 4 failures with Pr=1.04, and 

2 failures with Pr=1.07. This number of failures is minimally sufficient to show trends in the 

response because significant variability in time to failure might be expected from any one test. 

Just as important as the number of failed specimens is the time when the specimens failed. 

Because of the predicted safe period, the strength decay model predicts that the number of 

failures will occur much later in the test period than would be predicted by the classic model. 

The reliability predictions for this proposed test plan are plotted in Figure 15 to show how 

portions of the reliability curves pass through the observable region. Testing a lower SR or 

Table 1. Suggested strength decay study. 
23 

higher Pr would tend to increase the reliability predictions so that they fall outside the 

observable region. Testing at a higher SR would increase the proof load level so high that a large 

majority of specimens would fail during proof loading. 

Tight control of the load level will be particularly important in the proposed study because all of 

the load levels are close together and close to the nominal strength. It would also be wise to 

avoid removal of the specimen from the test fixture between proof loading and stress rupture 

testing so that changes in gripping do not influence the results. Following proof loading, 

specimens should be visually inspected, and any damage that would normally cause a specimen 

to be rejected should be noted. It is recommended that all surviving specimens be tested, but the 

data should be analyzed with and without the specimens that have obvious damage. The strength 

decay model should be evaluated by first fitting the strength data to obtain α and s 

ˆ 

o, and then 

the stress rupture data without proof loading (or proof level Pr=1) should be used to fit the b and 

c parameters. The updated model can then be used to make new conditional reliability 

predictions. Adjustments to the proof levels can be made, if required, to obtain adequate numbers 

of predicted failures within one year. The proofed stress rupture tests can be performed, and the 

results compared to the updated model. Random scatter would allow some failures prior to the 

predicted first failure, but no failures would be expected before the end of the tsafe period. 

A total of 100 test specimens would be required for this study with 10 expected to fail during 

proof loading. This test program could be run on strands or COPV vessels, but because strands 

are relatively inexpensive compared to vessels, it is recommended that strand testing be 

performed before vessel testing is considered. 

Summary and Conclusions 

A new reliability model has been proposed for stress rupture of COPVs and similar composite 

structures. The reliability model is based on assumed strength degradation over time. The 

strength decay model can predict a similar response to the “classic model” for stress rupture 

under constant stress. Moreover, the model predicts that even though there is strength 

degradation during a proof load, there is significant increase in reliability gained by eliminating 

any weak vessels. The model suggests that most strength decay does not occur until the 

decreasing strength approaches the applied stress level late in life, and this behavior helps 

explain why strength tests following prolonged stress rupture testing have not exhibited a 

measurable reduction in strength. The model also is consistent with increased levels of acoustic 

activity late in the life of a COPV. The model predicts that there should be significant periods of 

safe life following a proof loading because of the time required for the strength to decay from the 

proof load level to the subsequent loading level. Validating reliability stress rupture models is 

very difficult because of the high degree of scatter exhibited in observed times to failure. 

Suggestions for testing the strength decay reliability model have been made which allows 

observations with a reasonable number of test specimens and within a reasonable time period. 

Should the reliability model be validated to be accurate, COPVs may be safely designed to carry 

higher levels of stress than is currently allowed, which will enable the production of lighter 

structures. 
24 

References 

1. Shaffer, J.T., "Stress Rupture of Carbon Fiber Composite Materials," in 18th International 

SAMPE Conference, 1986. 

2. Cameron, K.D. et al., "Shelf Life Phenomenon and Stress Rupture Life of Carbon/Epoxy 

Composite Overwrapped Pressure Vessels," NASA Engineering and Safety Center, RP-06-83, 

2006. 

3. Grimes-Ledesma, L., et al., "Composite Pressure Vessel Working Group (CPVWG) Task 3: 

Stress Rupture Test Approach," NASA Engineering and Safety Center, RP-09-00537, 2012. 

4. Abernethy, R.B., The New Weibull Handbook, 5th Edition, 2006. 

5. Moore, R.L., M.A. Hamstad, and T.T. Chiao, "Stress Rupture Behavior of Graphite 

Fiber/Epoxy Strands," Composite Materials and Structures, 3(1): p. 19-23, 1974. 

6. Hamstad, M.A., "Local Characterization of Fiber Composites by Acoustic Emission," in A 

Critical Review: Techniques for the Characterization of Composite Materials, Office of Naval 

Research: Cambridge, MA, 1981. 

7. 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," in 48th 

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 

2007. 

8. Reeder, J.R., "A Critique of a Phenomenological Fiber Breakage Model for Stress Rupture of 

Composites Materials," NASA Langley Research Center, NASA TM-2010-216721, 2010. 
REPORT DOCUMENTATION PAGE Form Approved

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Technical Memorandum

 4. TITLE AND SUBTITLE

Composite Stress Rupture: A New Reliability Model Based on Strength 

Decay

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 6. AUTHOR(S)

Reeder, James R.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

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

A model is proposed to estimate reliability for stress rupture of composite overwrap pressure vessels (COPVs) and similar composite

structures. This new reliability model is generated by assuming a strength degradation (or decay) over time. The model suggests that most 

of the strength decay occurs late in life. The strength decay model will be shown to predict a response similar to that predicted by a 

traditional reliability model for stress rupture based on tests at a single stress level. In addition, the model predicts that even though there is 

strength decay due to proof loading, a significant overall increase in reliability is gained by eliminating any weak vessels, which would fail 

early. The model predicts that there should be significant periods of safe life following proof loading, because time is required for the 

strength to decay from the proof stress level to the subsequent loading level. Suggestions for testing the strength decay reliability model 

have been made. If the strength decay reliability model predictions are shown through testing to be accurate, COPVs may be designed to 

carry a higher level of stress than is currently allowed, which will enable the production of lighter structures.

15. SUBJECT TERMS

COPV; Composite Overwrap Pressure Vessel; Reliability Model; Stress Rupture

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