A constitutive model to describe the creep lifetime of PBO braided cord has been developed and fit to laboratory data. The model follows an approach proposed for p-aramid cord in similar applications, and has a Boltzman-type representation that arises from consideration of the failure phenomenon mechanism. The data were obtained using a hydraulic-type universal testing machine, and were analyzed according to Weibull statistics using commercially-available software. The application of concern to the author is NASA's Ultra- Long Duration Balloon and other gossamer spacecraft, but the motivations for the related p-aramid works suggest broader interest

Sterling, W. J.

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NASA Goddard Space Flight Center 
A CONSTITUTIVE MODEL FOR 
CREEP LIFETIME OF PBO BRAIDED CORD 
W. J. Sterling 
NASA Goddard Space Flight Center's Wallops Flight Facility 
Wallops Island, VA 23337 
Abstract mathematical form that arises from the mechanism 
favored herein is generally adaptable to others. 
A constitutive model to describe the creep lifetime of 
PBO braided cord has been developed and fit to 
laboratory data. The model follows an approach proposed 
for p-aramid cord in similar applications, and has a 
Boltzman-type representation that arises from 
consideration of the failure phenomenon mechanism. The 
data were obtained using a hydraulic-type universal 
testing machine, and were analyzed according to Weibull 
statistics using commercially-available software. The 
application of concern to the author is NASA's Ultra- 
Long Duration Balloon and other gossamer spacecraft, 
but the motivations for the related p-aramid works suggest 
broader interest. 
Background 
NASA's Ultra-Long Duration Balloon (ULDB) 
design presently uses braided cord made of PBO (zylonm 
HM) fibers in non-redundant load tendons. This material 
was chosen on the basis of unique properties that enable 
the balloon design; however, it also presents significant 
challenges. Other works discuss most of these challenges 
[I-41. The focus of this paper is on the tendon lifetime 
under anticipated flight conditions, which are 
approxin~ated as steady-load, or creep. 
Creep rupture is the phenomenon wherein a material 
fails after finite time of static loading below the ultimate 
tensile strength as measured under dynamic tensile 
conditions. It is essentially a physical deterioration of 
strength while under creep conditions. The time from 
initial loading until material failure is called the creep 
lifetime. For applications requiring a steady, or near- 
steady, load for a given duration at extremely low failure 
probability, mean field approaches such as averaging are 
excluded because the critical events occur in the low- 
strength extremities of the material strength probability 
distribution tail [5-71. Alternately, the critical events can 
occur in the high-stress extremities of the structural load 
probability distribution tail, but the focus here is on the 
material properties. 
Fortunately, the literature discusses that creep 
lifetime for p-aramid ( ~ e v l a r ~ ) ,  a cousin to PBO, may be 
represented by a mechanistic mathematical model of the 
material failure phenomenon. Although the failure 
mechanism(s) may be different for PBO, the 
Constitutive Model Development 
Creep failure of fiber bundles is believed to occur as 
the result of accumulating fiber breaks that eventually 
manifest as rapid local deformation leading to a 
catastrophic event [5,8]. The local deformation may cause 
rapidly increasing strain (i.e., tertiary creep), that is not 
evident on the macroscopic scale. In materials such as 
high-modulus polymer and glass fibers, observable creep 
is very small even at failure. 
The failure time variability of individual fibers 
subjected to the same stress is typically very high 
[7,9,10], but is also typically well-represented by Weibull 
statistics [5,11]. The variability is likely due to random 
imperfections arising from manufacturing and processing 
which may be especially prevalent in braided cords. The 
stress on an individual fiber falls to zero at the instant of 
failure, but may return as the fiber re-engages via friction 
with its neighbors. Concomitant with a fiber failure is the 
over-stressing of its neighbors through shear transfer. 
When a statistically significant collection of failures occur 
within a localized region, or when local collections of 
failures merge, a cascade of failure events ensues and 
leads rapidly to bundle failure. The level of yarn failure 
that may lead to rope breakage may be less than 1 % [8]. 
According to statistical bundle theory [12], the static 
strength of a bundle of fibers is less than the average 
strength of the individual fibers, and under the same static 
stress the time to failure of a bundle is less than for an 
average fiber. The static failure stress of ropes tends to 
decrease as the cross-sectional area (number of fibers) 
increases, becoming asymptotically constant [6]. 
Therefore, the logical way to represent lifetime data is to 
normalize the applied stress by the tensile strength [13], 
and that representation has been widely used [6,7,9-11, 
141 even though the molecular mechanisms for short-term 
and long-term failure may be significantly different [14]. 
There are two basic theories for stress rupture in 
polymers, molecular processes and fracture mechanics. 
Zhurkov developed the most widely-used molecular 
model [6], which describes the failure for polymers below 
their glass transition temperature as due to mechano- 
chemical scission of polymer backbone bonds [15-161. 
https://ntrs.nasa.gov/search.jsp?R=20070017848 2019-08-31T00:55:33+00:00Z
The scission energy comes from two sources, heat and 
stress, which implies that at higher temperature less stress 
is needed for rupture, and further suggests that creep 
lifetime increases as temperature decreases. Recent 
publications strongly support the Zhurkov theory [14,17] 
and mention that this theory has been independently 
developed by others [lo, 171. 
On the other hand, fracture mechanics theory 
suggests the rupture may be due to weak intercrystalline 
affinity [18-211, agglomeration of amorphous regions [22- 
241, or some combination thereof 19,181. However, these 
broad mechanisms are unlikely to be mutually exclusive, 
and they lead to similar mathematical representations 
differing only in the rate-controlling step of many kinetic 
phenomena [6,10]. Because the Zhurkov theory is more 
widely-accepted and has a well-developed basis in 
physics, it will be highlighted here. 
Not surprisingly, the Zhurkov model proposes a 
Boltzmann-type mechanism whereby the failure time (h) 
is presented as an exponential function of barrier energy, 
tb = t, exp[(U - Bo)/kT], (1) 
where k is the Boltzmann constant, T is the absolute 
temperature, U is the activation (or bond dissociation) 
energy, and B relates the applied stress, o, to the bond 
stress. As suggested above, the applied creep stress will 
be normalized by the dynamic tensile failure stress in the 
analysis. The pre-exponential factor, t,, has been 
described as the characteristic time for atomic vibrations 
[15-161 and the bond thermal oscillation period [25], and 
its value has been estimated to be lo-" to 10-l3 s [17,25]. 
Zhurkov's representation can be rearranged to 
where a = 6 2  t, + U/kT and K = -B/kT. We will adopt the 
mechanistic model expressed in Eq.2, and utilize a and K 
as fitting parameters, noting that although we are not 
working directly with the fundamental physical quantities, 
we have maintained contact with them, and given the 
appropriate data, we could determine them. Since Eq.2 
has such a common and general form, it should not be 
surprising to discover that it could also be used to 
represent alternate mechanistic models. A landmark study 
on stress-rupture of p-aramid [ I l l  applied the form of 
Eq.2, although the authors apparently did so without 
knowledge of the Zhurkov theory. 
Statistical Representation of Data 
Although often neglected in the engineering 
community, it is widely documented that the Normal 
distribution, and the accompanying use of the mean and 
standard deviation as characterizing metrics, is not likely 
to be appropriate for describing the capabilities of 
materials used in strength- or lifetime-critical applications 
[e.g., 3,26-281. It is more appropriate in these cases to use 
extreme value distributions such as Weibull and Log- 
Normal becausz these reduce exposure to unreasonable 
representations such as zero or negative strength and 
generally match the data shape distribution better. 
The Weibull distribution will not be discussed further 
here, except to state its form, 
P(k) = 1- exp[-ttdrl)'l, (3) 
where tb is the time to failure, P is the failure probability, 
and the fitting parameters are q, the distribution scale, and 
p, the distribution shape [ l  11. 
The next step in the development is to bring the 
statistical representation of the failure data together with 
the mechanistic failure model. For each creep stress level, 
the probability density function is fitted to the data using 
commercially-available software [29,30]. The variation in 
the fitting parameters with stress is then represented by 
the mechanistic model for the failure phenomenon 
[6,11,14,25]. 
When the fitting parameters are expressed in terms of 
the mechanistic failure model, 
11 = a, exp(-~,fl (4) 
and 
P = ap exp(--Kpf), (5) 
where f is the ratio of creep stress to ultimate tensile 
stress. The failure time is then seen to be predictable for 
desired stress and failure probability levels. When the 
natural logarithm of q is plotted versus f, K, arises as the 
negative slope and a, is exp(intercept). The parameters 
for p are analogously determined. 
Laboratory Procedure 
Creep and tensile experiments were performed using 
an Instron 8505 hydraulically-driven universal testing 
machine (UTM). Specimens were fabricated from 48,000 
denier braided cord by tying Brumrnel splices in each end, 
and were attached to the UTM via load pins. The ultimate 
tensile strength (UTS) in dynamic loading was 
determined via tests at constant strain rates over 0.015-19 
%/min. Creep lifetime measurements were made at 
several high fractions of the UTS. Lower load levels 
require considerably longer test times, which are more 
likely to exceed the reliable conduct of this test type. 
Tests at lower fractions of UTS are being pursued. Further 
details about the PBO braided cord material and test 
methods have been documented elsewhere [2,4]. 
Results 
The creep lifetime data are presented in Figure 1 on 
Weibull axes. In an analogous paradigm to logarithmic 
axes, data which fit the Weibull distribution well appear 
linear when plotted on Weibull axes. The software used to 
generate this plot and fit the data to the Weibull 
distribution 129,301 showed good fitting metrics for the 
three samples. Table 1 gives the Weibull fit parameters 
for each load level. These parameters are then plotted on 
semi-log axes according to Eqs. 4-5 in Figs. 2-3. Linear 
least squares fitting for lnq (Fig. 2, Eq. 4) yields 
lna,,=44.05, ~,,=0.43, and ~*=0.9997. The linear least 
squares fitting parameters for lnp (Fig. 3, Eq. 5) are 
lnap=2.12, q=0.03, and ~ ~ = 0 . 9 9 9 8 .  It is evident that the 
data support the theory. 
The goal of this work is to provide a guide for 
designers and engineers who intend to use this material in 
a static, or near-static, manner. Figure 4 provides such 
advice as a plot of lifetime to be anticipated as a function 
of loading and acceptable failure probability. The 
interpretation of Figure 4 is that for an application 
requiring approximately one month duration (lntb~15) 
with acceptable failure probability of 1% (P=0.01), the 
maximum loading should be 60% (f=0.6) of the cord 
ultimate tensile strength. Of course, interpretations 
beyond the available data should be done cautiously. 
Analysis of additional lifetime data from experiments at 
lower creep stress levels is underway. 
Summary 
A Boltzman-type phenomenological model for creep 
rupture, when coupled with a Weibull statistical analysis 
of laboratory data, provides a useful constitutive 
representation for the creep lifetime of braided PBO cord. 
This model will be useful to NASA's ULDB project, and 
also perhaps to other aerospace and civil engineering 
applications. 
Acknowledgements 
Ron Allred of Adherent Technologies in 
Albuquerque made helpful suggestions. Walt Thomas of 
NASA-Goddard was helpful with the Weibull statistical 
analysis. The data were obtained by technicians in 
NASA's Balloon R&D Laboratory. This work was funded 
by NASA's Balloon Program Office. 
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Key Words 
Braided cord, creep lifetime, creep rupture, ~evlar@, 
PBO, p-aramid, ULDB, zylonm. 
Load (% of UTS) Eta (q), secs Beta ( ) 
0.493 
2,33 1 0.579 
18,539 0.675 
Table 1. Weibull fit parameters (Eq. 3) for 
creep lifetime data shown in Fig. 1. 
tSQ.?%m 
0 Sonm 
C 
D 
F 
94 
.oi .03 .I .3.5 1 2 3 no m soloo somooo 3oooioooo iooooo ioooooo 
Time at Load (s) 
Figure 1. Creep lifetime data presented on Weibull axes. CDF = cumulative distribution 
function. The heavy broken grid line at CDF = 63.2% corresponds to q for all Weibulls. 
-0.2 - 
-0.4 -- 
23 
" 0.6 -- 9- 
s 
- 
-0.8 -- 
-1.0 
100 
90 
80 
h 
& 70 
r 
60 
50 
. . 
40 
0 5 10 15 20 25 
In(lifetime in seconds) 
Figure 4. Lifetime probability plot showing laboratory data. 
12 
10 -- 
ii * - i  
Y 
s 
- 6-- 
4 -- 
- 
' * ' ' ; " " : " " : " ' '  
' 
- 
2 , ~ m ~ ~ ; - ~ ' m : m ~ ~ ' ; m ~ ~ 8  
75 80 85 90 95 
Loading as fraction of UTS, f (%) 
Figure 2. Weibull scale parameter plotted according 
to failure model. 
75 80 85 90 95 
Loading as fraction of UTS, f (%) 
Figure 3. Weibull shape parameter plotted 
according to failure model. 


A Constitutive Model for Creep Lifetime of PBO Braided Cord

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