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Modeling the statistical lifetime of glass ber/polymer matrixcomposites in tension

S. Leigh Phoenix

Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA

Abstract

In this paper we present a viewpoint on modeling the lifetime of glass ber/polymer matrix composite structures loaded primarily

in tension along the ber axis. In many applications such components may sustain, over many years and in deleterious environ-

ments, stress levels that are a signicant fraction of their ultimate tensile strength. Thus the failure phenomenon of concern is creeprupture. Ideally, a comprehensive model should incorporate such features as environmentally driven, statistical degradation

mechanisms in the glass ber (such as stress corrosion cracking), creep and microcracking of the polymer matrix, slip at the ber/

matrix interface near ber breaks, local residual stresses from processing, including their complex micromechanical interactions.

Such a model should yield overall distributions for lifetime in terms of the overall applied stress eld, the overall volume of material,

and boundary eects. Parameters of the model should reect subtle scaling relationships among microstructural variables (e.g., ber

packing geometry), parameters of the statistics of ber strength and degradation, matrix and interface creep exponents, rate factors

in the stresscorrosion chemistry, and applied stress level. Particular attention must be paid to the character of the extreme lower

tails of the strength and lifetime distributions since these are crucial in establishing load levels that result in the extremely high

reliability levels important in life-safety applications. For example, the model should be able to predict the steady load level in a

composite specimen with an eective loaded volume that yields a given lifetime (e.g., 25 years) at an extremely low probability of

failure (e.g., 106). This essentially rules out mean eld approaches so prevalent in the mechanics and physics community. A model

of this sort would also be useful in the development of strategies for eective accelerated testing and data interpretation using

special timetemperature scalings and master curves. Lastly, the model would have value in guiding strategies for quality control,

materials processing, and component architecture during manufacture. Of course, such a comprehensive model is well beyond thepresent state of the art. Nevertheless, a surprising amount of progress has been made in developing the necessary conceptual and

computational framework including the micromechanics, chemistry and physics of the fundamental failure mechanisms. In this

paper we will review some of the relevant literature and suggest directions that should be fruitful in yielding useful models. 1999

Elsevier Science Ltd. All rights reserved.

Keywords: Lifetime distribution; Creep-rupture; Glass ber composites; Micromechanics

1. Introduction

In many applications in civil engineering [16], a

composite component must sustain, for many years,

stress levels that are a signicant fraction of its ultimatetensile strength, and often in deleterious environments.

Various applications in new construction are: (i) rein-

forcement of concrete using 1D rods, 2D grids and 3D

networks, (ii) tension members for prestressing concrete,

(iii) bridge cables, stays, hangers and roof support ca-

bles, (iv) pultruded beams and bridge superstructures,

and (v) storage tanks and piping. Interest also exists in

using composites for repair and rehabilitation of dete-

riorating structures. Further applications also include

pressure vessels and ywheels in aerospace and auto-

motive applications, and centrifuges in medical and

nuclear power. A generally recognized problem is creepfracture (also known as creep rupture, stress rupture,

and static fatigue), a catastrophic failure event that

typically occurs with little or no warning [6]. Thus, es-

tablishing load levels that yield long lifetimes at ac-

ceptable reliability levels becomes a major challenge. To

this end considerable research is being carried out in the

US [7] and internationally [8]. Many recent works report

basic lifetime tests on the performance of composite

components in various environments and over limited

time-frames (e.g., [912] in [8]). These works, however,

rarely consider governing micromechanical, statistical

Composite Structures 48 (2000) 1929

E-mail address: [email protected] (S. Leigh Phoenix)

0263-8223/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 6 9 - 0


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and chemical mechanisms at the ber scale, and rarely

draw on the vast experimental and modeling literature

developed over the past 50 years in connection with

national defence and space exploration. Often crucial

information on properties of the constituent materials,

and test conditions is not documented. Thus, it is

problematic to extrapolate the results to other settings.

A common methodology is to use a `brute force'

testing approach. In this method, a large number of

lifetime tests are performed on basic composite speci-

mens at each of the several load levels, and the results

are tted to a lifetime model based on phenomenology

such as a Weibull distribution. Scale and shape param-

eters are chosen to reect sensitivity to stress level

through power law or exponential laws (producing lin-

earity on loglog and semi-log coordinates, respective-

ly), which sometimes may contain an Arrhenius factor

to account for temperature or environmental eects.

While such tests are very important, their utility is

greatly diminished in the absence of an accurate modelto allow extrapolation of the data to other time and size

scales, chemical concentrations, and especially load

levels that result in low failure probabilities. Studies in

the literature suggest some key lessons: The precise

values of creep-rupture parameters cannot be predicted

through simple averaging rules based on bulk mea-

surements on the bers and epoxies; in fact, similar

systems can yield puzzlingly dierent results, pointing to

the need for understanding the basic micromechanisms.

Glass ber/polymer matrix composites have been of

interest for military and aerospace applications for

about 50 years, and a vast literature has developed. This

literature covers failure mechanisms in glass bers,

polymer creep and interface failure mechanisms includ-

ing environmental aspects. In addition there exists ex-

tensive lifetime data on failure of composite specimens

under long-term loading. Let us now review various

aspects of this literature.

2. Review of experimental literature

2.1. Experiments on single glass bers

The mechanisms of failure in E-glass, S-glass andother glass bers, with respect to short-term strength

and long-term static fatigue conditions, have been ex-

tensively studied both theoretically and experimentally,

and considerable data have been carefully generated.

The classic works of Taylor [13], Stuart and Anderson

[14], Charles [15,16], Mould and Southwick [17], and

Charles and Hillig [18] have resulted in a fundamental

understanding of the chemical and kinetic mechanisms

involved, including the eects of stresscorrosion in

humid and alkali environments. Kies [19], and Schmitz

and Metcalfe [20,21] carried out extensive studies of the

aw types, their sources during processing and handling,

and especially their statistical distributions. Hillig and

Charles [22] further advanced understanding of stress

dependence in surface reactions and how aws geo-

metrically evolve to form cracks. Metcalfe et al. [23]

studied spontaneous aw growth in alkali environments,

and provided some spectacular micrographs of periodic

cracking along glass bers. In the ber optics area, more

recent work has considered the protection of the surface

of glass bers using polymer coatings. Schonhorn et al.

[24], for example, showed how to improve the perfor-

mance of optical glass bers in static fatigue by coating

with a uv-curable epoxy-acrylate coating, but interest-

ingly certain degradation exponents remained un-

changed. Many key ideas in the failure of glass bers are

reviewed by Kelly and Macmillan [25].

Based on this literature we believe it is possible to

build a statistical, mechanistic model of the failure of

individual glass bers under steady and mildly cyclic

stresses that is much more advanced and comprehensivethan has been attempted to date. Such a model can in-

corporate eects of temperature, alkali (and acid) spe-

cies and concentrations, probability distribution shapes,

and scaling laws for lifetime (scale parameter or mean)

versus stress level. Much of the ber lifetime data can be

resolved fairly linearly on power-law coordinates with

slope exponents ranging from about 30 on downward as

the environment becomes more aggressive (e.g., higher

temperature, increased moisture and alkaline concen-

tration). Some of the above works show how data for

various cases and materials can be reduced onto one

master or universal fatigue curve. Surprisingly, manyrecent studies on the failure of glass ber/polymer ma-

trix composites have made virtually no reference to this

extensive literature.

2.2. Experiments on unidirectional glass ber/epoxy

composites

Extensive experiments on the strength and stress

rupture of S-glass/epoxy unidirectional composites

(mostly resin impregnated strands), were carried out at

Lawrence Livermore National Laboratory (LLNL) in

the late 1960s through the 1970s. The results are reported

in several publications and internal reports [2631], andsome data is shown in Fig. 1. (Kevlar and carbon ber

composites were also studied [32,33].) The most extensive

experiments were on epoxy-impregnated strands con-

taining several hundred laments in two basic epoxies.

Data are available for hundreds of strands loaded for

over 5 years at several load levels above 33% of the short-

term strength. (The experiments were unfortunately de-

stroyed by an earthquake in 1979.) Strength retention

tests were performed on specimens removed after sur-

viving for various times in an attempt to study degra-

dation in strength versus time [28]. Strength retention

20 S. Leigh Phoenix / Composite Structures 48 (2000) 1929


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was also performed on unloaded specimens that had

been stored under ambient conditions for up to 4 years,

with only about a 4% drop in strength, most of which

occurred during the rst year. While the database gen-

erated was massive, the modeling employed was minimal

with respect to relating strand performance to that of

virgin glass bers and the epoxy matrix.

Studies have also been performed at laboratories

outside of the US. In Israel, Lifshitz and Rotem [34],

performed experiments on glass ber/epoxy and glass

ber/polyester composites, and built a simple microme-

chanical model to explain the results. Lifshitz [35] re-

viewed the literature on stressrupture in many

materials including single E-glass bers tested at 50%

RH and various temperatures. In England, Kelly and

colleagues [36,37] studied the failure of bundles of glass

bers and composites by stresscorrosion mechanisms

in hot, wet environments. In these cases models built on

micromechanical and statistical mechanisms were de-

veloped to explain the results. In the mid 1980s work

was conducted in England by several workers [3843]

including much work on polyester matrix composites inaggressive environments, including both acids and bas-

es, and at various temperatures. These works underscore

the importance of bermatrix interactions, including

the importance of matrix cracking in allowing aggressive

chemical species to enter the composite and attack the

ber surface.

In the US during the 1970s and early 1980s long-term

stressrupture of glassber composites appears to have

received little attention. A compilation by Hilado [43]

has no papers devoted to the subject, though eects of

moisture on strength and fracture energy are covered.

2.3. Experiments on creep behavior of polymer matrices

Lifshitz and Rotem [34] showed that matrix creep in

shear between the bers plays an important role in the

breakdown of the composite, by altering over time the

stress-redistribution from broken to surviving bers.

Long-term creep data for polyesters and epoxies are

dicult to nd, but there are references to other poly-

mers and general timetemperature scaling principles.

Findley and Tracy [44] reported creep data for PVC

under loadings lasting 16 years. Kibler and Carter [45]

gave some creep results for 5208 epoxy resin at various

temperatures from 296 to 435 K, and t a power-law

creep function (in the context of linear viscoelasticity)

with corresponding exponents increasing from 0.1 to 0.4.

Master curves and shift factors for net creep compliance

were also given. Kibler [46] developed results for creep of

various graphite epoxy laminates from which creep pa-

rameters for the epoxy alone could be extracted. Visco-

elasticity theories for such applications were given by

Schapery in a review paper [47] and in a special appendix

to Kiblers report [46]. In the case of crack growth con-nected to local polymer creep and breakdown at the tip,

relevant theories were given by Schapery [48]. A well-

known monograph on molecular-based aging and creep

mechanisms in polymers is due to Struik [49].

3. Micromechanics and statistics of mechanisms in creep

fracture

Creep fracture of unidirectional brous composites

results from evolving microstructural damage in the

Fig. 1. Time to failure quantiles (% probabilities of failure) for S-glass/epoxy strands tested in stress rupture at LLNL in 1970s [27]. The last two data

points were added by the author from data generated just before an earthquake destroyed the experiments.

S. Leigh Phoenix / Composite Structures 48 (2000) 1929 21


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form of statistically accumulating ber breaks over

time, eventually resulting in rapid localization at some

location forming a catastrophic crack. When such

composites are loaded at the stress levels involved, at a

ber volume fraction of 65%, for example, composite

specimens such as resin impregnated strands or rods,

can have mean tensile strengths of the order of 1.752.5

GPa (250350 ksi). Under, ideal, ambient laboratory

conditions, composites loaded at 40% of these values,

namely 0.71.0 GPa (100140 ksi) can have mean

lifetimes of the order of 10 years. The variability in

time to failure, however, is typically very large with

coecients of variation in the order of 100500% de-

pending on the ber type, as indicated in Fig. 1. In the

nal stages of life, within such structures, a rapid in-

crease in strain rate (tertiary creep) may occur in a few

highly localized regions, but at the macroscale this is

very dicult to detect. In fact, the amount of overall

creep observed is very small and even a secondary

creep stage may be obscured. This is because creep ismostly due to break opening displacements since the

matrix carries negligible tensile load and the brittle -

bers creep negligibly. Continuous graphite bers, for

example, typically show no creep at room temperature

[50]. (When the load-bearing bers are aligned o-axis

in a composite, the situation is quite dierent as matrix

creep and failure may dominate, as shown by Ragha-

van and Meshii [51,52].)

The high variability in strength found in brittle bers

is well modeled by WeibullPoisson statistics. This

variability, due to various random aws on the ber

surface and in the ber interior, means that many ber

breaks are inevitable in the composite as time passes or

as load is increased, especially in glassbers, where aw

growth due to stresscorrosion mechanisms occurs.

Initial ber breaks may be ber discontinuities due to

processing or breaks due to initial loading, and at loads

well below the composite mean strength are typically

widely spaced relative to the ber diameter. A few,

however, may be in small clusters. These breaks create

local stress concentrations in both the matrix and the

neighboring intact bers as the matrix, through shear,

transmits the tensile loads of broken bers to neigh-

boring bers.

Near these initial breaks, especially those in smallclusters, the local tensile stress levels in the ber can

approach 5 GPa or even higher, whereas the maximum

shear stresses in the matrix are of the order of 35 MPa,

more than 100 times lower. At such stress levels the

matrix will creep in shear, or, progressively debond from

the ber (as a propagating cylindrical mode II crack)

followed by a time-dependent slip. The eective shear

strains even before the slip can reach as much as 30% or

40%. Thus, the length scale of overloading of neigh-

boring intact bers increases over time, and the proba-

bility of subsequent failure of these bers also increases.

The resulting additional ber breaks further increase

local ber and matrix stress concentrations, and further

accelerate the local matrix shear creep, interface de-

bonding and slip processes, again increasing the local

stresses and probabilities of failure of neighboring ber

elements. Generally there will be many small, stochas-

tically growing clusters of ber breaks distributed

throughout the loaded composite. Eventually one of

them will become large enough, possibly through cluster

linking, to initiate a catastrophic crack.

In summary, there is an interaction of such factors as

the statistics of ber aw growth, matrix creep, interface

debonding and slip, local ber packing, ber volume

fraction and overall composite volume under load. Lo-

cal stress level, temperature, moisture concentration,

and acid or alkali concentration, will aect the rates of

these processes in dierent ways. Overall composite

breakdown rates and statistics crucial to life-prediction

will reect nontrivial interactions of these rates and

shape parameters of the statistical ber aw distribu-tions as they change over time. One can expect extremal

statistical processes over the loaded volume to play a

crucial role. This means that mean eld concepts will

play a minor role in understanding failure.

4. Current models for creep fracture

A detailed mathematical description of the above

viewpoint can be found in Phoenix et al. [53] and Otani

et al. [54] who developed a model of failure backed up

with experimental results on model microcomposites ofseven carbon bers (Hercules IM6) in an epoxy. It was

shown that, whereas the carbon bers underwent neg-

ligible creep rupture, that of the composites was ap-

preciable; the lifetime scale parameter versus stress level

on a loglog plot showed a power-law exponent of

about 55, about one-fth of the 294 value for individual

bers, as shown in Fig. 2. Moreover, this composite

exponent was shown to depend explicitly on the Weibull

shape parameter for ber strength, q, the creep/debond

exponent for the matrix, b, and the critical ber break

cluster size, k following kak 12qab.A related aspect was the distribution for lifetime of

the composite as compared to the bers. Farquhar et al.[51] plotted lifetime data for individual IM6 graphite

bers, on Weibull coordinates for various normalized

stress ratios (stress divided by Weibull scale parameter

for strength) as shown in Fig. 2. The t was excellent

and the Weibull shape parameter for lifetime was about

0.02 representing huge variability (>1000%). In com-

parison, the lifetime distributions for the 7-ber, IM6

graphite ber/epoxy microcomposites [53,54] showed

Weibull shape parameter values more than 10 times

larger, as shown in Fig. 2. In fact, the lifetime distri-

bution was not truly Weibull but showed bimodal



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behavior as the model predicted; the Weibull shape

parameter for the upper tail was about 0.2 and for the

lower tail was about 0.4. These higher values corre-

sponded to greatly decreased variability in compositelifetime versus the bers, and the variability decreased

even further deep into the lower tail of the lifetime dis-

tribution especially at lower loads. The kink in the bi-

modal distribution was shown to be caused by a change

in the critical cluster size at shorter times.

In connecting their results to other works, Phoenix

et al. [53] referred to experimental work on graphite -

ber/epoxy strands at Lawrence Livermore National

Laboratory in the 1970s and 1980s [32], at Oak Ridge

National Laboratory in the 1980s, and in Japan [55],

again following a direct experimental approach. Slightly

larger power-law exponents were measured in those

cases (probably because of reduced boundary or surface

eects due to having many more bers, reducing the

eects of slip), but the values were still far lower thanvalues above for single bers. Also the Weibull expo-

nents for lifetime were comparable to the values for the

7-ber composites.

In glass ber/polymer matrix composites, stochastic

aw growth in the individual glass bers also plays a

critical role [3442], making the failure process and

modeling much more complicated than in the carbon

ber composites. The breakdown parameters depend

subtly on interactions of the statistics of ber aw

growth, matrix creep and debonding mechanisms.

Similar processes also occur in metal matrix and glass

Fig. 2. Lifetime data for Hercules IM6 graphite bers and 7-ber/epoxy: (a) Weibull scale parameter for lifetime versus stress level for single bers on

power-law coordinates; (b) corresponding data for 7-ber microcomposites; (c) distributions for lifetime of single bers at stress ratios (stress divided

by Weibull scale parameter for strength) 0.925, 1.00 and 1.05, plotted on Weibull coordinates; (d) corresponding data for 7-ber microcomposites at

the stress ratio 0.83.



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matrix composites reinforced with ceramic bers. In

recent modeling eorts [5659] stress-redistribution

among bers was assumed to follow global load-shar-

ing. (See [59] and references therein to Curtins earlier

landmark work.) Calculation of the mean lifetime has

been carried out, but not the variability or the distri-

bution shape. Thus, distributions for lifetime (especially

the reliability associated with lower-tail behavior), and

size eects have not been obtained as yet. These latter

issues may be treated, however, by extending recent

work on time-independent, global load-sharing models

by Phoenix et al. [60].

5. Basis for improvements in modeling

5.1. Computational micromechanics techniques

Up to now statistical models for creep fracture in-

volving micromechanics have used highly idealizedstress-redistribution rules. This has been done in large

part to simplify the probability calculations but also

because more sophisticated micromechanical calcula-

tions are computationally too intensive, especially using

standard techniques such as the nite element method.

One extreme has been equal load-sharing (ELS) among

bers [3436], which has recently been extended to the

concept of global load-sharing (GLS) [59]. The other is

local load-sharing (LLS) where the stress of failed bers

is redistributed equally onto its nearest surviving

neighbors following geometrically prescribed patterns

[53,54]. Matrix creep has usually been captured by in-creasing with time the characteristic length of the local

stress transfer region, using simple scalings from prob-

lems with an isolated ber break in an array of bers.

These rules have led to considerable insight through

scaling relationships involving ber Weibull moduli,

matrix creep exponents, load-sharing constants and ber

break cluster sizes, but for larger 3D composites they are

not realistic enough to yield precise quantitative pre-

dictions of the lifetime scale parameters and distribution

shapes.

In creep fracture, various micromechanical features

occur, that require more sophisticated treatment. As

mentioned, when a ber breaks the stress at the berdiscontinuity drops to zero, but over a characteristic

length recovers gradually along the ber to the remotely

applied ber stress. Also corresponding to this length is

an overstress length on neighboring bers, where nom-

inally the maximum overstress occurs directly adjacent

to the ber. The shear deformation near broken bers,

may reect elastic, plastic and viscoelastic eects in the

matrix and frictional sliding eects at the interface act-

ing together to determine the growth of the character-

istic length of stress transfer in time, and the actual

shape of the overstress proles on neighboring intact

bers. This characteristic overstress length will grow to

become many times the length based on initial elasticity,

but the stress concentration peaks may be lowered

somewhat also (though on more distant neighbors they

will increase). Interestingly, these peak values are un-

changed when only matrix creep under linear visco-

elasticity occurs; decreases in the peak turn out to result

from matrix deformation that is nonlinear in stress such

as plasticity and time-dependent slip. Nevertheless the

probability of additional ber failures occurring locally

in time will increase, resulting in clusters of breaks that

increase in size.

To improve the modeling, there have recently been

concerted eorts at Cornell University and elsewhere to

develop computational micromechanics techniques that

are much faster and more realistic. The most successful

framework has been built upon the idealized, shearlag

framework of Hedgepeth [61,62]. (See the monograph of

Chou [63] for a review and many applications.) In these

models the bers carry the tensile loads and the matrixprimarily transmits loads in shear, allowing load trans-

fer from ber to ber near breaks. Extension of these

models to include additional stress tensor information

may seem desirable and is reasonably straightforward,

but numerical studies have shown that the gains in ac-

curacy and realism are surprisingly small and rarely

worth the greatly increased computational cost. In fact,

the shearlag approach yields realistic and accurate

stress and displacement results that are consistent with

those from experiments using micro-Raman spectros-

copy and nite element calculations [64,65]. But the

greatest advantage is that the shearlag model canmodel stress-redistribution in complex, random ber

break patterns including crack-like formations. In the

latter case, when compared to classical continuum re-

sults for cracks in orthotropic media, the stress results

agree remarkably down to the length scale of one ber

diameter [66]. Also shearlag models show promise of

modeling complex process zones ahead of crack tips not

amenable to classical continuum treatments since cru-

cial, discrete statistical features are lost (which surpris-

ingly do not smooth out as the crack grows large relative

to the ber diameter).

The simplest, purely elastic version developed at

Cornell is called break inuence superposition (BIS)[66], and it can handle rapidly in 2D hundreds of arbi-

trarily placed ber breaks [67,68] on a small worksta-

tion. A more advanced version called quadratic

inuence superposition (QIS) [69] uses quadratic order

corrections to allow for localized matrix yielding, in-

terfacial debonding and frictional sliding around many,

arbitrarily located ber breaks. The QIS technique can

compute the extent and shape of the yield and debond

zones, and the stress and strain distributions every-

where. In the time-dependent case, the newest version is

called the viscous break interaction (VBI), which can



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handle ber break arrays in viscous and viscoelastic

matrices following a power-law creep function [70]. This

new model in 2D builds on inuence functions based on

an earlier work at Cornell [71,72]. All these versions

show promise of being up to three orders of magnitude

more powerful than full discretization schemes such as

nite dierence or nite element methods. Finally we

note that Curtin and coworkers [73] used lattice Green's

functions to solve problems similar to those treated by

the BIS and QIS method. However, high accuracy re-

quires ne discretization along the bers, reducing

computational eciency.

6. Stochastic failure models for bers and bundles

In addition to the computational micromechanics

aspects described above, there is also long-standing

experience in stochastic modeling of the failure of ber

bundles, and chains of bundles under ELS, GLS [7480] and LLS [81,82] among bers. Experiments have

also been performed on bers and composites to sup-

port the basis for these models [8385]. Several works,

for example, have dealt with the failure of bundles

where the failure of the individual bers depends sto-

chastically on their individual load histories [7477,81]

and reects ber strength degradation kinetics. This is

a crucial aspect to modeling failure in glass bers. As

discussed, an important aspect has been to capture

time-dependent creep in the matrix and debonding at

the interface, and how it aects stress transfer mecha-

nisms and probabilities of local ber failure. These

works incorporate this feature crudely [53,54,70]. Also,

an important theme has been the treatment of long

chains of nite bundles. In these cases, the mathemat-

ical framework of the statistical theory of extremes is

crucial [76,77,8183].

7. Future prospects in modeling

The goal of building a comprehensive model will be

accomplished by incorporating: (i) ber mechanical

properties and aw statistics at the length scale of local

ber load transfer, including any stochastic stresstimedependency, (ii) matrix plasticity and creep in shear

around ber breaks, (iii) interface debonding and vis-

cous frictional sliding in terms of local shear stress, (iv)

local ber and matrix packing geometry (2D planar, 3D

hexagonal and random) and associated residual stresses

from processing, and (v) multiple matrix cracking in-

sofar as it aects penetration of the environment to the

ber surface. Major aspects will be to use these theo-

retical models and supporting numerical analyses to

determine key scalings, such as: (a) the shapes of re-

sulting strength and lifetime distributions, eective

Weibull shape parameters (in cases where the Weibull

distribution is an accurate approximation) as a function

of load level and distribution, and composite volume,

(b) sensitivity of lifetime to load level in power-law (log

log), exponential (semi-log) and other functional

frameworks motivated by the physics and micromech-

anics of stochastic aw growth in the composite, (c)

critical cluster size and `crack threshold' parameters. An

important goal should be to uncover new functional

forms for strength and lifetime distributions when rele-

vant, rather than assume Weibull forms a priori. This is

particularly important when modeling sensitivity to

stress level and lower tail behavior.

A long-term goal should be to implement the model

in a Monte Carlo framework to: (i) validate key ana-

lytical scalings and determine certain proportionality

constants not accessible by analytical methods alone,

and (ii) perform many more replications (say 1000

samples per stress level) on realistic structures than can

be handled by physical experiments alone. The ultimateobjective should be to generate computational tools to

design material microstructures and determine pro-

cessing routes towards enhancing lifetime and reliability

through proper quality control tests and handling pro-

cedures for the glassbers, and tailoring of bermatrix

interface and local matrix creep properties. For valida-

tion, predictions should be compared to experimental

results on actual composite specimens (resin impreg-

nated strands or coupons) as well as on microcomposite

specimens [53,54].

For example, with respect to the ber we already have

stochastic models for its failure based on Weibull/Pois-son statistics and fracture mechanics concepts for aws

[25,36,37,51,81,84,85], but these do not take full ad-

vantage of early knowledge on chemical mechanisms of

stresscorrosion [1318,22,23] and aw statistics [19

21]. These ideas should be revisited to better couple

chemistry, temperature, aw statistics and crack-growth

mechanisms into a stochastic fracture model for bers.

This is essential to understand the chemical interactions

of the glass and the environment at the bermatrix

interface. Versions of the model should be comprehen-

sive and suitable for numerical simulation codes for

composite creep fracture, but simplied, parametric

versions will be necessary and desirable for asymptoticand scaling analysis. One goal should be to assess the

extent to which the simplied models can be used

without signicantly compromising the accuracy of

predictions.

With respect to the matrix, as mentioned earlier, at a

ber break the load is transmitted to neighboring -

bers, and the cross-linked polymer matrix locally un-

dergoes large shear stresses and strains. As time passes

the matrix creeps and eventually work hardens, re-

sulting in interface failure and a slowly propagating

debond or shear crack [86]. Furthermore the extent of



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interface sliding will depend on friction controlled by

the balance in normal stresses resulting from thermal

shrinkage during curing (compression) versus swelling

from moisture absorption (tension), these also de-

pending on the ber packing geometry and volume

fraction. This process needs to be better modeled. Also

moisture and aggressive chemical species (alkalis) may

penetrate the matrix to the ber surface by diusion

through the matrix, or transport along the debonded

ber/matrix interface, or along transverse matrix

cracks. We believe the tools are available to model

these eects.

With respect to stress and strain evolution around

large arbitrary arrays of ber breaks in 3D, recent

computational mechanics techniques are sophisticated,

weighted superposition methods where the calculations

are tied to the amount of damage (ber breaks, non-

linear matrix deformation zones) rather than to the full

composite size. Solving multiple ber break problems

involves an appropriately weighted, superposition ofcertain basis or inuence functions obtained from cer-

tain unit solutions. An important advance is the very

recently developed version, VBI to handle ber break

arrays in viscous and viscoelastic creeping matrices fol-

lowing a power-law creep function [69]. We believe this

method is adaptable to a viscoelastic matrix with non-

linearities involving a power law in both stress and time,

or to an interface slipping in time. It also can be ex-

tended to include 3D ber arrays, variable ber spacing

and arbitrary sequences of ber failures. This is needed

to eciently calculate evolving ber and matrix stress

and strain proles at any stage of the composite failure

process. In particular it is necessary to model what

happens in growing clusters of breaks as they coalesce to

form a catastrophic crack. We should note that there is a

strong indication that when these nonlinear processes

are particularly strong as is often the case, GLS becomes

a very accurate model once the eective bundle size is

determined [58,59]. A basis exists for a model that could

be adapted for this task [80].

The stochastic models for failure of ber bundles

and brous composites, discussed earlier [7482], have

shortcomings that need to be addressed. Much more

realistic stress-redistribution calculations in 3D are

necessary and these can be achieved by the methodsjust discussed. Second, there is a need to pursue scaling

concepts with respect to volume eects in the strength

distributions that have been pursued in the physics

literature [87,88]. Perhaps most important is the need

to combine stochastic decay in ber strength with

matrix creep in one unied framework, leading to

lifetime distributions, size scalings and stress depen-

dencies that are realistic and can be used to interpret

real composite lifetime data in the context of the

mechanisms involved. Thus, using enhanced computa-

tional capability for micromechanics, probability anal-

ysis and combinatorics in an extreme value setting, it

will be possible to develop more advanced functional

forms of failure distributions, including size and stress

gradient eects, eects of stress level, and eects of

temperature and environment.

With respect to accelerated testing, questions to

consider through modeling are: To what extent it is

possible to be misled by apparent good performance

early in tests? To what extent do paradoxical phe-

nomena occur? How sensitive are such tests to factors

known to eventually reduce long-term lifetime accord-

ing to the models above or to data in the literature?

And lastly, can parameters be estimated from acceler-

ated tests on simple composite specimens that can be

used in predicting the lifetime of larger specimens in

service?

Finally such models could be used to assess manu-

facturing approaches, materials processing, quality

control and component architecture insofar as they af-

fect durability and reliability. Questions are: Whatquality control tests on the ber, matrix and composite

strands or coupons are necessary to ensure reliable

components? Is proof loading useful as a strategy to

weed out weak components and condition the mate-

rial, or, does it introduce large number of ber breaks

that subsequently accelerate the creep fracture process?

Should yarns and rovings be clustered in the material or

should the bers be distributed as much as possible

throughout the material? Should components be con-

tinuous or arrays of smaller components as in bridge

strands, and what is the optimal number? If material or

components are removed from service and residualstrength tests performed can this be converted into a

meaningful measure of remaining life? In large appli-

cations should material be put on test in laboratories

under conditions similar to the actual application, and

should material be periodically removed for testing? In

quality control tests of strength coupons, what signs in

the failure modes and distributions are indicative of

material that will have poor reliability? In engineering

practice these may well be the most relevant questions.

We believe modeling plays a crucial role in answering

them.

Acknowledgements

This work was supported under a grant from the US

National Science Foundation, No. CMS-9800413.

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