Fatigue Behavior of Polymers

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Introduction

Polymers and polymer-based composites are increasingly displacing more tradi-
tional materials in a wide variety of engineering applications, and estimates by
the National Institute of Standards and Technology place the total annual loss of
engineering components due to fracture at more than $110 billion. Failure may
occur for a variety of reasons. Most notably, failure can occur if the applied stress
exceeds the strength limits of the material. Failure may also be attributed to mate-
rial property deﬁciencies resulting from improper processing conditions that lead
to defects (eg, gas or shrinkage pores, inclusions, and weld-and-ﬂow-line defects).
Surface defects can result from mechanical processing or handling. These include
scratches, gouges, and other markings that introduce stress concentrations. Fi-
nally, defects can nucleate and grow in a material during service as a result of
repeated loading at stress levels below the nominal yield strength, even when the
material has satisfactory properties and contains no adventitious defects. When
these failures occur as a consequence of a hygrothermal or mechanically induced
cyclic loading condition, they are referred to as fatigue failures.

In general, a variety of mechanisms may contribute to the failure of ac-

tual components in service. These may include chemical degradation or oxidation:
a chemical mechanism that may induce cross-linking and chain-scission. Alter-
nately, other physical processes may alter the state of the polymer (eg, surface
active agents in the presence of stress may induce crazes due to local diffusion of
the agents near defects). These aspects are not discussed in this article.

Failure of a component due to fatigue generally occurs through a two-step

process. The ﬁrst step involves the initiation of microcracks or other damage at
inhomogeneities or defects in the material. This damage can initiate and evolve

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at nominal stress levels far below the yield or tensile strength of the material.
This step is commonly referred to at the initiation stage. The second step involves
the growth of damage through the coalescence of microcracks and the subcritical
propagation of these small cracks to form large cracks and ultimately cause com-
ponent failure. This step in the failure process is referred to as the propagation
stage. For most polymeric materials, the initiation time can be orders of magni-
tude greater than the propagation time. The initiation of such microcracks and
their subsequent propagation form the subject of this article. This article places
emphasis on the fatigue behavior, and not necessarily on aspects of fatigue crack
propagation (FCP), which is considered to be a topical subset of the fracture behav-
ior of polymers and is out of the scope of this article (see F

RACTURE

). Nevertheless,

some aspects of FCP and fracture are discussed as they relate to speciﬁc cumula-
tive damage theories and lifetime prediction methods. Earlier reviews in polymer
fatigue and FCP have appeared (1–7).

General Terminology

A common aspect to all fatigue failures is that they occur as a consequence of an
imposed cyclic hygrothermal and/or mechanical loading process. In general, any
cyclic loading condition can be imposed on any specimen geometry. However, the
most common loading conditions for mechanically induced fatigue involve uni-
axial (tension and/or compression) or ﬂexural loading conditions. Although the
stress state changes with specimen geometry and loading condition, the com-
mon approach is to describe the fatigue behavior in terms of the magnitude of
some imposed cyclic stress. Two common specimen geometries used for fatigue
studies include the dogbone-shaped tensile bars (ASTM D638) (8) for tension–
tension fatigue experiments or ﬂexure specimens (ASTM D671-71) (9) for tension–
compression fatigue studies. These two specimen geometries are schematically
shown in Figure 1. The ﬂexural fatigue specimen in Figure 1 is tapered along
its gage length to produce a nominal stress distribution that is constant along
the gage length of the specimen. The ﬂexure geometry is tested in a cantilever
beam conﬁguration. This conﬁguration produces a bending moment across the
specimen thickness that is linearly distributed along the beam length. By linearly
altering the beam bending moment of inertia (via the tapered width), the stresses
along the specimen gage length become constant. For polymer-based composites,
ﬂexural fatigue using the geometries illustrated in Figure 1, are quite common.
Thus, both the tensile and ﬂexure fatigue geometries produce stress distributions
that are nominally constant along their gage lengths and are below those imposed
at the grips. These attributes are critical for accurate measures of fatigue resis-
tance of a material to ensure that specimen geometry does not alter the result by
promoting premature failure at the grip locations.

It should be noted that there are subtle differences when comparing data

generated in uniaxial tension and ﬂexural fatigue tests. A uniaxially applied load
nominally imposes a uniform normal stress throughout the cross section of the
specimen, with a plane strain condition occurring at the specimen’s center. In
contrast, the normal stresses in a ﬂexure test are linearly distributed across the
specimen, with the maximum occurring at the outer surfaces of the specimen, and

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Gage

area

(a)

(b)

Gage

area

Fig. 1.

Schematic of common specimen geometries used in fatigue testing of polymeric

materials. (a) Tensile test specimen. (b) Flexural test specimen.

are zero at specimen’s neutral axis (typically at the center for most specimens).
Consequently, ﬂexural fatigue tests are more affected by surface ﬂaws while uni-
axial tests are more affected by ﬂaws within the specimen’s interior. Also, because
of the high internal damping and low thermal conductivity characteristic of most
polymers, hysteretic heating and the associated effects are more prevalent in uni-
axial tests than in ﬂexure tests.

In mechanically induced fatigue tests, the stress or strain is oscillated about

some mean stress (strain). If the peak or maximum stress is deﬁned by

and the

minimum stress by

, then the stress ratio is often deﬁned by R where

(1)

The type of cyclic is typically a sinusoidal, triangular, or square wave with

the amplitude, mean stress, and frequency all adjustable parameters to consider.
Some of these parameters are shown in Figure 2.

Numerous studies have shown that the cyclic lifetime of a test bar or com-

ponent varies inversely with the magnitude of the applied cyclic stress, strain,
or deﬂection. As a consequence, various test procedures have been developed to
obtain mechanical property information useful in fatigue design analysis. Long
ago, W¨ohler (10) developed a procedure wherein a notched or unnotched specimen
is subjected to a cyclic stress (

σ ) and the number of loading cycles (N) to cause

failure is recorded. This procedure is repeated for a range of stress levels, and the
data are presented in the form of a

σ –N curve (11,12). Examples of σ–N fatigue

curves are shown in Figure 3 (12).

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Stress cycle

Stress

amplitude

Mean stress

Compression

Tension/compression

Tension

Time

Stress
(strain)

Fig. 2.

Important parameters for fatigue experiments.

17 min

2.8 h

28 h

12 days

115 days

Fatigue life (10 Hz)

SAN

POM

PA-6,6

PA-6

ASA

Number of cycles

Fle

xur

al stress amplitude

, MP

Fig. 3.

W¨ohler curves for different thermoplastics, generated via ﬂexural fatigue loading

at 10 Hz (12). To convert MPa to psi, multiply by 145.

The cyclic lifetime of a given polymer decreases with increasing applied stress

consistent with greater damage accumulation per cycle at higher stress. Some
of the plots in Figure 3 exhibit a relatively well-deﬁned plateau, suggesting an
endurance limit for a particular polymer below which fatigue damage does not
signiﬁcantly accumulate. By contrast, other engineering plastics reveal a steadily
decreasing allowable cyclic stress with increasing number of loading cycles. Stress
cyclic lifetime curves are used to deﬁne the permissible operating stress (or strain)
in order to preclude fatigue failure for an anticipated number of cyclic excursions.
For example, if it is desired that a component withstand at least N

loading cycles

in a given lifetime, the part should be designed so that the applied cyclic stress is
less than

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Fatigue tests are both costly and time consuming because of the number

of samples that must be tested in an expensive test machine for days or weeks
until fracture. Also, since fracture is a stochastic process by nature, repeated tests
should be conducted at each stress level considered. Finally, the number of cycles to
failure, N

, gives no clear indication of the fatigue limits for a given application of

a polymeric material because in many cases, the deﬁnition of failure is ambiguous.
For example, failure may be deﬁned as a decrease in specimen stiffness or other
irreversible changes rather than actual specimen fracture.

Fatigue Behavior

In general, fatigue failure occurs by one of two possible modes: a mechanically dom-
inated mechanism or a thermally dominated mechanism. A mechanically domi-
nated process involves the nucleation of damage and its subsequent growth in the
form of crazes, microcracks, and ultimately large cracks that are responsible for
specimen failure. This process occurs at relatively lower stresses and frequencies
and results in a higher fatigue life. This regime is commonly referred to as the high
cycle regime and the failure mode of the polymeric material is quite often brittle
in nature (13–16). This regime is also referred to as the true fatigue response of
the material.

At relatively higher stresses and frequencies, hysteretic heating affects the

physical and mechanical properties of the polymer. This effect is a combined con-
sequence of the fact that most polymers demonstrate high internal damping and
low thermal conductivity. In this process, heat generated from mechanical fa-
tigue cannot be dissipated to the surroundings and the polymer temperature rises
throughout the test. Failures usually occur rather quickly and the failure mode is
usually ductile (13–16). This regime is referred to as the low cycle regime and is
not considered to be an intrinsic response of the material to fatigue loading con-
ditions. If one polymer exhibits a greater tendency than another for hysteretic
heating under cyclic loading conditions, the relative ranking of the two poly-
mers may differ greatly when the tests are conducted at both high and low cyclic
frequencies.

Identifying whether a particular set of fatigue conditions is mechanically

or thermally dominated can be observed in the evolution of the hysteresis loops.
This is shown characteristically for the case of polyacetal in Figure 4 (13). In the
low cycle region, hysteresis loops show a progressive increase in compliance and
irreversible work as the material is fatigued. In contrast, the high cycle hysteresis
loops show that the material initially becomes slightly stiffer and less viscoelastic
(as evidenced by the decrease in irreversible work) as the cycles increase. This is
fatigue-induced embrittlement and occurs prior to craze initiation and subsequent
crack growth. A W¨ohler plot of polyacetal showing the thermally and mechanically
dominated regions is shown in Figure 5 (13). Note that a discontinuity occurs in the
σ–N at the stress level where the transition occurs. This is critically important to
recognize since data in the thermally dominated regime suggest that an endurance
limit has been reached. However, as the stress level is reduced, a transition in the
failure mode occurs from ductile to brittle, and failure is measured at a ﬁnite
cycle number below the apparent endurance limit. Hence, extreme care should

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Polyacetal Fatigue Behavior

Cycles

Thermally dominated region

Mechanically dominated region

Stress

, MP

0.02

0.04

0.06

10 100 1000

10,000 100,000

1,000,000

Strain, cm/cm

Mechanically Dominated Region

Stress

, MP

50 100 200

300

Strain, cm/cm

Thermally Dominated Region

Stress

, MP

0.00

0.04

0.08

0.12

Fig. 4.

Plots of hysteresis loops for polyacetal fatigued in the low cycle (thermally dom-

inated) and high cycle (mechanically dominated) regions. The numbers located by each
hysteresis loop indicate the cycle number (13). To convert MPa to psi, multiply by 145.

be exercised when using fatigue data for design considerations to ensure that the
proper failure mode is observed from the accelerated tests.

Hysteretic Heating and the Low Cycle Regime.

At relatively high fre-

quencies and stress levels, many polymers may become overheated as a result of
the accumulation of hysteretic energy generated during the loading cycles. Such
energy dissipation produces a signiﬁcant temperature rise within the specimen,
which, in turn, causes the elastic modulus of the material to decrease. As a result,
large specimen deﬂections contribute to even greater hysteretic energy losses with
further cycling. In the limit, an autoaccelerating tendency for rapid specimen heat-
ing and increased compliance is experienced. Such thermal failure corresponds to
the number of loading cycles at a given applied stress range that bring about
an apparent modulus decay to 70% of the original modulus of the specimen (10–
13). Extreme care should be taken in ranking the fatigue resistance of polymeric
materials since the test conditions may effect the measurement of the true or
the intrinsic fatigue resistance of the material. In general, results like those pre-
sented in Figure 3 should only be used to rank materials once it is assured that
the tests are run properly to ensure the appropriate failure mechanisms. These
test results can be applied in design only when all design factors including mag-
nitude and state of the stress, part size and shape, temperature, heat transfer

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Polyacetal

Thermally dominated region

Mechanically dominated region

Stress

, MP

Cycles

Fig. 5.

Plot of fatigue

σ–N curve for polyacetal resin, illustrating both thermally dom-

inated and mechanically dominated regions. Note in the thermally dominated region an
apparent endurance limit appears (13). To convert MPa to psi, multiply by 145.

conditions, cyclic frequency, and environmental conditions are comparable to the
test conditions.

Studies (17–19) have shown that the energy dissipation rate under cyclic

loading conditions varies directly with the test frequency, the loss compliance
of the material, and the square of the applied stress. Neglecting heat losses to
surrounding environment the temperature rise in the sample per unit time can
be calculated as

π f J

( f

,T)σ

(2)

where

T is the temperature change/unit time, f the frequency, J

the loss com-

pliance,

σ the peak stress, ρ the density, and c

the speciﬁc heat. Previous studies

(15,16,20) have shown that for all stress levels above the endurance limit, the poly-
mer heats to the point of melting at failure. The temperature rise is stabilized at a
maximum below the point of thermal failure. It should be noted that for conditions
associated with thermal failure, cyclic lives can be enhanced by intermittent rest
periods during the cyclic history of the sample (20–22). Temperature rise resulting
from adiabatic heating can thus be dissipated periodically. Therefore, component
life can be enhanced via cooling in thermally dominated fatigue applications (23).

The transition from thermal to mechanical failure has been described in

terms of a changeover stress level, which depends on test frequency, mean stress,
cyclic waveform, and specimen surface area-to-volume ratio (24). An empirical
relationship between the uniaxial changeover stress and test frequency is shown
in Figure 6 (24) and is given by

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Uniaxial ther

mal f

atigue changeo

er stress

, MP

Frequency, Hz

= 0.63 mm

−1

= 0.4 mm

−1

= 0.4 mm

−1

100

Fig. 6.

Changeover stress from mechanical to thermal failure as a function of specimen

surface area-to-volume ratio

β and test frequency (24). To convert MPa to psi, multiply by

145.

/V(C

− C

log

f )

(3)

where

is the changeover stress, A the specimen surface area, V the specimen

volume, and f the cyclic frequency. C

and C

are empirical constants and related

to the heat capacity of the polymer and properties of the environment that the
polymer is fatigued in. A similar methodology is also presented in Reference 16.

High Cycle Fatigue and Polymer Embrittlement.

In the high cycle re-

gion (ie, at stress levels and frequencies low enough such that hysteretic heating
is stabilized), failure of the material occurs through the nucleation of damage and
subsequent propagation of cracks. In general, the total fatigue life of a specimen
is a combination of an initiation (N

) and a propagation (N

) stage. The initia-

tion stage is associated with the cycles required to initiate damage and develop
a defect (crack) of a size such that the threshold stress intensity is reached. The
propagation stage is associated with the number of cycles required to propagate
the crack under subcritical conditions from the threshold size to the critical size
(such that failure is imminent). Thus far, all discussions have associated with
the total cycles (N) and

= N

+ N

(4)

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Fig. 7.

Fracture surface in polyacetal resin specimen subjected to tensile fatigue. The

penny-shaped crack initiated at a void or other inhomogeniety at its center (25).

If a defect is present in a component at the outset of its service life, N corre-
sponds largely to the propagation stage of crack extension. Depending on the size
and shape of the defect, N

represents either a small or a large fraction of N.

Depending on the polymer and processing conditions, damage may nucleate rel-
atively early and the total life is dominated by the subcritical fracture resistance
of the polymer. Figure 7 (12) illustrates such a case for poly(methyl methacrylate)
(PMMA), where the time for craze initiation is relatively short compared to the
total life of the specimen. In other cases (more typical of engineering thermoplas-
tics with controlled processing conditions), the initiation stage governs the total
fatigue life. Table 1 (13) shows a comparison between the initiation and propaga-
tion life for two engineering thermoplastics: polyacetal and nylon-6,6 exposed to a
sinusoidally applied maximum cyclic stress of 30 MPa at 2 Hz. When comparing
the cycles for initiation and propagation in Table 1, it is clear that the early stages
of damage development involve mechanisms and kinetics not well described by
processes that govern the subcritical fatigue crack growth (discussed in a later
section).

Table 1. Comparison between Fatigue Initiation and Propagation Lifetimes for
Polyacetal and Nylon-6,6

Material

, MPa

·m

, MPa

·m

Nylon-6,6

2.0

6.0

5.4

× 10

− 9

6.8

9000 2

× 10

∼2 × 10

Polyacetal

2.5

3.1

1.3

× 10

− 17

20.0 186000 5

× 10

∼4.8 × 10

Ref. 13.

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It is generally well accepted that damage initiates at preexisting defects

or other inhomogeneities within the material. This is typically illustrated in
Figure 7 for a polyacetel resin specimen (25). Figure 7 shows the fracture sur-
face from an injection molded ASTM D638 tensile bar subjected to a tensile cyclic
fatigue. The initiation of the penny-shaped crack occurred at a void or other in-
homogeniety and propagated subcritically under cyclic fatigue to a critical length
prior to sudden failure. The specimen was subjected to approximately 5

× 10

cycles to cause failure and the number of cycles required to propagate the penny-
shaped crack to the critical size is of the order of 2

× 10

cycles. Thus, the initiation

of the penny-shaped crack at the defect site governed the total life of the polymer.

If the preexisting defects in the material are well below the threshold size of

the polymer, the initiation process may involve a variety of morphological and mi-
crostructural changes. One phenomenon observed in some polymers involves the
process of mechanically induced embrittlement. In this process, the polymer be-
comes slightly stiffer and less viscoelastic as evidenced by the hysteretic response
of the material (13). This is illustrated for the case of polyacetal in Figure 4. Notice
that the hysteresis loops in the high cycle region become slightly stiffer and the
corresponding irreversible work decreases as the material is continually fatigues.
This occurs while damage in the form of crazes, cavitation, and microcracks would
all decrease the stiffness of the material. This suggests that additional morpho-
logical changes may occur during the fatigue of a polymer. Potential mechanisms
of this process are discussed in a later section.

Classical Modeling Studies

W¨ohler (10) curves are commonly used to rank polymers and identify endurance
limits (see Fig. 3) for particular applications. However, they are of little use beyond
this without the implementation of a theory that allows for the prediction of fatigue
life under more arbitrary loading histories. Predictive methodologies that allow
for lifetime estimates have been developed to address this issue. These theories
are generally referred to as cumulative damage theories and may or may not
incorporate aspects from fracture mechanics or physical characteristics of the
materials. Some early theories developed originally for metals (26,27) recognize
that total fatigue life is dominated by the initiation stage (see eq. 4) and therefore
do not consider the propagation stage in lifetime estimates.

One of the earliest, and most popular, cumulative damage theory was intro-

duced by Minor (26), and is now referred to as Minor’s Law. Minor proposed that
if a material would last a 100 cycles at a particular stress level, then 1/100 of its
life is consumed in every cycle at that stress level. This basic postulate provides
a framework whereby the

σ –N curve (eg, Figs. 3 and 5) can be used to estimate

the fatigue life of a material subjected to an arbitrary loading history. Consider
that an arbitrary loading history can be fractionated into m blocks of cycles, each
under constant fatigue conditions. Then, Minor’s law can be written as

= 1

(5)

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In equation 5, n

is the number of cycles within a block applied at stress level

and N

is the total number of cycles to failure at that stress level (obtained from

the

σ –N curve). If validated for a particular material, equation 5 can be used for

any arbitrary loading condition. In this way, the fatigue data generated in the
laboratory is referred to as an accelerated test and equation 5 is used much in the
same way that time–temperature superposition is used to predict the viscoelastic
deformation of a material. Note, however, that any predictive methodology like
that presented in equation 5 must be validated before it is employed for lifetime
estimates. In the case of Minor’s law, the sequence of the loading history should
have no effect on the total lifetime. That is, if one specimen is loaded for a selected
portion of its life at stress level

and the remaining part of its life at

, then

the same total life would be achieved if the specimen is ﬁrst loaded at stress level
σ

followed by

Another common theory was proposed by Manson (27) and is referred to

as the universal slopes equation. In this model, the plastic strain or permanent
deformation, is considered as a measure of the damage imposed in the material.
On this basis, the true plastic strain amplitude can be used as a measure of the
fatigue behavior. Moreover, the fatigue curve can be predicted in terms of the
monotonic stress–strain curve. This empirical approach was initially statistically
correlated with many metals and takes the following form:

ε =

N
D

− 0.6

+ 3.5

− 0.12

(6)

where

ε is the total strain, D is the ductility, E is the elastic modulus, and σ

the ultimate tensile strength. The ductility term D is calculated by

= ln

(7)

In equation 7, A

is the original cross-sectional area and A is the ﬁnal cross-

sectional area at specimen failure. The coefﬁcients 3.5,

−0.6, and −0.12 in equa-

tion 6 are based on a statistical ﬁt of over 29 metals. In that sense equation 6
universally predicts the fatigue behavior of a wide range of metals by using data
obtained from a monotonic tensile test.

Opp and co-workers (14) attempted to extend the universal slopes equa-

tion to predict the behavior of polymer fatigue. However, in their attempts they
showed that a form of equation 6 was not useful for predicting polymer behavior
and instead developed a model to predict the low cycle fatigue behavior based
on hysteretic heating. And in this sense is consistent with those presented in
equation 3 and References 15–17.

Other classical methods to predict the fatigue behavior in complex or ran-

dom load patterns involve the so-called rainﬂow and Markov methods (28). These
methods utilize any form of a cumulative damage theory to predict the lifetime in
random loading situations.

McKenna and Penn (29) developed a model for predicting the fatigue life

of polymers and applied it to an amorphous glass, PMMA, and polyethylene. The
model is characterized within the framework of a cumulative damage model. They

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show that the mean failure times in constant rate of stress experiments can be
successfully predicted from a model using a time to fail function obtained from
constant stress experiments. In this manner they were able to bridge the gap
between creep and fatigue loading conditions. In their model, the time to fail, t

is related to the stress history

σ (ξ) by the following equation:

[(

ξ)]

= 1

(8)

where

(

σ) is the mean time to failure in constant stress experiments. Thus, each

increment of time d

ξ, during which the load is σ(ξ), is weighted inversely as the

lifetime

(

σ ), which the specimen would have had under a constant stress σ.

Once

(

σ ) is determined from creep (constant load experiments), equation 8

can be used to predict the failure under any stress history. The form of

(

σ ) that

was found to describe the data in Reference 29 is

(

σ ) = Ae

−Bσ

(9)

where A and B are constants. Equation 9 uses the form that is derived from molec-
ular considerations by Tobolsky and Eyring (30), Coleman (31,32), and Zhurkov
(33). Given the form for time to failure under constant stress in Equation 9, an
explicit expression for a constant rate of stress can be explicitly obtained by inte-
grating equation 8. The ﬁnal expression takes the following form:

ln( AB

σ +1)

(10)

In equation 10, t

becomes the time to failure at a constant stress rate

σ . Similarly,

for a sinusoidal stress history

σ (t) = p+ q sin(ωt)

(11)

the time to failure becomes

ˆt

( p)

(Bq)

(12)

In equation 12, I

is the zero-order Bessel function, B is the constant from

equation 9, and ˆt

( p) is the time to fail at a constant stress p.

Using equation 12, data from creep tests can be used to predict the failure

in sample subjected to a sinusoidal load history. Figure 8 presents a comparison
between experimental results and theoretical predictions (29) for PMMA by us-
ing equation 12. In this case the prediction underestimates the lifetime in some
cases by nearly an order of magnitude but is generally very good. In the case of
polyethylene (not shown) the model overestimates the experimentally observed
fatigue life. Another practical limitation to the methodology presented in equa-
tions 8, 9, 10, 11, 12 is that creep failure data is needed to predict the fatigue

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Dead load

Peak stress, MPa

Time to break, s

Fatigue data

Fatigue prediction from
additivity of damage

Fig. 8.

Comparison of failure times obtained in zero-tension sinusoidal fatigue exper-

iments with those predicted by equation 9 for PMMA at 0.164 Hz, T

= 297 K (29). To

convert MPa to psi, multiply by 145.

lifetime. Experimentally, creep experiments take considerably longer to conduct
than fatigue experiments to failure.

Other approaches have been suggested to more formally deﬁne the form of

damage that occurs during fatigue (34–37). These methods model damage as an
array of microcracks (34,35) that inevitably require a tensoral deﬁnition of damage
(36,37). These methodologies allow for multiaxial stress states to be considered.
Thus far, these theories have had only limited applications in polymeric materials.
One difﬁculty that arises from such a framework is the necessity to sum over an
array of damage using Green’s functions and hence linearity in the constitutive
behavior of the material is required. Nonetheless, these approaches may be useful
in applications where the nonlinearity imposed from the damage array dominates
the material response.

Fatigue Damage Initiation

Physical and Morphological Changes.

Some studies have discussed

the microstructural and morphological changes that occur in polymers as a conse-
quence of fatigue loads. These effects may occur before the formation of noticeable

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damage in the form of voids or microcracks occur. These changes may be critical
to understanding the nucleation of damage as well as its growth kinetics. Bouda
(38) found that the effects of fatigue on glassy polymers can manifest either as
a strain hardening or softening, depending, among other things, on the partic-
ular material being investigated. Changes in the dynamic viscoelastic behavior
were also reported as well. Similarly, Takahara and co-workers in numerous in-
vestigations (39–41) measured changes in the viscoelastic response of a variety
of different polymer systems. Both changes in the storage and loss moduli were
observed during fatigue. More recently, Sakurai and co-workers (42) conducted
a comprehensive investigation of changes induced in polyurethanes subjected to
fatigue. They presented dramatic evidence of microstructural changes occurring
after a fatigue load was applied. These changes were attributed to changes in
crystallinity in addition to dissociation of the hard-segment microdomains.

A recent paper (13) presents results from in situ measurements of both the

dynamic viscoelastic properties and energetics (ie, potential energy, strain energy,
and irreversible work) of nylon-6,6 and polyacetal specimens subjected to fatigue.
The measurements of dynamic viscoelastic behavior illustrate how the bulk ma-
terial properties may be changing during fatigue. Measurements of the strain
energy, potential energy, and irreversible work also indicate material changes but
can also be more directly used in calculating energy release rates for crack growth.
Thus, the evolution of energy release rates may also be an indicator of when crack
(damage) growth occurs in the specimen. A plot of the dynamic viscoelastic be-
havior is shown in Figure 9 and the corresponding evolution of the energetics is
given in Figure 10.

The transition from the low cycle to high cycle fatigue occurs between 48 and

50 MPa at a frequency of 2 Hz for polyacetal and this transition is clearly evident
in both graphs. In the low cycle regime, the materials become softer and more vis-
coelastic as evidenced by the dramatic drop in storage modulus and corresponding
increase in tan

δ. Also, both the potential energy and irreversible work increase

considerably in the low cycle regime. All of these results are consistent with the
softening of the material due to hysteretic heating.

In the high cycle regime, the polyacetal becomes more compliant (see

Figs. 9 and 10) as well, but the amount is more subtle than in the low cycle regime.
More interesting, the material also becomes less viscoelastic as evidenced by the
drop in tan

δ and irreversible work. This is in contrast to low cycle regime and

suggests that other morphological changes are occurring in addition to damage.
Subsequent studies (13) also showed that the crystallinity increased slightly in
both nylon-6,6 and polyacetal.

A similar study on polycarbonate showed that cyclic stress induced physical

aging on polycarbonate (43). In this work it was noted that both the Young’s moduli
increases while the loss modulus decreases. This related phenomenon is referred
to as fatigue aging. In contrast, thermophysical aging is the process of the spon-
taneous approach of the glassy state toward thermodynamic equilibrium. This
phenomenon has been reviewed extensively by Struik (44) (see A

GING

HYSICAL

This spontaneous change is accelerated perceptibly when the temperature of the
glass is increased toward the glass transition. Physical aging causes an increase
in density, relaxation modulus, and other properties. Physical aging alters the
relative kinetics of crazing and shear banding processes, so that a normally tough

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211

Polyacetal

Dynamic Viscoelastic Behavior

′

750

500

1000

1250

1500

Cycles

′′

100

150

200

Cycles

tan

0.05

0.00

0.10

0.15

0.20

tan

Modulus

, MP

Modulus

, MP

Fig. 9.

Plots of dynamic viscoelastic response during fatigue of polyacetal over a range

of maximum stress levels. All tests were conducted at a frequency of 2 Hz at room tem-
perature (13).

56 MPa; —– 54 MPa; - - - - - 52 MPa; — - 50 MPa; — - - 48 MPa;

— – 45 MPa. To convert MPa to psi, multiply by 145.

polymer becomes very brittle upon aging (45,46) apparently because the shear
yielding process becomes retarded.

A common method employed to detect physical aging in glassy polymers

involves the use of differential scanning calorimetry. Careful studies of the ther-
mograms show an enthalpic overshoot as the polymer is heated through the glass-
transition region. This overshoot occurs both with thermophysical aging and fa-
tigue aging but the peak temperature of the endotherm is slightly increased for the
fatigue aged samples (43). Figure 11 shows how the endothermic overshoots are
affected by the stress amplitude for polycarbonate samples exposed to different

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2500

2000

1500

1000

500

Energy

, kJ/m

Cycles

2500

2000

1500

1000

500

Energy

, kJ/m

Cycles

Potential Energy

Polyacetal

Energy Density Evolution

Strain Energy

Energy

, kJ/m

Cycles

Irreversible Work

500

400

300

200

100

Fig. 10.

Plots presenting the energy density evolution for polyacetal over a range of stress

levels. All tests were conducted at a frequency of 2 Hz at room temperature (13).
56 MPa; —– 54 MPa; - - - - - 52 MPa; — - 50 MPa; — - - 48 MPa; — – 45 MPa. To convert
MPa to psi, multiply by 145.

thermal histories before loading. In all cases, the fatigue tests were stopped after
8000 cycles. For samples exposed to low or moderate annealing prior to fatiguing,
an increase in the enthalpic overshoot appeared and was relatively insensitive
to the amplitude of stress. For the highly annealed material, the fatigue actually
reduced the endothermic overshoot (ie, reversed the physical aging). Similarly
Figure 12 shows how the enthalpic overshoot evolves during the early stages of
fatigue.

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213

Crazing zone

Incubation zone

Stress amplitude of fatigue, MPa

Enthalp

y o

ershoot b

y dsc

, J/g

1.25

1.50

1.75

2.00

2.25

2.50

Fig. 11.

Differential Scanning Calorimetry enthalpy overshoot of samples of different

previous physical aging histories plotted as a function of cyclic stress amplitude for a ﬁxed
8000 cycles (43).

aged at 130

◦

C for 24 h;

aged at 130

◦

C for 12 h;

aged at 130

◦

C for

6 h. To convert MPa to psi, multiply by 145.

Early Stages of Damage Formation.

A variety of studies have indicated

that void formation, cavitation, or incipient nano-sized cracks occur in the polymer
during the early stages of fatigue. The nucleated voids then continue to grow
and coalesce while the fatigue process continues. The subcritical crack growth
kinetics has been measured over a wide range of polymers (discussed in the next
section). Once the ﬂaws reach a threshold ﬂaw size (of the order of 10

− 3

m for many

polymers), the subcritical crack growth kinetics are generally well described for
many polymers. However, the sizes of these ﬂaws that nucleate and grow during
a large portion of the fatigue life are quite often below the threshold ﬂaw size
for subcritical crack growth kinetics to govern. This section discusses research
devoted to damage at this length scale.

Studies by Zhurkov (33,47) introduced the idea that breakage of primary

chemical bonds plays a major role in the fracture of polymers. He found that the
time to failure under a resistant uniaxial stress of PMMA and polystyrene below
their glass-transition temperature could be expressed in terms of a thermally
activated process involving three kinetic parameters:

= t

e(U

− γ σ

)

/RT

(13)

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Number of cycles of fatigue

Enthalp

y o

ershoot b

y dsc

, J/g

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

0.200

0.175

0.150

0.125

0.100

0.075

0.050

0.025

0.000

5000

10,000 15,000 20,000 25,000 30,000

Fig. 12.

Differential Scanning Calorimetry enthalpy overshoot as a function of cycle num-

ber at a given stress amplitude. Note the different scales in the data (43).

aged at 130

◦

for 40 h and fatigued at 20 MPa;

quenched from 165

◦

C and fatigued at 31 MPa. To convert

MPa to psi, multiply by 145.

In equation 13, t

is the time to failure, U

is interpreted as the activation en-

ergy for chain scission, t

is interpreted as the inverse of the molecular oscillation

frequency, and

γ is a structure-sensitive parameter. Experimentally he (47) ob-

served the formation of platelet-shaped nanoscopic voids, which he referred to as
incipient cracks, that form very early in the polymer once a tensile load is ap-
plied. Both the incipient cracks and their number density were measured from
small-angle x-ray scattering studies on samples that had tensile loads applied to
them. The characteristic sizes and number densities of the incipient cracks are
included in Table 2. Zhurkov showed that the incipient cracks formed at the very
early stages and the remaining life was associated with the coalescence of these
incipient cracks to ultimately cause failure.

LeGrand and co-workers (48) conducted positional small-angle x-ray scat-

tering studies on high impact polystyrene exposed to fatigue loading conditions.
Their results clearly indicated that ﬂaws of some type are nucleated and grow as a
result of increasing fatigue. They also suggested that the nucleation and growth of
ﬂaws appears to follow a simple exponential law. However, the results regarding

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215

Table 2. Characteristics of Incipient Cracks in Polymers

Microcrack

Diameter

longitudinal,

transverse,

Density

p/p

Polymer

, cm

− 3

calculated

measured

Polyethylene

× 10

—

Polypropylene

32–35

× 10

—

Poly(vinyl chloride)

—

× 10

—

Poly(vinyl butyral)

—

× 10

—

Poly(methyl

170

× 10

0.4

× 10

− 2

0.3

× 10

− 2

methacrylate)

Polycaproamide

9–25

× 10

1.8

× 10

− 2

2.0

× 10

− 2

Ref. 47.

how material properties or loading conditions affect the growth rates were clearly
inconclusive in this study.

Positron annihilation lifetime spectroscopy studies (49) polycarbonate sam-

ples subjected to fatigue. Positron annihilation lifetime spectroscopy was used to
probe structural changes in terms of the hole volume and number density of fa-
tigued samples. The ortho-positronium (o-Ps) pickoff annihilation lifetime

, as

well as intensity I

, were measured as a function of cyclic stress. It was found that

, the longest of the three lifetime components, increases with fatigue cycles. The

holes where the o-Ps can localize become larger upon fatigue aging. Commensu-
rate with increasing hole size was an indication of hole coalescence from the o-Ps
annihilation intensity. It was suggested that this process was the precursor to
crazing. Complimentary transmission electron microscopy indicated that isolated
voids of the order of 100 nm could be seen (50).

Tensile dilatometry (51–54), which is commonly used to monitor cavitation

in rubber-modiﬁed glassy polymers, was extended to fatigue application to moni-
tor the volume changes that occur during fatigue loading conditions (55). In this
method, axial and transverse strain gages are attached to the sample during a
uniaxial fatigue test. From the axial and transverse strain measurements the
total volume change (

)

can be calculated from

= (1 + ε

)(1

+ ε

)

− 1

(14)

where

is the axial strain and

is the transverse strain. The volume change can

then be further decomposed into an elastic part and an irreversible part denoted
by the subscripts e and i respectively in equation 15 below:

(15)

The total volume change can be measured at the peak load during the fa-
tigue test and the irreversible part can be measured at the unloaded portion of
the cycle. In this way the evolution of both the elastic and inelastic volume change
can be monitored during a fatigue test. Figure 13 shows how the irreversible

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Fig. 13.

Irreversible volume change evolution for polyacetal samples fatigued over a range

of stress levels (55).

σ = 56 MPa, N

= 10

cycles;

σ = 54 MPa, N

= 3 × 10

cycles;

σ = 48 MPa, N

= 3 × 10

cycles;

σ = 45 MPa, N

= 10

cycles. To convert MPa to psi,

multiply by 145.

volume change evolves as a polyacetal specimen is fatigued over a range of stress
levels. The results show that the irreversible volume increased as the sample
was fatigued over a broad range of stress levels. Further, the Poisson’s ratio (see
Fig. 14) increases in the low cycle regime but decreases in the high cycle regime.
The decrease in Poisson’s ratio is characteristic of an increase in porosity in the
material.

Recent studies on PMMA have shown that craze initiation appears at a rel-

atively early stage in the fatigue life of the material (12). The total fatigue life
is compared to the onset of craze initiation in Figure 15. In this particular case
the lifetime is governed primarily by the growth of crazes until a defect (crack)
approaches the threshold size.

More direct morphological studies were conducted on isotactic polypropylene

(iPP) to isolate the mechanisms of fatigue damage and their kinetics (56). The
damage imposed to samples fatigued in the high cycle regime occurs in the form
of crazes. These are shown typically in Figure 16 for the case of iPP subjected to
6

× 10

cycles.

Figure 17 shows a sequence of optical micrographs illustrating the evolution

of damage. During fatigue, small crazes initiate and grow homogenously through
the material. However, as the fatigue test continues, small crazes coalesce and
the arrangement of crazes becomes neither even nor randomly distributed. This
becomes especially evident in the samples with a high level of fatigue damage.
Clear patterns arise in the form of cascades of crazes (see high magniﬁcation in
Fig. 17) that are clear evidence of craze–craze interactions where the stress in-
tensity at one craze tip is shielded by the formation of another craze. Also, careful

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217

Fig. 14.

Poisson’s ratio evolution for polyacetal samples fatigued over a range of stress

levels (55).

σ = 56 MPa, N

= 10

cycles;

σ = 54 MPa, N

= 3 × 10

cycles;

σ = 48

MPa, N

= 3 × 10

cycles;

σ = 45 MPa, N

= 10

cycles. To convert MPa to psi, multiply

by 145.

inspection of mature crazes indicates that as the crazes mature, much smaller
crazes initiate and grow around each main craze in the form of a process zone,
much like that which has been observed in crack propagation in polymers. Ulti-
mately the sample becomes saturated with crazes that resemble a planar array of
damage that is evidently completely dominated by the craze–craze interactions.

50 s

8.3 min

1.4 h

14 h

Fatigue life (2 Hz)

Craze initiation

Stress amplitude

, MP

Fracture

Crack propagation

Number of cycles

PMMA

Fig. 15.

Plot comparing total fatigue life and time for craze initiation for PMMA (12). To

convert MPa to psi, multiply by 145.

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Fig. 16.

Scanning electron micrograph of fatigue-damaged iPP. The arrows denote the

direction of applied load (56).

Eventually, of the crazes approaches the threshold size and subcritical crack prop-
agation ensues until failure.

The fatigue damage described in Figure 16 has also been quantitatively de-

scribed and the energetics of craze growth has been evaluated (56). This analysis
involved ﬁrst approximating the stress intensity of a single craze by treating the
craze as a microcrack of equivalent size with the craze ﬁbrils replaced by a closure
stress (traction) across the microcrack face. If a single craze is considered in an
inﬁnite plate, the stress intensity factor K

tot

for the craze can be written as (56)

tot

= K

∞

+ K

= (σ

∞

− σ

)

√

πᐉ

(16)

where

∞

is the remotely applied stress,

is the closure traction produced by

the craze ﬁbrils, and

is the craze radius. By substituting an expression for the

effective stress across the craze face as

σ = σ

∞

−σ

, the stress intensity due to

a single craze takes the same form as that of a single crack. Hence, the potential
energy change of a body containing a single craze would be given by

= 2

ᐉ

ᐉ =

ᐉ

tot

ᐉ =

πσ

ᐉ

(17)

In equation 17,

denotes the potential energy change in the specimen, b is the

specimen thickness, G is the energy release rate, and E

is the storage modulus

of the undamaged iPP.

In accordance with a Grifﬁth-type criterion for the growth of a craze

− 2γ = 0 for the necessary condition for craze growth), the following expression

for the speciﬁc energy

γ to form a craze is given by

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219

Fig. 17.

Optical micrographs of fatigue damage in iPP tensile base, showing the progres-

sion of damage with increasing cycle number (56).

= 2γ

tot

πᐉ

= 2γ

(18)

Rearranging terms in equations 18 yields the following relationship:

ᐉ

(19)

Thus far, the relationships apply to an isolated craze embedded in a specimen. To
evaluate the energy release associated with an ensemble of crazes, the interaction
of the crazes must be considered. In Reference 56, a damage parameter, similar
in fashion to that introduced by Bristow (57), for a two-dimensional case was
considered:

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ρ =

1
A

= 1

ᐉ

2
i

≈ nᐉ

(20)

In equation 20, A is some representative area and n becomes the number density
of crazes of radius

. The representative area A is related to the representative

volume V by V

= bA. Figure 18a describes how the number density of crazes and

mean craze radii (from Fig. 17) change with fatigue life, while Figure 18b shows
how the damage density

ρ, as deﬁned by equation 20, evolves. Note from the

graphs in Figure 18 that the number density of crazes decreases with increasing

Fig. 18.

Graphs describing the evolution of damage between 10

and 10

cycles. (a) Craze

number density (circles and short dashes) and average craze radius (squares and long
dashes) vs cycle number. (b) Damage density (eq. 20) vs number of cycles (56).

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221

cycle number while the craze length continually increases. This indicates that
coalescence is a dominating process during the stage of the fatigue process. Also,
the damage density

ρ continually increases as the fatigue process continues.

For a dilute suspension of cracks (crazes), Bristow (57) showed that the ef-

fective modulus E is linearly affected by the damage density in accordance with
the following relationship:

= 1 − πρ = 1 − nπᐉ

(21)

In 1993, Kachanov (58) showed that the following nonlinear approximation
(eq. 22) actually remains highly accurate at high crack (craze) densities, provided
the crazes are homogeneously distributed.

+ πρ

+ nπᐉ

(22)

Kachanov showed that equation 22 holds at high crack (craze) densities since

the competing interactions of shielding and ampliﬁcation result in the following
expression. In Reference 56, an expression for the change in the potential energy
density (potential energy change per representative volume) associated with craze
growth can be derived by combining equations 16, 17, 18, 19, 20, 21, 22:

= n

ᐉ

= n

ᐉ

tot

ᐉ

1
2

πn

ᐉ

= γ (2nᐉ + πn

ᐉ

)

(23)

The latter expression in equations 23 describes the speciﬁc energy for craze growth
in terms of changes in the potential energy density during the fatigue test. Using
equations 23 along with measured information regarding the number density
of crazes and craze radii (Fig. 17), the speciﬁc energy for craze growth,

γ , was

evaluated for iPP. The results indicated that the energy required to grow a craze
is of the order of 13 J/m

and was generally constant over the decade of cycles

measured.

Fatigue Crack Propagation

Background and Overall Considerations.

Many aspects of FCP includ-

ing both experimental approach and theory evolve from concepts well established
in the area of fracture mechanics. Many treatments dealing with both the theory
(59–63) as well as associated treatments to polymeric materials (2,5,64,65) have
been published. The basic tenent of fracture mechanics is that the strength of
most real solids is governed by the presence of ﬂaws. Many theories have been

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developed mathematically to describe the stresses in the presence of a crack as
well as the energy released during crack propagation. Grifﬁth (59) introduced a
criterion based on an energy principle and argued that when the change in the
elastic potential of a brittle glass equals or exceeds the energy required to create
new crack surface area, the crack will propagate uncontrollably. It has been shown
(60–63) that the Grifﬁth criterion can be related to the elastic energy release rate
and stress intensity factor for an elastic body. Thus, two parallel criteria arose to
describe the critical conditions for the failure of a body containing a crack (defect).
These criteria are written as follows:

≥ G

≥ K

(24)

In equations 24, whenever the energy release rate G

becomes greater than or

equal to the critical energy release rate G

, the fracture (failure) occurs. Similarly,

when the stress intensity factor K

exceeds the fracture toughness K

, the fracture

occurs. It should be noted that the subscript I in equations 24 refer to mode I failure
(opening mode), and similar criteria can be written for modes II (shearing mode)
and III (tearing mode). The criteria in equations 24 are conceptually different but
are mathematically identical for a linear elastic body since the energy release rate
and stress intensity factor are related by

(25)

where E

= E under plane stress conditions and E

= E/(1 − ν

) under plane strain

conditions. Both the energy release rate and stress intensity factor are a function of
the specimen global geometry, applied stress level, and crack length. Consequently,
the stress intensity (and/or energy release rate) will increase even if the stress level
remains constant when the crack length increases. Thus, specimens with larger
ﬂaws have a lower nominal strength than specimens in which the ﬂaw sizes are
smaller. This type of approach is commonplace in many engineering applications
today and has proven successful to describe the fracture behavior of a wide range
of materials.

For a given stress level, if the ﬂaw (crack) size is large enough such that the

critical condition(s) expressed in equation(s) 24 is reached, then immediate failure
is expected. If however, the stress level or ﬂaw size is reduced, then immediate
failure is postponed but subcritical crack growth may occur. Subcritical crack
growth refers to the slow progressive growth of a crack prior to reaching a critical
condition. One of the most common processes to accelerate this process is to apply
a fatigue load to the body. Consequently, the associated study of subcritical crack
growth under fatigue load is referred to as fatigue crack propagation (FCP). Many
studies and texts have been devoted to measuring FCP kinetics of polymers and
an excellent summary is provided in Reference 2.

One of the earliest attempts to describe the fatigue behavior of materials in

general is referred to as the Paris Law (66–68). The Paris Law is an empirical
relationship that exists between the crack growth rate d

/dN and the change

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223

in stress intensity factor by

ᐉ

= A(K

)

(26)

In equation 26,

= K

Imax

− K

Imin

, and K

Imax

relates to the stress intensity

factor associated with the maximum stress level in a given cycle and K

Imin

is the

corresponding stress intensity factor associated with the minimum stress in a
given cycle. The coefﬁcients A and m in equation 26 are considered material con-
stants. Paris Law (eq. 26) and variations therein have been widely used to describe
the crack growth kinetics of a broad range of materials (eg, Ref. 1). However, the
application of equation 26 has shown to be applicable only over a certain range of
stress intensities (which relates to a certain range of crack sizes).

Many studies have veriﬁed the log–log linear relationship between d

/dN

and

K, as predicted by equation 26, but others have shown sigmoidally shaped

FCP plots. Crack growth rates almost always decrease to vanishingly low values
as

K approaches a limiting threshold value K

and increase markedly at

very high

K when K

Imax

approaches K

. The asymptote at low

K conditions

describes FCP rates that diminish rapidly. As such, a limiting stress intensity
factor range

is deﬁned and represents a service operating limit below which

fatigue damage does not obey the Paris Law.

Figure 19 illustrates a typical sigmoidal-shaped crack growth behavior as

illustrated by regions I, II, and III. In region I, a crack is below the threshold size
and thus does not follow the growth kinetics as described by equation 26. Thus,
region I is many times referred to as the nucleation stage. In region II, equation
26 applies. Region II is associated with subcritical crack growth and equation 26

Region I
nucleation
stage

10−

0.1

Region II
subcritical
crack
growth

Region III
rapid fracture

Fatigue crack growth rate, mm/cycle

Cyclic stress intensity factor

∆K, MPaⴢm

1/2

∆K

crit

∆K

Fig. 19.

Illustration of typical sigmoidal curve in d

/dN vs K plot of FCP data. Three

regions are identiﬁed: region I associated with damage nucleation; region II associated
with subcritical crack growth; and region III associated with unstable, rapid failure.

224

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1.0

0.8

0.6

0.4

0.2

K, MPa m

1/2

, mm/cycle

Fig. 20.

FCP rates vs

K for several engineering plastics and metal alloys (39). A, LDPE;

B, epoxy; C, PMMA; D, polysulfone; E, polystyrene; F, PVC; G, poly(phenylene oxide); H,
polycarbonate; I, nylon-6,6; J, HI-nylon-6,6; K, poly(vinylidene ﬂuoride); L, acetal resins;
M, 2219-T851 aluminum alloy; N, 300M steel alloy.

can be integrated to predict remaining life of a body containing a crack in this
region. Region III is associated with unstable, rapid fracture once the ﬂaw has
grown to a critical size.

Characteristic plots of the FCP data are shown in Figure 20 for several

engineering plastics and metal alloys (69). The samples A–L are polymeric ma-
terials (both amorphous and semicrystalline) while M and N are metal alloys.
Figure 20 demonstrates the universality of equation 26 to describe the subcritical
crack growth kinetics of a wide range of materials. The data also generally suggest
values for the threshold ﬂaw size and critical stress intensity factor for each mate-
rial. Notice that some semicrystalline materials show resistance to crack growth
approaching that of metals.

Several studies have determined the threshold stress intensity for engineer-

ing plastics, eg, polycarbonate (5). The d

/dN values are in excellent agreement

for growth rates

> 10

− 5

mm/cycle but differ signiﬁcantly for lower values. Addi-

tional threshold data have been reported for PMMA as a function of stress ratio R,
where R

= K

min

max

(70). Other studies have been conducted to characterize the

near threshold crack propagation behavior (12,71). Figure 21 (12) shows a

σ–N

curve for polyethersulfone, including data at the near threshold level. Similarly

Vol. 6

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225

−2

−3

−4

−5

−6

0.1

Polyethersulfone

atigue cr

k g

wth r

ate

, mm/cycle

Cyclic stress intensity factor

∆K, MPaⴢm

1/2

Fig. 21.

Fatigue crack growth data for polyethersulfone, including data in the near thresh-

old regime (12).

Table 3 (71) deﬁnes the threshold stress intensity and the range over which the
Paris equation is valid for ultrahigh molecular weight polyethylene (UHMWPE)
subjected to different processing conditions. From Table 3, it is clear that subtle
processing conditions can alter the threshold level and the onset of FCP.

Unusual crack growth behavior has been reported in some semicrystalline

polymers with FCP rates decreasing with increasing

K at intermediate levels

(72–76). This transient response has been related in low density polyethylene
(LDPE) to a change in the scale of the fracture micromechanism (72), whereas
others attribute this kinetic transient to localized crack-tip heating, reasoning that
such heating would reduce the yield strength and result in a large increase in the
depth of the crack-tip damage layer (74). Chudnovsky and Moet (75,76) developed
another theory to accommodate for this behavior changes. This theory, referred to
as the crack layer theory, is formulated within the framework of thermodynamics
of irreversible processes. The crack layer theory treats the crack and surrounding
damage as a single entity. The rate of crack propagation becomes a thermodynamic

Table 3. Fatigue Threshold Stress Intensity Factor and Paris Regime or UHMWPE
Subjected to Different Processing Conditions

GUR 415

(threshold), MPa

·m

Paris regime, MPa

·m

Compression molded

1.8

1.80–2.8

Compression molded

γ -air

1.2

1.20–1.8

Extruded 90

◦

orientation

1.7

1.70–2.3

Extruded 0

◦

orientation

1.3

1.30–1.9

Extruded 0

◦

γ -air

1.01

1.01–1.7

Extruded 0

◦

γ -peroxide

1.12

1.12–1.3

Ref. 71.

226

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ﬂux within the theory, with complimentary thermodynamic forces being described
through the energy release rate. This theory is capable of predicting the stick-slip
character observed in some growth rates.

In the crack layer theory (75,76) the movement of the crack and surround-

ing damage is decomposed into elementary movements: translation, rotation,
isotropic expansion, and distortion. In this way, the damage surrounding the tip
can evolve in a general sense. These elementary movements become thermody-
namic ﬂuxes. The reciprocal forces contain an active part (energy release rates
associated with each movement) A

, and a resistive part (an energy barrier) R

Within the context of classical irreversible thermodynamics, the entropy of the
system, S

, can be expressed in terms of a bilinear form of forces and ﬂuxes shown

in equation 27.

= I

( A

− γ R

)

(27)

In equation 27, (A

− γ R

) represent the thermodynamic forces for each elementary

movement and

are the corresponding reciprocal ﬂuxes. The active parts corre-

spond to the energy release associated with each movement. The resistive parts
constitute the materials resistance to these movements and contain the intrinsic
material property,

γ , the speciﬁc enthalpy of damage (eg, crazing). The ability of

this formalism to account for size and shape changes in the process zone during
crack propagation produces changes in the translational crack propagation rate.
Thus, non-Paris FCP can be modeled using this approach.

Effect of Test and Process Conditions.

Many studies have evaluated

the effects of test frequency on the FCP rates in polymers. Although hysteretic
heating explains a decrease in fatigue resistance with increasing cyclic frequency
in unnotched polymer test samples (eg, eq. 3, Fig. 6, Ref. 24), the fatigue resis-
tance of notched polymers is exceedingly complex. FCP rates can decrease, re-
main unchanged, or increase with increasing test frequency (77–79). Figure 22
illustrates the effect that cyclic frequency has on the FCP rate of poly(vinyl chlo-
ride) (PVC). In the case of PVC, the crack speed decreases with increasing fre-
quency, but the threshold stress intensity K

and the fracture toughness K

remain unchanged. Also notice from the similarity of slopes in the FCP data
in Figure 22 that the material coefﬁcient m as deﬁned in the Paris equation
(Eq. 26) is insensitive to test frequency. Similar studies (78,79) on polycarbonate
showed that cyclic frequency had virtually no effect on its FCP kinetics. In con-
trast, the FCP kinetics in nylon-6,6 over a range of frequencies shows that higher
crack speeds occur at higher frequencies. Consequently, no universal behavior is
evident.

As expected, test temperature will have a dramatic effect on the FCP kinetics

of polymers owing considerably to their temperature-dependent viscoelastic na-
ture. Indeed, studies of polystyrene and acrylonitrile–butadiene–styrene (ABS)
have shown that FCP rates for given

K level generally decrease with decreas-

ing test temperature (80). By contrast, a minimum FCP resistance was noted
in polycarbonate and polysulfone at intermediate test temperatures (81,82). A
complex test-temperature response was also noted in studies on the inﬂuence

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−5

−4

−3

−6

−5

1 Hz

10 Hz

100 Hz

dᐉ

, mm/cycle

dᐉ

, in./cycle

0.2

0.4

0.6 0.8 1

0.2

0.5

∆K, ksiⴢin.

1/2

∆K, MPaⴢm

1/2

Fig. 22.

Effect of cyclic frequency on the FCP rate in PVC (77–79).

of test temperature and absorbed water on FCP in nylon-6,6 (83). Of particular
interest was the possibility to normalize the data (83) through empirical use of
time–temperature superposition principles.

The effect of thermal history and process conditions can affect both the un-

notched fatigue behavior (43) through physical aging (44) or through changes in
the crystal morphology. Similarly the FCP behavior of semicrystalline polymers
can also be affected through differences in thermal history (84–88). Some studies
suggest that resistance to crack propagation increases with decreasing spherulite
size (84,85), while others claim spherulite size has no effect (86,87). A recent re-
port (88) showed that annealed specimens to have a lower resistance to crack
initiation and subsequent propagation. Although the same fracture mechanism,
in which the brittle crack gradually becomes more ductile, prevailed in both cases,
the annealed material proved to be weaker. This was attributed to differences in
the failure mechanisms at the root of the crack. The annealed sample showed a
voided and ﬁbrillated crack tip while the quenched sample showed a nonﬁbrillated
structure.

Effect of Molecular Architecture, Morphology, and Microstructure.

Striking effects of molecular weight on FCP rates in notched PMMA (89) and PVC
(90) have been noted (91,92), and similar strong effects have been reported for

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× 10

log d

ᐉ/d

(

Fig. 23.

Functional relationship between FCP rates in PMMA and PVC and reciprocal of

molecular weight at 10 Hz for

K = 0.6 MPa·m

(89). Data for PVC displaced to the right

by a factor of 2.

polycarbonate (93) and polystyrene (94). In general, the Paris law is followed over
a wide range of

K. It was found (94,95) that this data followed the following

empirical relationship:

ᐉ

= A

(

(28)

where A

, B

, and n are constants depending on the material and test conditions.

Thus, even though values of static toughness may change relatively little over a
given range of molecular weight, small increases in the molecular weight M

may

result in up to several orders of magnitude decrease in crack growth rate. Also, the
effect is greater at lower molecular weights. This is shown for the case of PMMA
and PVC in Figure 23 (89).

It has also been found (92) that the addition of small amounts of high or

medium molecular weight to PMMA results in greater resistance to both fatigue
and static loading, whereas the addition of low molecular weight has a deleteri-
ous effect. Since the FCP response can be correlated semiquantitatively with the
proportion of high M

species estimated by gel-permeation chromatography (96),

it has been suggested that resistance to cyclic disentanglement is favored by high
M

species. While the short time scale of static fracture tests permits relatively

little disentanglement, the longer time scale involved in fatigue allows the fracture
process to reﬂect more strongly the energy dissipation concerned in breakdown
of the entanglement network (2). In this sense, the fatigue process resembles the

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229

longtime relaxation processes in time-to-failure tests (97,98) in which strong ef-
fects of molecular weight are also seen. Hence it is likely that during cycling,
the entanglements in craze ﬁbrils involving low M

species are progressively

broken down, but the longer molecules permit considerable strain hardening to
occur in the ﬁbrils (2,4,99). This conclusion is supported by fractographic evi-
dence obtained for specimens of PMMA containing high and low molecular weights
(100).

The FCP resistance in semicrystalline polymers is typically higher at higher

molecular weight. Examples include high density and ultrahigh molecular weight
polyethylene (101–103), acetal resin (104,105), nylon-6,6 (104), and polypropylene
(103). The effect of molecular weight on the FCP rates for these polymers is shown
in a comparative plot in Figure 24. In general, however, the sensitivity of the crack

−6

−5

−4

−3

−2

dᐉ

, mm/cycle

0.4

0.6 0.8 1

5 6

∆K, MPaⴢm

1/2

Fig. 24.

Effect of molecular weight on fatigue crack growth rates in high density polyethy-

lene (solid line): M

= 45,000 (A), 70,000 (B), 200,000 (C) (101,102); nylon-6,6 (dashed line):

= 17,000 (D), 34,000 (E) (104); and acetal resins (dotted-dashed line): M

= 30,000 (F),

40,000 (G), 70,000 (H) (105).

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growth rate to molecular weight is less than that in amorphous polymers. This
may be due to the effect of the crystal structure stabilizing the polymer through
tie chains and the like.

Some attempts have been made to develop quantitative theories to rational-

ize the effects of molecular weight and have met with some success, at least with
PMMA, PVC, and polystyrene (94,95,106). Thus it has been proposed that the
FCP rate can be expressed as the product of two functions, one involving

K and

the other characteristic of the relaxation process in the plastic zone (95):

ᐉ

= f (K) ×

(29)

where f (

K) reﬂects the effect of applied stress and τ

, the relaxation time, re-

ﬂects the resistance to chain disentanglement. Increased FCP resistance has been
correlated with increased characteristic stress-relaxation times in PVC (107) and
a rubber–polystyrene interpenetrating polymer network (108). Considering that
the relaxation time follows a Zhurkov-type rate process (33), the following may be
expected:

= A

exp

−Vσ

(30)

where A

is a constant and V is the activation volume. Assuming that the stress

involved is a localized average mean stress in the plastic zone, the following equa-
tion can be derived:

ᐉ

= f (K) exp

∞

(M)

(31)

In equation 31,

= σ

+ R)/2, where σ

is the yield stress and R is the

load ratio. V

∞

is the activation volume for inﬁnite molecular weight M, and

ψ(M)

reﬂects the fraction of molecules that can form a mechanically effective entangle-
ment network. In terms of molecular weight,

ψ(M) is deﬁned as ψ(M) = (V

∞

/V)

− W∗)/(1 − W∗), where W is the weight fraction of molecules when M > M

; M

is the minimum value of M required for a stable network; and W

∗ is the minimum

weight fraction of those molecules required to form a stable craze. The value of
(1

− W∗) is consistent with the void volume of mature crazes in PMMA and PVC

(95).

The model applies to PMMA and PVC and gives values of V that agree fairly

well with values for static fracture. Thus a combination of rate process theory
with fracture mechanics can usefully relate molecular and continuum ideas in
rationalizing fatigue fracture. Vertical shifting of FCP curves gives master curves
relative to a very high molecular weight specimen (

ψ ∼

= 1) as long as K

independent of M, ie, when M is greater than some limiting value. This is shown
in Figure 25 for the case of polystyrene (94). The shift factor a

is related to V and

ψ(M):

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231

−7

−6

−5

−4

−3

−2

log d

ᐉ/d

, mm/cycle

0.1

Log

∆K ∗

Fig. 25.

Master curve for FCP in polystyrene (10 Hz, sine wave) as a function of stress

intensity factor (94). Curve includes data for 12 specimens varying greatly in molecular
weight and its distribution, and encompasses both the threshold of crack growth and the
approach to catastrophic fracture.

exp(V

)

exp(V

∞

)

(32)

An alternative, simpler approach has been proposed in which crack growth

is essentially considered the reverse of crack healing (105). In effect, disentangle-
ment is taken to be the reverse of interdiffusion of molecules at the surfaces of two
planes in close contact. Thus, assuming scaling laws for characterizing molecular
reptation hold, it has been suggested that d

/dN is proportional to the quotient of

the distance of interpenetration x to t

∞

, the time required for complete molecular

diffusion and interpenetration. Since x

∝ M

and t

∞

∝ M

, then

ᐉ

∝ M

− 2.5

(33)

Both equations 32 and 33 give a good agreement when applied to data in
Reference 105.

A few controlled studies on the effects that cross-linking has on the FCP resis-

tance have been done. When irradiation was used to increase M

in polystyrene,

keeping M

constant, the fatigue life was affected very little by the total dose

applied. In fact, the FCP behavior did not alter signiﬁcantly even up to the gel
point and beyond. The FCP resistance of cross-linked polystyrene is lower than

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0.7:1

1.1:1

1.4:1

1.2:1

0.8:1

1.0:1

1.6:1

1.8:1

2.0:1

10−

dᐉ

, mm/cycle

0.4

0.6

0.8

1.0

∆K, MPaⴢm

1/2

Fig. 26.

FCP behavior in Epon 828/MDA epoxy as a function of amine/epoxy ratio (num-

bers labeling curves) at 10 Hz (113). Dotted line represents other data (110).

that of polystyrene or rubber-modiﬁed polystyrene (109). A typical epoxy resin is
much less resistant to FCP than thermoplastic polymers (2,110). This observa-
tion is generally consistent with the ability of cross-linking to inhibit segmental
mobility, crazing, and consequent energy dissipation.

Most systematic studies of cross-linked polymers conﬁrm the above hypoth-

esis, but one study reveals an interesting exception. As the percentage of a mul-
tifunctional acrylate cross-linking agent in PMMA increased from 0 to 11, the
FCP rates increased by over 2 orders of magnitude (at

K = 0.5 MPa·m

); the

slopes of the d

/dN curves also increased (111). Similar deleterious effects of cross-

linking are described in a study of static and fatigue performance in bisphenol
A epoxies cured with methylenedianiline (112). These results are presented in
Figure 26. Note that as the amine/epoxy ratio decreased from 2:1 to 1:1 (i.e., from
a nearly linear to a densely cross-linked resin) K

decreased and FCP rates in-

creased by over an order of magnitude at a given

K. While an excess of amine

increased FCP resistance, an excess of epoxy had the opposite effect. This differ-
ence can be explained by considering that the epoxy is difunctional and the curing
agent is tetrafunctional. When an excess of di-epoxy exists, the resulting network
has an effective increase in molecular weight between cross-links, approaching
that of a rubbery network. However, when an excess of curing agent exists, the

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233

molecular weight between cross-links is not necessarily increased and many ter-
minal uncross-linked ends are produced. Thus, the results reﬂect this difference
in network architecture. Note, however, that it would be interesting to compare fa-
tigue results at similar temperatures relative to their glass transition rather than
at one temperature since the glass transition is inversely related to the molecular
weight between cross-links.

Fractography.

The fatigue-fracture-surface micromorphology of engineer-

ing plastics is complex. In general, the surface may reveal a range of roughness,
and this roughness is usually associated with either a change in stress intensity
factor or stress states. In the interior of a specimen, the stress state is more conﬁn-
ing than near the edges, and careful inspection of fracture surfaces usually reveals
this effect. For example, the surface shown in Figure 7 (25) illustrates greater tex-
ture near the initiation site where the nucleation and growth process below the
threshold level occurs. Once subcritical crack growth is established, the fracture
surface produces a smoother and uniform texture. At the critical size, rapid frac-
ture occurs and the crack speed approaches the stress wave speed in the material.
A clear distinction appears at the critical radius in Figure 7, denoting this sudden
change in crack speed. Note in this case, that the surface beyond the critical size
is notably rougher than the other part of the surface (see also F

RACTOGRAPHY

Figure 27 illustrates a fracture surface on another semicrystalline poly-

mer, an aliphatic polyketone terpolymer (114). When comparing Figure 7 with
Figure 27, some generic features are noticed, but there are also some contrasting
differences. Much like that observed in Figure 7, Figure 27 contains a clear penny-
shaped crack whose center locates the nucleation site. However, in stark contrast
to that observed in Figure 7, Figure 27 shows a distinctly more textured fracture
surface up to the critical radius (see A in Fig. 27). Just at the regime where the
crack reached a critical size, the fracture surface becomes remarkably smoother,
followed by a more textured region, and ultimately a very textured region near
the edge (C in Fig. 27). The region near the edge (Fig. 27C) illustrates how the
boundary condition near the free edge affects the texture and indicated a tearing
of the last ligaments during the last stage of failure.

In many polymers, fatigue striations can be observed at relatively high

levels. These striations correspond to the successive location of the advancing
crack front after each loading cycle or local crack jump. There is a clear distinction
between macroscopically observed clamshell markings, which represent periods of

Fig. 27.

Scanning electron composite micrograph of fatigue-induced fracture surface in

aliphatic polyketone terpolymer: (A) initiation site and subcritical crack growth, (B) critical
radius, and (C) texture associated with remaining ligaments (114).

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growth during which hundreds or thousands of loading cycles may have occurred,
and fatigue striations, which represent the extent of crack-front advance during
one load excursion. Each fatigue clamshell marking may, therefore, represent the
extent of crack advance corresponding to thousands or tens of thousands of indi-
vidual load or strain excursions. The microscopic measurement of the FCP rate, ie,
striation width, is in excellent agreement with macroscopically determined crack
growth rates (115). This strong correlation between macroscopic and microscopic
growth rates in polymeric solids reﬂects the fact that 100% of the fracture surface
in this

K regime is striated—an observation not typically found in metal alloy

systems. In the latter instance, other fracture mechanisms such as cleavage, mi-
crovoid coalescence, and intergranular failure are observed in various locations
on the fracture surface.

It is common for signiﬁcant damage to evolve under the crack surface as

well. This occurs because the high stress in the vicinity of the crack tip promotes
the formation of damage. This damage consumes signiﬁcant energy and is largely
responsible for the high apparent toughness that many polymers exhibit. The
damage in the vicinity of the tip can take many different forms and is largely
associated with the polymer. Common features include crazing in most semicrys-
talline polymers and some glassy polymers (eg, polystyrene). Glassy polymers
with higher cohesive energy densities promote shear banding. Stress state will
also play a role in this process. An excellent treatise describing these morpholog-
ical features can be found in Reference 65. The damage that occurred below the
surface shown in Figure 27 is illustrated in Figure 28 (114). The damage in this
polymer occurs as an array of crazes, with the highest density near the surface
of the main craze. As the distance from the fracture surface is increased, the size
and density of crazes reduce. Earlier stages of this same process are illustrated
for iPP in close-up micrograph in Figure 17 (56).

Fig. 28.

Scanning electron micrograph of damage occurring below crack surface in an

aliphatic polyketone polymer (Ref. 114). Note that the highest density of crazes occurs
near the crack surface.

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235

Fig. 29.

Scanning electron micrographs of crack tip in plane strain (above) and plane

stress (below) conditions for an aliphatic polyketone terpolymer (114).

The damage mechanisms produced in unnotched fatigue specimens can be

compared to those where a controlled defect (razor sharp notch) is introduced into
fatigue specimens. Figure 29 (13) illustrates the damage that surrounds the crack
tip both in the interior of the specimen (plane strain condition) and at the outer
surface (plane stress condition). In the plane strain condition, a high concentra-
tion of collinear crazes is apparent and is consistent with the damage produced at
a natural defect (Fig. 28). Note that in both the plane strain and plane stress con-
ditions, the damage surrounds the crack. This damage is referred to as a process
zone since it is closely associated with the fracture process. In the case of plane
stress (Fig. 29), the damage is inclined with respect to the fracture surface. These
inclined features are shear bands and occur in the constraint-free regime near the
surface.

Other factors can affect the type of fatigue damage, thereby changing the

FCP and fracture toughness of the material. This is illustrated for the case of

β-

phase nucleated iPP (25) (see Fig. 30). When comparing the differences between
beta phase (Fig. 30) and

α-phase (Fig. 6), a clear difference in the fatigue damage is

noticed. In the

α-nucleated polymer (Fig. 6), highly ﬁbrillated crazes evolve orthog-

onal to the loading condition and are not altered by the spherulitic morphology.
In contrast, in the

β-nucleated polymer (Fig. 30), the crazes show no ﬁbrillation

and appear to follow the lamellar direction of the spherulite.

The state of stress can dramatically affect the failure mode and strength

of polymers. This effect alters the fracture surface characteristics when differ-
ent modes of fracture are applied. Figure 31 shows comparative micrographs of

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Fig. 30.

Scanning electron micrograph of

β-nucleated iPP subjected to 10

cycles (25).

fracture surfaces in carbon ﬁber epoxy resin composites tested in mode I (opening
mode) and mode II (shear mode) loading conditions. Notice that the mode I fracture
surface appears relatively smooth while the mode II surface has a characteristic
hackled surface. Also, the fracture energy for the mode II sample is greater than
that of the mode I sample.

Other Factors Affecting Total Fatigue Life

Effects of Chemistry in Homogeneous Polymers.

Without knowing

the relative importance of initiation and propagation stages of fatigue damage
accumulation, it is many times difﬁcult to identify the role of material character-
istics such as molecular weight and impact modiﬁer content on these two stages
of fatigue damage process. Some preliminary experiments have been reported to
identify the extent of the fatigue crack initiation process as inﬂuenced by molec-
ular weight, impact modiﬁer level (116), and stress level (6,117). Figure 32 shows
a three-dimensional plot illustrating the interactive effects of M

and rubber con-

tent (%) on fatigue crack initiation (FCI) lifetime (N

) in impact-modiﬁed PVC. The

rubber phase is methacrylate–butadiene–styrene (MBS) polymer. N

is deﬁned as

the number of loading cycles necessary to nucleate a crack 0.25 in length from a
semicircular surface notch with a radius of 1.59 mm. FCI lifetimes increase with
increasing molecular weight for all rubbery phase contents. Conversely, in high
M

PVC blends, rubber modiﬁcation lowers FCI resistance although the addition

of MBS in low M

PVC slightly improves FCI resistance. The addition of a rub-

bery phase to epoxy resin and nylon-6,6 also reduces FCI lifetimes and hastens
thermal fatigue, respectively.

In addition to the FCP behavior, the molecular weight and distribution pro-

foundly affect both thermal and mechanical properties of polymers (118–120).
Thus tensile strength of many polymers is very low or negligible when the aver-
age M is low (

<10

to 10

), but increases rapidly above M

, a characteristic value

of molecular weight, and then levels off (119–124). When M

> M

, the relationship

between strength and molecular weight may often be expressed by an empirical
equation of the form

Vol. 6

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237

Fig. 31.

Scanning electron micrographs of unidirectional carbon ﬁber epoxy resin com-

posites tested in modes I and II test conditions (25).

= A− B/M

(34)

where

is tensile strength, A and B are constants, and M

is an average molec-

ular weight. Often the number-average molecular weight M

, which strongly re-

ﬂects the presence of chain ends as defects, is appropriate (121,122). Other phys-
ical properties vary in a similar fashion (125–128); a sigmoidal shape over the
whole range of M may sometimes be discerned on a log–log plot. The initiation
and propagation of fatigue damage also depends strongly on molecular weight.
Indeed both static and fatigue fracture are controlled by similar viscoelastic pro-
cesses, although the balance of processes and intensity of expression in failure
rates differ signiﬁcantly (2). Energies and volumes of activation tend to be simi-
lar for molecular ﬂow and fatigue processes (108,109), and the fatigue resistance
of many polymers correlates well with fracture toughness (2). Thus an earlier
proposal (124) that the nature of entanglement networks controls M

and frac-

ture generally has been conﬁrmed and ampliﬁed. Much evidence suggests that
only molecules longer than the average length spanning a craze can form a duc-
tile network that can effectively resist crack propagation (113,125,126,129–132).
Equation 34 has been derived on this basis, with A

= σ

for inﬁnite M (132), and

the role of l

, the contour length between entanglements, has been shown to control

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Notch stress

= 50 MPa

Initiation lif

−

cycles

× 10

−5

Phr MBS

Fig. 32.

Three-dimensional plot revealing interactive effects of M

and rubber content

(MBS) % on FCI lifetime in PVC (116).

the maximum elongation possible for a polymer (133,134). In addition, high and
low values of the ratio of l

to the rms end-to-end distance between entanglements

tend to be associated with crazing and shear yielding, respectively. Experimental
evidence clearly shows that dense, stable crazes can develop only when M

> M

(113,135). Although molecular weight may or may not affect the crazing stress
(135–137), a direct correlation exists between higher M values and greater ability
to resist propagation of a crack through a craze.

The fatigue resistance depends strongly, and often dramatically, on molecu-

lar weight and its distribution (2,4). As M

increases from 1.6

× 10

to 2

× 10

three polystyrenes with narrow molecular weight distributions, the S–N (stress–
fatigue life) curves (obtained under conditions yielding negligible hysteretic heat-
ing) shift upwards and to the right, and the fatigue endurance limit increases
from

∼1/14 to 1/2 of the breaking stress (4,138) (see Fig. 33). Similar improve-

ments in fatigue life with increasing molecular weight have also been noted in
pulsed compression tests on polystyrene (104). Tensile strength increases by 25%
over the range of molecular weight, whereas fatigue strength increases by 100%
(4,138–140). Equation 34 holds for tensile strength, but a different dependence is
seen for fatigue strength; the ratio of the former to the latter varies from 0.1 to 0.3.
The

σ –N curve for a polydisperse polystyrene (M

= 2.7 × 10

) was not parallel

to the others in Figure 33, and yielded an endurance limit higher than predicted
on the basis of the other specimens (4,138,140). This ﬁnding is similar to that in
which the addition of only 1% of high molecular weight species (M

= 7 × 10

)

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239

Fatigue life,

Stress

, MP

Fig. 33.

Effect of molecular weight on fatigue life in polystyrene. M

= 1.6 × 10

(A), 3.3

× 10

(B), 8.6

× 10

(C), 2.0

× 10

(D). Specimen B has a broader distribution of molecular

weight than the others (4). To convert MPa to psi, multiply by 145.

was able to induce craze stability in a brittle, low molecular weight polystyrene
(M

= 10

) (113). These results have been attributed to increased entanglement

density, and hence, orientation hardening of crazed ﬁbrils, and decreased concen-
tration of defects in the form of chain ends in long molecules (4,122). At the same
time, increases in fatigue life are not noted when molecular weight is increased
by irradiation, so that polydispersity is essentially constant (140). Under such
conditions, even below the gel point, the beneﬁts of increased molecular weight
are clearly overwhelmed by the ﬂaws introduced by chain ends.

Effect of Diluents and Plasticizers.

Diluents, plasticizers (qv), resid-

ual solvents or monomers, or absorbed environmental agents greatly affect the
viscoelastic response of a polymer and related properties such as damping, modu-
lus, glass-transition temperature T

, and yield strength. This resultant changes in

properties can therefore affect fatigue behavior whether the diluents are dispersed
uniformly in the bulk or only at the surface. The effects of lower stiffness differ
depending on whether testing is conducted under constant stress or constant de-
ﬂection conditions. In the ﬁrst case, fatigue behavior may deteriorate because
of the increase in strain experienced by more compliant specimen. In the latter
case, the more compliant specimens (which experience lower stresses) may exhibit
superior performance. The increase in compliance may also increase the fatigue
life of a specimen by plasticizing the crack tip and blunting an otherwise brittle
failure. Conversely, it may reduce the fatigue life by reducing the strength of the
material in the presence of the crack or craze tip.

The

σ–N behavior in tension–compression at 7 Hz of plasticized PVC con-

taining 10 phr of dioctyl adipate exhibits several features characteristic of other
polymers (141). Under conditions of high strain amplitude, high ambient temper-
ature, and specimen cooling by natural convection, thermal failures are observed:

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a steady increase in damping (tan

δ) and a decrease in modulus was observed as

failure was approached. In contrast, brittle failure along with crack temperatures,
and an increase in storage modulus and a decrease in tan

δ prior to failure. How-

ever, when thermal failure conditions are coupled with forced-air cooling, brittle
fracture is also typical. Also, the criterion (142) for thermal failure is valid (141).

In PVC plasticized with dioctyl phthalate (DOP), FCP rates reportedly in-

creased as the DOP concentration increased from 0 to 10–20% (143), suggesting
that lower yield stress and modulus, and resultant tendency toward creep, may
play an important role. Quite different results were obtained in a later study at
a higher frequency, which may have hindered such creep (90). In this case, FCP
rates were relatively unchanged by the presence of DOP, although K

values were

somewhat reduced, especially at the lowest concentration, 6%. Evidently the an-
tiplasticizing embrittling effect of DOP at low concentrations was manifested more
in the static than in the fatigue response. It is believed that PVC can form an ef-
fective entanglement network and stable crazes in the presence of much DOP, and
the long time scale loading characteristic of fatigue may permit extensive cyclic
disentanglement.

With internally plasticized PMMA, i.e., a copolymer with butyl acrylate, still

other effects have been seen (89). As the concentration of butyl acrylate increases
to mole fractions of 0.1 and then 0.2, FCP rates increase dramatically and K

values decrease, presumably because increased ductility is associated with lower
strength and modulus. However, with a mole fraction of 0.3 butyl acrylate, local-
ized crack-tip heating and blunting develop from an increase in damping, and FCP
rates decrease, so that the overall behavior, with respect to FCP only, resembles
that of a commercial PMMA. Hence it appears that a plasticizing comonomer in
a chain signiﬁcantly facilitates the detailed local viscoelastic response and weak-
ens craze ﬁbrils, whereas a semimiscible external plasticizer may not necessarily
inhibit the ability to form strong craze ﬁbrils.

The plasticizing inﬂuence of absorbed water on the FCP response of

polyamides has also been studied (144–150). Earlier data were difﬁcult to in-
terpret because various experiments employed different test frequencies, mean
stress levels, amounts of imbibed water (0–8.5%), and different specimen conﬁgu-
rations. According to the principles of linear elastic fracture mechanics, specimen
conﬁguration should have no bearing on material fatigue response, since the prop-
agation of fatigue crack is assumed to be controlled entirely by the stress ﬁeld near
the crack tip and therefore to be a function of

K only. However, for the case of

engineering plastics that exhibit signiﬁcant viscoelastic heating at the prevailing
test conditions, variations in specimen conﬁguration have been shown to alter
FCP resistance as a result of changes in overall stress range (

σ) acting on the

specimen (150,151). For example, crack growth rates determined with the center-
cracked (CCT) specimens exceed those using wedge-opening load (WOL) samples
(151). In addition, greater hysteretic heating occurs in the CCT samples when
compared with those measured in WOL samples. These results are readily under-
stood in terms of the expression for the stress intensity factor range, which takes
into account the different geometries.

K = Y(σ )

√

ᐉ

(35)

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FATIGUE

241

In equation 35, the function Y accounts for the different specimen geometries
(64). Since Y is much smaller for the CCT specimen than for the WOL specimen
(64), it follows that

CCT

is much larger than

WOL

; the latter difference in far-

ﬁeld stress distribution alters the hysteretic heat generation capability and the
relative stiffness of these samples. Simply stated, the far-ﬁeld stresses must be in-
creased in the CCT specimen to achieve the same change in stress intensity as the
WOL specimen. These higher far-ﬁeld stresses, in turn, produce more hysteretic
heating.

Nylon–water interactions and their associated inﬂuence on mechanical re-

sponse are of considerable engineering importance since the absorption of mois-
ture occurs readily when the nylon component is in service. Roughly 2.5 wt% wa-
ter is imbibed after equilibration at 50% rh. Depending on the property and the
amount of water absorbed, effects may be beneﬁcial or deleterious. For example,
RT modulus increases marginally with the addition of 2.2% water in nylon-6,6 but
is signiﬁcantly reduced with 8.5% water (152). Such differences in elastic modulus
have been traced to the manner by which the water molecules are incorporated
in the nylon. At low absorbed moisture, the polar H

O molecules interrupt the

hydrogen bonds between amide groups in neighboring chain segments and create
water bridges between these groups. This tightly bound state is believed to per-
sist until all of the hydrogen atoms bond in the amorphous fraction group ratio
of 1/2 (153,154). At higher concentrations, weakly bound water is imbibed and
essentially serves as a diluent.

Fatigue tests on unnotched samples have demonstrated that the fatigue

strength (the stress corresponding to failure at a given number of cycles) is reduced
by as much as 30% when nylon-6,6 is equilibrated at 50% rh (152) (see Fig. 34). In
marked contrast, it has been found that fatigue growth rates in nylon-6,6 exhibit
a pronounced minimum at an absorbed moisture content of 2.6 wt% water, and

100

200

300

400

500

1000

2000

3000

4000

5000

6000

Dry, as molded

50% rh

Cycles to failure,

Stress

, 1b(wt)/in.

Stress

, kg(wt)/cm

Fig. 34.

Unnotched fatigue life in nylon-6,6 as a molded and equilibrated at 50% rh (155).

The test specimen is 0.775 cm thick.

242

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crack growth rates are higher in saturated material (8.5 wt% water) than in the
dry polymer (149,156). The decrease in crack growth rates with increasing water
content at low moisture can be attributed to enhanced chain mobility and an at-
tendant increase in crack-tip blunting associated with localized crack-tip heating.
In addition, fatigue resistance is believed to increase with the slight increase in
elastic modulus that accompanies the addition of up to 2.5 wt% water. Conversely,
the marked increase in crack growth rates at moisture

>2.5 wt% is believed to

result from the pronounced reduction of elastic modulus which accompanies high
amounts of loosely bound absorbed water (152,154).

The complex inﬂuence of absorbed water on FCP resistance in nylon-6,6 and

in nylon-6 (149) reﬂects a competition between the beneﬁcial inﬂuence of crack-
tip blunting and the deleterious effect of modulus degradation. This competitive
interaction is reﬂected in the fatigue-mechanism transition discussed below. The
blunting mechanism dominates the fatigue response at low moisture levels and
the decrease in modulus overshadows crack blunting at higher water contents. The
same deformation processes occur in fatigue tests on unnotched samples; however,
overall global stress levels in these samples are much larger than for the case of
precracked specimens. As a result, the greater amount of hysteretic heating in
nylon-6,6 samples equilibrated at 50% rh reduces specimen stiffness resulting in
more damage per cycle than that experienced inferior fatigue properties in the
σ–N test.

The inﬂuence of absorbed water on FCP resistance is expected to be more

signiﬁcant for high impact nylon-6,6 (HI-N66) since this blend exhibits greater
amounts of hysteretic heating than neat nylon-6,6 (157). In general, the effect of
water content on relative ranking in FCP resistance for various nylon-6,6 blends
is related to specimen temperature. At low water contents and low values of

heating is beneﬁcial, and blends with more rubber show the best FCP resistance.
For these conditions, heating is localized and contributes to crack-tip blunting,
which lowers the effective

K. With increasing water content, and/or K, greater

specimen heating occurs which is detrimental to fatigue resistance. Since rubbery
additions and extensive hysteretic heating bring about a substantial reduction in
E

, the principal cause for reduced FCP resistance in the rich blends containing

water, it is interesting to note that when

K is normalized with respect to the

elastic modulus, the richer blends exhibit greater resistance to cyclic strain (157).
In general, variations in FCP behavior of HI-N66 depend strongly on changes
in the dynamic storage and loss moduli resulting from hysteretic heating. Heat-
induced changes in modulus are more important than the absolute temperature
increase of the sample.

The absorption of water by PMMA also inﬂuences fatigue resistance in this

amorphous polymer (158). There is signiﬁcant reduction in fatigue lifetime with
up to 1% water content for extruded PMMA (M

= 79,300) tested at 27.6 MPa and

2 Hz (see Fig. 35). This is related to plasticization and reduction in resistance to
craze initiation and subsequent breakdown. At higher water content, no further
deterioration takes place even though the water molecules tend to cluster and
aggregate.

Surface Finish and Modiﬁcation.

Previous examples involve essentially

homogeneous polymer–diluent systems. Interesting effects with practical conse-
quences also occur when unnotched samples are exposed to various environmental

Vol. 6

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243

0.5

1.0

1.5

Water content, %

atigue lif

, cycles

2.0

2.5

3.0

Fig. 35.

Fatigue life vs water content in extruded PMMA (M

= 79,300) tested at 27.6

MPa and 2 Hz (158).

media (4,159). Because fatigue failure in many cases usually involves initiation
at the surface, such effects are to be expected.

Deleterious effects of atmospheric exposure on mechanical properties are

well known. Surface oxidation in rubbers results in embrittlement, surface crack-
ing, enhanced fatigue propagation, and reduced fatigue life. Even in a ductile
plastic, the formation of brittle surface layers and cracks invariably facilitates the
onset of crack propagation under static (51,160) or fatigue loading. For example,
when exposed to natural weathering (qv), the fatigue strengths of polystyrene,
ABS and acetal resin decrease both in an absolute sense and in a relative sense
with respect to tensile and blending strength.

The role of surface oxidation in the fatigue of polystyrene has been clearly

demonstrated (159). While control samples exhibit an average fatigue life of 25,000
cycles in air, lifetimes increase

∼90% by coating with a ductile metal (aluminum

or gold, 200 nm thick), and

∼100% by testing in nitrogen. Coating with ﬂexi-

ble, rubber polymers or with a low molecular weight oligomer of polystyrene also
increases fatigue life in polystyrene (159,161). The relative effect is greater for
unpolished specimens; interestingly, the coatings themselves are more effective
than polishing. It seems likely that these coatings reduce stress concentrations
at surface ﬂaws. Even though the natural rubbers used are susceptible to oxida-
tion, the polishing effect is dominant. On the other hand, coating high molecular
weight brittle polystyrene layer results in a decrease in fatigue life (4); the propen-
sity for facile initiation of craze cracks evidently overcomes any protective effect.
Of course, oxidation is favored by stress.

Although liquid environments often induce environmental-stress crazing

or crazing in glassy plastics, not all liquids are deleterious to fatigue perfor-
mance, and some are beneﬁcial. Typical liquids, eg, heptanes, esters, and alcohols,

244

FATIGUE

Vol. 6

decrease the fatigue life of polystyrene (162); the decrease is greater when sol-
ubility parameter is closer to that of polystyrene. The increased solubility and
plasticization lowers the crazing stress and hence facilitates crack development.
In contrast, with liquids known to increase crazing stresses in polystyrene, eg,
water, ethylene glycol, or glycerol, fatigue lifetimes increase

∼3×, 4×, and 9×,

respectively. This effect is attributed to enhanced hydrogen bonding, to ﬁlling of
surface voids and ﬂaws, or to surface tension changes.

Increase in fatigue life for nylon-6,6 exposed to petroleum ether, water, and

alcohols has been reported (163), and the effects of various liquids on fatigue
of polycarbonate have been described (164). Various vapors can be beneﬁcial to
fatigue life in polystyrene (4,44).

Thermal History.

This is an important parameter in determining vis-

coelastic and mechanical behavior (165). For example, in amorphous polymers,
quenching from temperature above T

increases the free volume and enthalpy, in-

creases yield stress, and decreases fracture toughness (see also A

GING

). With crys-

talline polymers, the morphology (qv) depends very much on the balance between
nucleation and growth corresponding to the rate and pattern of cooling from melt.

Although the fatigue behavior of semicrystalline polymers can be signiﬁ-

cantly altered by varying the thermal history, effects of such history on fatigue in
amorphous polymers are not well documented. Even though sub-T

annealing of

polycarbonate reduces the static fracture toughness, relatively little effect on FCP
response is noted (165). Similarly, only modest and difﬁcult-to-reproduce FCP ef-
fects of quenching and annealing on FCP behavior have been reported for PVC
(166,167). Although repetitive cycling can condition the state of a polymer, at least
under some conditions (168), the fact that yielding occurs within the plastic zone
implies the attainment of a state of mobility equivalent to a temperature high
enough to erase some or all of the effects of the applied history.

Semicrystalline Polymers.

As a class, semicrystalline polymers (qv) tend

to exhibit superior fatigue resistance, relative to that of amorphous polymers (qv),
both in terms of

σ –N behavior and FCP rates (2,4,169), as long as tests are con-

ducted at frequencies low enough to minimize excessive hysteretic heating. This
is so, even allowing for variations, due to molecular weight. Indeed even though
performance in a standard S–N test at 30 Hz may be relatively poor, many such
polymers behave very well under service conditions involving lower frequencies
and strains. Thus polypropylene is an excellent material for integral hinges.

Such inherently superior fatigue resistance is not fortuitous. Semicrystalline

polymers having adequate intercrystalline linkages can not only dissipate consid-
erable energy when the crystallites are deformed but also form a new and strong
crystalline morphology (170,171). Crystallization developed during straining of
amorphous polymers is responsible for the superior fatigue resistance of strain-
crystallizing rubbers (172,173).

Conﬁrmation of the general fatigue superiority of semicrystalline polymers

is implicit in studies of

σ –N behavior at frequencies low enough to minimize tem-

perature rises (169). While the ratios of the endurance limit to tensile strength for
typical amorphous polymers (polystyrene, PMMA, and cellulose acetate) are ap-
proximately 0.2, values for nylon-6,6, acetal resin, and polyteraﬂuoroethylene are
0.3, 0.5, and 0.5, respectively. At a low frequency, polyethylene (M

> 50,000) ex-

hibits no failure after 5

× 10

cycles, even at the relatively high alternating stress

Vol. 6

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245

of 21 MPa (300 psi) (169). Curves for polymers such as nylon-6,6, poly(vinylidene
ﬂuoride), and acetal resin appear to the right of the curve and data for nylon-12
also fall in this region.

Polymer Alloys, Blends, and Composites.

The fatigue behavior of

multicomponent polymers can be complex. On one hand, the addition of inho-
mogeneities produces stress concentrations that can reduce the fatigue life of a
structure dramatically. On the other hand, the heterogeneities may provide mech-
anisms of reinforcement, enhanced energy dissipation, or other features that may
dramatically improve the overall fatigue life of a material (see P

OLYMER

LENDS

With regard to particulate-ﬁlled or rubber-modiﬁed systems, the polymer

matrix must provide the load bearing capability. The geometry and elastic prop-
erties of the discrete phase, together with quality of the interface, can all affect
the stress concentration developed by the introduction of the particle. One can
consider that at nominally low cyclic stresses the matrix material in the vicinity
of the particles may be effectively cycling at much higher stresses. This, in turn,
may shift the apparent high cycle/low cycle fatigue transition (eg, see Figs. 4 and
5) to lower stress levels or cycles. Also, the initiation time for damage may be
effectively reduced by the introduction of the stress concentration. However, once
the damage initiates, these same mechanisms may reduce the subcritical crack
growth kinetics signiﬁcantly by dramatically increasing the size of the process
zone and increasing the energy dissipated within the process zone. For example,
cavitation of a rubber particle can change the failure mode of the matrix poly-
mer from a brittle mechanism (crazing or microcracking) to a ductile mechanism
(shear banding), thereby increasing the fracture toughness of the material by as
much as an order of magnitude (51).

The effect of a rubbery phase on the fatigue behavior is an example of such

complex behavior. As with an unmodiﬁed polymer, high stresses or frequencies
may generate enough hysteretic heating to induce failure by softening the matrix
while at lower stresses and/or frequencies a nominally brittle failure can occur.
This is shown for the case of ABS in Figure 36 (174). Although hysteretic heating
is common without the introduction of a second phase, the rubbery phase can
cause several speciﬁc effects. First, a signiﬁcant decrease in the yield stress and
modulus are typical along with an increase in the creep response. Moreover, the
reduction in ultimate elongation associated with cyclic softening may be greater
in rubber-modiﬁed polymers (175).

In another more recent study (176), FCP studies were conducted on rubber-

toughened polycarbonate. The results indicated that over all ranges of

K, the

crack growth rate was roughly a factor of 20 lower than that for unmodiﬁed poly-
carbonate. This was found over a range of mean stress levels. However, the thresh-
old stress intensity factor

dropped signiﬁcantly in the modiﬁed system as

compared to the unmodiﬁed system, indicating that the rubber-toughened system
may have a shorter initiation time since smaller defects may start to propagate
as cracks.

The behavior of polymer alloys and immiscible blends have gener-

ally the same complex behavior as rubber-modiﬁed systems. In the case of
polystyrene/polyethylene blends (177), the crack growth rates of the blends
were as much as 20 times lower than that of unmodiﬁed polystyrene. How-
ever, like the results in Reference 176, the results in Reference 177 show that

246

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Vol. 6

Number of cycles to failure,

Ductile

Brittle

Fatigue

Creep

Duration of tensile loading at fracture, s

Stress

± σ

, MP

Fig. 36.

Comparison of

σ –N curve under a square-wave loading with creep-rupture data

for a typical ABS polymer (175). Note that the creep-rupture time scale is equivalent to
that of the fatigue cycle scale in terms of time under tensile load. To convert MPa to psi,
multiply by 145.

the threshold stress intensity of the blend is lower than that of the modiﬁed
system.

In some cases, the heterogeneity may provide an effective reinforcement for

the matrix material (eg, chopped ﬁber composite). In this type of material, the
load is transferred to the ﬁber from the matrix through shear and the mode II
fracture strength of most polymers is much greater than the mode I strength
(eg, Fig. 31). Thus, transmitting the load from the matrix to the reinforcement
via shear is an efﬁcient approach. Once the load is transferred, the ﬁber may
provide the necessary strength and long-term durability. In order to realize this
type of reinforcement, adequate ﬁber lengths and high interface strength are
necessary.

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