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Fatigue in polymers
By: Ido Gal
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Table of context
1. Definition of Fatigue .........................................................................................3
2. Fatiuge models.......... ........................................................................................
7
3. Linear elastic fracture mechanics............................................................. 13
4. von Mises yield criterion.......................................................................... 21
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List of figures
Figure 1 - Figure 1- Applied stress versus failure time (static fatigue) for high density
polyethylene at various temperatures........................................................................ 3
Figure 2 Stress amplitude versus log Nf. for several polymers. ..............................4
Figure 3 - Stress amplitude versus log Nf for a polyacetal copolymer in different
frequency .................................................................................................................. 5
Figure 4 - Fatigue crack growth rate da/dN versus Ki for several polymers6
Figure 5 - Maximum specimen temperature as a function of cycles applied.8Figure 6 - True strain e - Sum ofPlastic ( p) and Elastic (el) Strains.. ................ 10
Figure 6a,b,c - Acetal: a-strain Vs Nf ,bstrain Vs energy per cycle and c temp rise
Vs strain .14
Figure 7a,b,c - Cast PMMA: a-strain Vs Nf ,bstrain Vs energy per cycle and c temp
rise Vs strain..14
Figure 8a,b,c - Nylon 6/6 with 5% MoS2: a-strain Vs Nf ,bstrain Vs energy per cycle
and c temp rise Vs strain..15
Figure 9a,b,c Glass-reinforced nylon 6/6: a-strain Vs Nf ,bstrain Vs energy per
cycle and c temp rise Vs strain15
Figure 10a,b,c - Polycarbonate: a-strain Vs Nf ,bstrain Vs energy per cycle and c temp rise
Vs strain16
Figure 11a,b,c PVC: a-strain Vs Nf ,bstrain Vs energy per cycle and c temp rise
Vs strain. ................................................................................................................. 16
Figure 12 - Log of limiting stress Vs log of crack length ........................................... 17
Figure 13 - Schematic view of crack growth during the crack tip plastic zone
formation ................................................................................................................. 19
Figure 14Extensive growth oflast pair of slip bands .............................................. 19
Figure 15- Fatigue crack growth rate data for polycarbonate. ................................20
Figure 16 - Scanning electron micrograph of a polycarbonatc fatigue crack in the
DeC region ............................................................... Error! Bookmark not defined.1
Figure17 - the stress state at a point P ahead of the crack .......Error! Bookmark not
defined.1
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Fatigue definition:
Fatigue is the fracture undergone by a material when it is subjected over a long
period of time to stresses that are lower than its strength.
Under certain conditions (temp, time etc) the micro cracks existing in a material
grow slowly, and, with this, K iincreases until it reaches the critical value kic
At that moment, a sudden brittle fracture occurs in the part that had been
supporting constant or alternating loads over a long period of time
Two types of fatigue can be distinguished
Static fatigue Dynamic fatigue
Static fatigue
Static fatigue occurs under conditions of constant load in which the stress applied is
less than that needed to produce fracture, F, under conditions of monotonically
growing load (stress-strain tests). Static fatigue is represented by curves of the
applied stress Vs the time required for failure.
Figure 1 is an example of polyethylene at various temperatures.We can see that when stress increase the time for failure is decrease.
We can distinguish two failure mechanisms
ductile fracture- high stresses and short times
brittle fracture- low stresses and long times
The ductile-brittle transition is shifted to longer times as the temperature decreases
Chemically aggressive environments favor brittle fracture
Figure 1- Applied stress versus failure time (static fatigue) for a sample of high density polyethylene at various
temperatures. The inflection shows the point of change from brittle failure to ductile failure
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Dynamic fatigue
Dynamic fatigue is the failure or fracture of a material under cyclic loads .
It is obvious that for a given stress amplitude, the time to failure is shorter than in
static fatigue.
Dynamic fatigue curves is stress amplitude Vs the logarithm of the # of cycles to
failure.
The time to failure increases with decreasing stress amplitude
The curves have a sigmoidal (S, C shape), but at intermediate stress a linear relation is
obtained
For large numbers of cycles (10^7) the curves become horizontal meaning it is the
edge stress amplitude that the material can be cyclic it is also known as endurance
limit
Figure 2 - Stress amplitude versus logNf. for several polymers. Nf= cycles to failure
The fatigue of polymers is strongly dependent on the load frequency
Thermal fatigue failure - viscoelastic behavior of polymers provokes some heatdissipation + lower thermal conductivity can cause increases in the temperature high
stresses and high test frequencies.
Mechanical fatigue contains:
Initiation developed from surface or internal defects or flaws
Propagation of a crack the fatigue crack grows by a small amount in each cycle, this
stage seems to control the fatigue life
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Growth of a crack (Propagation)
For brittle polymers the propagation rate is proportional to the stress intensity factor range
m andAare constants for different polymers that depend on (temperature, frequency, stress ratio,
and the characteristics of the polymer such as molecular weight and crystallinity)
In Figure 14.42 (double logarithmic) plots ofda/dNVs Kz are presented for several polymers.
The behavior is linear but not always becauseA andmare not truly constants.
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Fatigue modelit is not always possible to obtain normal life-tests of the mechanical components of
new products because of time issues
Mathematical simulation and mechanical model techniques is often taking in
considerationThe technique of establishing the fatigue characteristics of a material usually
utilized by plastics materials suppliers is described in ASTM D-671. This technique,
copied directly from metallic fatigue tests.
from elastic beam theory, the maximum fiber stress in the beam can be calculated
given the intensity of the driving force
several deficiencies exist in this type of test the viscoelastic nature of polymers
causes creep or cold flow in the beam, thereby increasing the beam deflection
This type of test originated for metals where the sinusoidal oscillation of a
metallic specimen generally causes very slight temperature changes due
to hysteresis loss (damping)
The high thermal conductivity of metals coupled with low damping tan delta 10^-4
With polymers, due to the time-temperature dependence of the mechanical
characteristics, a much more complex situation exists. Repeated flexure of a plastic
beam can cause significant temperature increases, even at low frequencies, to the
point where actual melting may occur. The low thermal conductivity and relatively
high damping of most polymers at normal temperatures of use are responsible for this
situation.
Six types of structural plastics were chosen to represent the field. These materialsincluded:
A. Cast polymethyl methacrylate
B. Rigid polyvinyl chloride
C. Polycarbonate
D. Acetal homopolymer
E. Glass-reinforced type 6/6 nylon
F. Type 616 nylon with 57% MoS2
All testing was performed using an MTS low-cycle hydraulic fatigue tester
(frequencies from 0.1 to 100 cps)All polymers, at certain strain levels will exhibit a tendency to self heat with
the specimen temperature in the stressed zone rising gradually to a threshold level
where it increases very rapidly
( Figure 5). These studies have supported some observations by Riddel, Koo, and
OToole (1) that other strain levels exist where this temperature rise stabilizes due
to heat loss and failure generally is of a flaw propagation type.
The threshold temperature level appears to be related to the glass transition or any
transition which may lie between the test ambient and Tg. Elevated temperature
can increase or decrease fatigue life depending on both strain level and polymer type.
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Fatigue models
The advent of low-cycle fatigue tests has led to the development of fatigue models
and techniques
For predicting the strain-cycle failure curve.Many fatigue models developed, all of which are based on the fatigue behavior of
metals.
There are 3 widely-known fatigue models:
1. strain models2. energy models3. empirical models
Strain modelsCoffin model- Coffin felt that plastic strain, or permanent deformation, was a measure of
damage.
By that, Coffin proposed that the true plastic strain amplitude be used as a measure offatigue behavior.
Metals results tend to confirm this hypothesis.
Coffin modelMore for metals
Where A and C are constants,
e - Strain range
p - Force
N - # of cycles.
Further testing by Coffin led to the conclusion that the constant A = 0.5.
Defining the stress- strain curve of the material as one-quarter cycle,
Coffin establish strain model for fatigue:
D = true ductility of the material.
The application of Coffins method to polymers is hampered greatly by the sensitivity of
the polymer to Temperature and strain rate
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EMPIRICAL MODEL
Manson- predicted the fatigue curve from the stress-strain curve.
This model developed by statistically correlating static properties with actual fatigue data for
several dozen metals.
3 line drawn on log-log paper through specified points:
1- Plastic strain
2- Elastic strain
3- Total true strain range (1+2)
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ENERGY MODEL
Assumption: total hysteresis energy to failure is a constant. If the hysteresis energy is
denoted by Q, the failure criterion is:
There are some models who to apply it by:
constant be obtained by letting the stress-strain curve result in a failure at one-
half cycle
use a nonlinear stress-strain equation and obtain the energy in terms of strain
considering the fact that there is a limiting energy which represents the
endurance limit, and uses the amount of energy exceeding this limit instead of
total energy
the energy model is better than the other two because it have more physics basis but still
not good enough for polymer because it does not show size effects, rate effects,
temperature effects, or the effects of relaxation and creep.
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COMBINED ENERGY MODEL
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Polycarbonate: a-strain Vs Nf ,bstrain Vs energy per cycle and c temp rise Vs strain
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The Short Fatigue Crack Problem
The high fatigue crack growth rates and reduced threshold values of cracks less
than a critical length (l*) can best be explained in terms of linear elastic fracture
mechanics (LEFM). LEFM relates the stress intensity factor at a crack tip (Ki) to thefar field applied stress ( ) and the crack length (I):
1
For long cracks (I> l*) the threshold value of is a constant, in accord with
LEFM, which requires that the stress state be dependent on alone as seen
at figure 1
For short cracks (I< l*) this is not true and a deviation from the straight line is
observed, The threshold values are reduced and the small cracks grow faster
than comparable long cracks with the same (according to LEFM).
Figure 12 Log of limiting stress Vs log of crack length
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Demands that the plastic zone size Rp be small compared to the crack length l:
The plastic zone is given by (35, 36)
Where is the yield stress Equations 1 and 3 thus combine to yield
Which is a necessary condition for LEFM to hold.
Ki short cracks require higher applied stresses (Equation 1), hence a higher( / ) ratio. For very short cracks, therefore, the limits of applicability of
LEFM could conceivably be exceeded.
Discontinuous crack growth (DCG)-
In at least four ductile amorphous polymers, however, the fatigue crack tip zones
differ for short and long cracks that are nominally at the same Ki
The long cracks have a single preceding craze, while the short fatigue cracks exhibit
an unusual zone called the "epsilon" plastic zone
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The remarkable feature of this plastic zone is that the crack tip plastic flow is
contained within three narrow planes (rather than just one), one in the crack growth
direction in the form of a craze and two others as slip
bands symmetrically placed above and below the craze plane (Figure 2).
The development of epsilon zones in short cracks was identified at four polymers:
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polycarbonate, Radel
polysulfone,
Polyestercarbonate copolymer
Ardel polyarylate block copolymer.
The fatigue crack growth behaviors of short and long polycarbonate cracks were
compared by Takemori
Obtained from a 3-D surface grown part-through crack with a clamshell profile
(Figure 16)
Long cracks showed DCG for Ki between 0.4 to 0.8 MPa . Because of the
difficulty in measuring crack growth rates in small part-through cracks, data on the
growth rate of short cracks were only obtained for Ki > 0.87 MPa , although the
overall range for epsilon development in this crackspanned from 0.5 to 1.3 MPa
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Slip band formation in short cracks may be due to a change in stress state at the crack
tip (from the LEFM stress state), which favors shear flow.
Materials that obey von Mises yield criterion, yielding occurs when the octahedral
stress approach critical value.
Figure 17 von Mises yield criterion
Large differences in the diagonal elements of the stress tensor( ) enhance octand
promote shear yielding.
these stress differences must now be added to the .
The term may conversely be reduced due to the shortness of the crack and the
resultant proximity to the free surface.in 3-D surface cracks, the surface crack opening is flanked by plane stress surface
shear zones that "suck in" on the surface.
This can provide further stress relief in , thus perhaps explaining why epsilon cracks
are most readily seen in 3-D surface cracks of the clamshell (figure 16)
Fig 17 the stress state at a point P ahead of the crack
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1. References
1.1. Rice, J. R. 1967. Fatigue Crack Growth, ASTM STP 415, pp.
247-311. Philadelphia : ASTM. 542 pp
1.2. D. A. OPP, D. W. SKINNER and R. J. WIKTOREK 1968 A Model
For Polymer Fatigue
1.3. Michael T. Takemori 1984 POLYMER FATIGUE
1.4. Haward, R. N., ed. 1 973. The PhYSics of Glassy Polymers. New
York : Wiley. 620
1.5. Takernori, M. T., Kambour, R. P., Matsumoto, D. S. 1983. Polymer
Commun.1.6. Hertzberg, R. W., Manson, J. A. 1 980 Fatigue of Engineering
Plastics. New York
1.7. Radon, J. C. 1980. Int. J. Fract
1.8. Andrews, E. H. 1969. Testing of Polymer ed. W. E. Brown,
1.9. Hertzberg, R. W., Skibo, M. D., Manson J. A. 1979
.