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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

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