Fatigue: Its Measurement & Applications

Suresh Gulati

October 9, 2006

Definition

Materials, when stressed above threshold level, experience flaw growth and become weak, i.e. less durable. This phenomenon is called FATIGUE.

Types of Stress

• Constant Stress (static fatigue)

• Constant Stress Rate (dynamic fatigue)

• Cyclic Stress (cyclic/dynamic fatigue)

Fatigue Mechanism Summary

SiO2 + H2O +Stress Si (OH)4

Hence, stress becomes the catalyst, i.e. no stress, no fatigue !

Strong oxide bonds become weak hydroxyl bonds !!

Necessary Elements for Fatigue

1. Stress

2. Flaw

3. Water Vapor

4. Time

Role of Water Vapor

• Small molecule (< 3 Ao )

• Easily fits the cavity of 6-member SiO2

tetrahedra

• Need only few molecules of H2O

• Hence fatigue can occur at low RH

• Higher the humidity, higher the fatigue

• No fatigue in vacuum

Role of Stress

• Stress is the catalyst

• Higher the stress, higher the fatigue

Role of Temperature

• Vibrational energy of H2O molecule vs. T

• Chemical reaction of SiO2 with H2O vs. T

• Vibrational energy of glass forming oxides,SiO2, Na2O, CaO, B2O3, etc. vs. T

• Fatigue vs. Coeff. Thermal Expansion

• Fatigue at subzero temperature

Role of Stress Duration

• (SiO2 + H2O + stress) reaction requires time

• Short duration of time means less fatigue

• Long duration of stress means more fatigue

Measurement of Strength

• 4-point Bend Test (uniaxial)

• Tensile Test (uniaxial)

• Ring-on-Ring Test (biaxial)

• ASTM Standard

• Test Duration

• Test Environment

St. Venant Flexure

• M/I = E/R = /y

• MOR = f = 1.5 P (L – l ) / (b t2)

P = load at failureL = support spanl = load spanb = width of speciment = thickness of specimen

Strength Distribution

• Gaussian

• Weibull

Pf = 1 – exp [ - ( / o)m ]

Pf : failure probability at stress o : stress at Pf = 0.63

m : slope of (lnlnPf vs. ln )

Weibull Distribution

• Unimodal (single value of m, i.e. uniform flaw)• Multimodal (multiple values of m, i.e. multiple

families of flaw)

• Physical meaning of m: approx.(mean f / std.dev)

• Fatigue is difficult to measure for multiple families of flaw

Measurement of Dynamic Fatigue

d/dt

(MPa/s)

N

(MPa)

m t

(s)

3.0 20 75 15 25

0.3 20 68 18 250

0.03 20 59 16 2500

0.03 20 45 19 25000

0.003 20 30 17 250000

Weibull Distribution Plot

4000

4500

5000

5500

6000

6500

7000

7500

8000

1

2

5

10

20

40

60

80909598

99.5

PPAD plot: 05/31/06

Data Summary Set1: (19 Specs) Mean=7.01E3; Stdev=253; m=33.4; S

0=7.12E3

Set2: (19 Specs) Mean=6.05E3; Stdev=315; m=23.0; S0=6.19E3

Set3: (19 Specs) Mean=5.46E3; Stdev=321; m=21.0; S0=5.6E3

Set4: (19 Specs) Mean=4.89E3; Stdev=298; m=20.7; S0=5.01E3

Set5: (4 Specs) Mean=4.47E3; Stdev=120; m=44.7; S0=4.52E3

WeiPPAD 5.010 (01/29/03) (XOnProb) (MedianEst)

WEIBULL DISTRIBUTION

Fai

lure

Pro

ba

bili

ty

STRENGTH (psi)

Weibullized Eagle SG dynamic fatigue at each rate, fastest rate (#1) has the strongest distribution.

Computation of Fatigue Constant

• Let = median strength at (d/dt)1

• Let = median strength at (d/dt)2

• Then

1 / 2 = [ (d/dt)1 / (d/dt)2 ]1/(n +

1)

lnvs. ln (ddt) Plot

100µ 1m 10m 100m 1 10 100 1k 10k3k

4k

5k

6k

7k

8k

9k

RaMS plot: 4/13/2006

Power Law: n = 19.195% c.i.: 17.6, 20.981 Specimens [D = 3.71]

Str

en

gth

(p

si)

Stress Rate (psi/s)

1

n+1

Dynamic Fatigue Constants for Silicate Glasses (ambient environment)

Glass Code Composition Avg. n Value CTE (25-300C)

9061 CTV Panel 14 98

0080 SLS 16 80

7740 Pyrex 27 32

7059 Color Filter 28 38

1737 LCD 24 37

1723 SAS 30 36

7940 FS 37 5.5

7971 ULE 45 0.3

Plot of n vs. CTE

10 15 20 25 30 35 40 45 500

20

40

60

80

100

CTE ~ 1/n

CT

E (

10-7/C

)

Fatigue Constant, n

Physical Meaning of n

n Value Strength loss due to 10X lower (d/dt)

15 13.4 %

20 10.4 %

25 8.5 %

30 7.2 %

35 6.2 %

40 5.5 %

Measurement of Static fatigue

• Apply static (constant) stress to 4-point bend specimen and measure time tf till it fails

• Plot lnvs. tf

• Estimate n value from

1n tf1 = 2

n tf2

n = [ (ln tf2 – ln tf1) / (ln 1 – ln 2) ]

Plot of ln vs. ln tf

** *

* * * *

* * *

* * *

ln (time)

ln(stress)

*

*

**

1/n

1

Comparison of Static vs. Dynamic n Values

• Dynamic n > Static n

• Code 9061 glass: nd = 22, ns = 14

• CO diesel filter: nd = 30, ns = 15

Allowance for Fatigue Damage( LD Diesel Filter )

• Recall n tf = constant• New Filter: 1 = o, tf = to = 1 sec.• AT Filter for LDD application:

n = 70, Life = 200,000 KmRegen. Freq’y: 300 Km

max Duration: 75 sec. each regen.Find fatigue factor & usable strength

LD Diesel Filter (cont’d)

• Total regens = 200,000 Km / 300 KM = 667• Total stress duration = 667 x 75 s = 50,000 s.

• Useable strength = 2 as shown below

2n x 50,000 = 1

n x 1

2 = 1 [ 1 / 50,000 ]1/n

= 0.86 1 = 0.86 o

Fatigue Factor = 0.86

Allowance for Fatigue Damage( HD Diesel Filter )

• CO Filter for HDD applicationn =30, Life = 700,000 KmRegen. Freq’y = 400 Km

max Duration = 100 sTotal Regens = 1750Total Stress Duration = 175,000 s

2 = [ 1/175,000 ]1/30 = 0.67 o

Fatigue Factor = 0.67

Allowance for Stressed Area( HD Diesel Filter )

• Assume 9” diameter x 12” long filter

• Assume m = 15

• Area Factor = [ Aspec / Afilter ]1/m

• Aspec = 0.75” x 1” = 0.75 in2

• Afilter = 3.14 x D x L = 113 in2

• Area Factor = 0.72

Allowance for Acceptable Pf

( HD Diesel Filter )

• Failure Probability Factor = ( Pf )1/m,

• m = 15

• FPF = 0.54 for Pf = 0.0001 (0.01 % fail)

• FPF = 0.46 for Pf = 0.00001 (0.001% fail)

Final Useable Strength of CO Filter for HD Diesel Filter

• Useable Strength = o x FF x AF x FPF

= 0.26 o for Pf = 0.01%

= 0.22 o for Pf

=0.001%

Threshold Stress(fatigue free stress)

• At certain value of stress, known as threshold stress, the critical flaw does not propagate

• Materials can sustain threshold stress without becoming weak, hence we may call that “happy stress”

Static Fatigue Test for Estimating Threshold Stress

• Use 4-point bend test at 400C• Measure MOR of 15 specimens• Assume n =ndyn and estimate allowable stress (eas) for 9 x 12 filter

at Pf = 0.001%• Apply static stress = eas on 15 specimens

= eas + 50 psi on 15 specimens

= eas + 75 psi on 15 specimens• Hold above static stresses for 100 hours at 400C• Record any premature failures• Remove stress• Measure MOR of surviving specimens for each set• Compare MOR distributions before and after static stress• Expect max. loss of strength at highest static stress and none at eas

Static Fatigue Fixture with Multiple Specimens

Static Fatigue Fixture in 400C Oven

Comparison of Strength Distributions after Static Load Test CO-E1(axial) Static Load then Residual MOR at 400oC

100100 150 200 250200

250

300

350

400

450

500

Baseline

static breaks 664

Res

idua

l Str

engt

h (p

si)

Static Stress (psi)

Strength Data for CO 200/12

• Baseline: 353 psi (s = 0 )

• MOR res : 296 psi ( s = 150 psi )

• MOR res : 243 psi ( s = 175 psi )

• MOR res : 187 psi ( s = 200 psi )

Crack Growth during Static Loading

ar / ai = [ MORi / MORres ]2

• ar / ai = 1.42 @ 150 psi

= 2.11 @ 175 psi

= 3.56 @ 200 psi

Estimated n value: ~15

Static vs. Dynamic n Values

• Static value is lower than dynamic value due to longer time for stress corrosion reaction in humid environment

• Which n value should be used? This will depend on stress/time history for a given application

Applications of Fatigue Effect

• Color TV Panel

• Space Shuttle Windows

• Fiber Optics

• Catalytic Converters

• Diesel Filters

• LCD Panel

Acknowledgements

• John Helfinstine

• Janto Widjaja