Tensile Strength of Composite Fibers

Author: Brian RussellDate: December 4, 2008SMRE - Reliability Project

Objective:Using data provided by “Reliability Modeling, Prediction, and Optimization”, Case 2.6, “Tensile Strength of Fibers” I will explore the tensile strength of silicon carbide fibers after extraction from a ceramic matrix.

Description of System:• Estimate fiber strength after incorporation into the

composite.• Fiber strength is measured as stress applied until

fracture failure of the fiber. • The objective of the experiment was to determine the

distribution of failures as a function of gauge length of the fiber after incorporation into the composite.

Methodology used for Analysis: • Data will be imported to Minitab so that

mathematical manipulation can be performed to produce transfer functions.

• Using Excel, Monte Carlo simulations was performed to simulate a larger population

• Equations were manipulated using Maple to produces the appropriate Reliability functions and display the data graphically.

• The Results of the Monte Carlo was compared to the Maple Results

321

0.6

0.4

0.2

0.0

C1

PD

F

2.01.00.5

99.9

90

50

10

1

C1

Perc

ent

321

100

50

0

C1

Perc

ent

321

4.5

3.0

1.5

0.0

C1

Rate

Correlation 0.969

Shape 3.11972Scale 1.92163Mean 1.71903StDev 0.603230Median 1.70863IQR 0.844805Failure 50Censor 0AD* 0.924

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for Length of 265mmLSXY Estimates-Complete Data

Weibull

10.01.00.1

99.9

99

908070605040

30

20

10

5

3

2

1

C1

Perc

ent

Shape 3.119Scale 1.922N 50AD 0.773P-Value >0.250

Probability Plot of C1Weibull - 95% CI

This Minitab plot shows that the response at length 265 mm fits a Weibull well with shape of 3.119 and scale of 1.922.

The scale parameter is: a = 1.992The shape parameter is: b = 3.119So the Weibull function that fits this data isF=1-exp(-(t/a)^b)F:=1-exp(-(t/1.992)^3.119)To perform the Monte Carlo Simulation in Excel, this expression is first transformed to: t=-1.992*ln(1-3.119(F))

Minitab Response for Fiber Length 265

mm

Minitab Response for each fiber length

321

0.6

0.4

0.2

0.0

C1

PD

F

2.01.00.5

99.9

90

50

10

1

C1

Perc

ent

321

100

50

0

C1

Perc

ent

321

4.5

3.0

1.5

0.0

C1

Rate

Correlation 0.969

Shape 3.11972Scale 1.92163Mean 1.71903StDev 0.603230Median 1.70863IQR 0.844805Failure 50Censor 0AD* 0.924

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for Length of 265mmLSXY Estimates-Complete Data

Weibull

4321

0.6

0.4

0.2

0.0

25.4

PD

F

521

99.9

90

50

10

1

0.1

25.4

Perc

ent

4321

100

50

0

25.4

Perc

ent

4321

6

4

2

0

25.4

Rate

Correlation 0.997

Shape 4.86156Scale 3.10592Mean 2.84712StDev 0.669072Median 2.88037IQR 0.918007Failure 64Censor 0AD* 0.391

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for 25.4LSXY Estimates-Complete Data

Weibull

5432

0.6

0.4

0.2

0.0

12.7

PD

F

5432

99

90

50

10

1

12.7

Perc

ent

5432

100

50

0

12.7

Perc

ent

5432

2

1

0

12.7

Rate

Correlation 0.977

Loc 1.10693Scale 0.215072Mean 3.09585StDev 0.673604Median 3.02507IQR 0.880738Failure 50Censor 0AD* 1.084

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for 12.7LSXY Estimates-Complete Data

Lognormal

5432

0.6

0.4

0.2

0.0

5

PD

F

5432

99.9

90

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1

5

Perc

ent

5432

100

50

0

5

Perc

ent

5432

6

4

2

0

5

Rate

Correlation 0.980

Shape 7.19240Scale 3.72481Mean 3.48921StDev 0.571679Median 3.53976IQR 0.765493Failure 50Censor 0AD* 0.775

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for 5LSXY Estimates-Complete Data

Weibull

Monte Carlo Analysis

265 25.4 12.7mm 5mm

F:=1-exp(-(t/1.992)^3.119) F:=1-exp(-(t/3.106)^4.862) F:=1-exp(-(t/3.32943)^5.85190) F:=1-exp(-(t/3.72481)^7.19240)t=-1.992*ln(1-3.119(F)) t=-3.106*ln(1-4.862(F)) t=-3.32943*ln(1-5.85190(F)) t=-3.72481*ln(1-7.19240(F))

1.784042003 2.817144112 3.081436469 3.486495448

Using the Minitab functions transformed in Excel:

Reliability equations for all four lengths were calculated using Maple

(data for 265mm shown here)

Cumulative Distribution Function Reliability Function

Probability Density Function

Hazard Function

Distribution Plots from Maple

MTTF Using Maple

The data shows that as the fiber length increases, the Mean Time To Failure (MTTF) decreases. A fiber of length 5mm has a MTTF of 3.5 seconds compared to a fiber of length 265 inches has a MTTF of 1.8 seconds.

Monte Carlo analysis was performed using the following equations in Excel:265 25.4 12.7mm 5mm

F:=1-exp(-(t/1.992)^3.119) F:=1-exp(-(t/3.106)^4.862) F:=1-exp(-(t/3.32943)^5.85190) F:=1-exp(-(t/3.72481)^7.19240)t=-1.992*ln(1-3.119(F)) t=-3.106*ln(1-4.862(F)) t=-3.32943*ln(1-5.85190(F)) t=-3.72481*ln(1-7.19240(F))

1.784042003 2.817144112 3.081436469 3.486495448

The MTTF values in Excel match the values calculated in Maple.

Fiber Length (mm) Maple Excel Monte-Carlo

265 1.781963 1.779479205

25.4 2.847213 2.893593614

12.7 3.084489 3.081029868

5 3.489212 3.482595755

Results:

As the length of composite fibers increases

from 5 mm to 265 mm,

the tensile strength decreases.

References: Reliability Modeling, Prediction, and Optimization, Wallace R. Blischke and D.N. Prabhakar Murthy, published 2000 by Wiley-Interscience Publication