1. Tensile Strength of Composite Fibers Author: Brian Russell Date: December 4, 2008 SMRE - Reliability Project

2. 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.

3. 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

4. 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 = 1.992 The shape parameter is: b = 3.119 So the Weibull function that fits this data is F=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

5. Minitab Response for each fiber length

6. Monte Carlo Analysis Using the Minitab functions transformed in Excel:

7. Reliability equations for all four lengths were calculated using Maple (data for 265mm shown here)

8. Cumulative Distribution Function Reliability Function Probability Density Function Hazard Function Distribution Plots from Maple

9. MTTF Using Maple

10. 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: The MTTF values in Excel match the values calculated in Maple. Fiber Length (mm)MapleExcel Monte-Carlo 2651.7819631.779479205 25.42.8472132.893593614 12.73.0844893.081029868 53.4892123.482595755

11. 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