4: Weibull Distributed Strength

Introduction to Weibel Distributed Strength

• All materials contain flaws of various sizes and orientations.
• For brittle materials, the flaw size distribution directly affects component strength.
• The two-parameter Weibull distribution is suitable for modeling strength distribution in brittle materials.

Weibull Distribution Overview

• Cumulative Distribution Function (CDF): Represents the probability of failure at a specific stress level.
• The probability of failure can be expressed as an exponential function of applied stress ($ ext{sigma}$).
• Two parameters of the Weibull distribution:
  • Weibull Modulus (m):
    • Shape parameter, unitless exponent.
    • Significantly impacts the predictability of material strength.
  • Scale Parameter (θ):
    • Represents the scale of stress, has units of stress (e.g., megapascals).

Understanding Probability of Failure and Survival

• Probability of Survival:
  • Calculated as $1 - P( ext{failure})$.
• Visual representation:
  • Graphs of CDF for different Weibull modulus and scale parameter values.
  • For example, with a Weibull modulus of 5 and a scale parameter of 405 MPa,
    • At 300 MPa stress: 20% chance of failure.
    • Above 600 MPa stress: Almost 100% chance of failure.

Impact of Weibull Modulus on Strength Variability

• Increasing Weibull modulus ($m$) leads to steeper CDF curves indicating:
  • Reduced variability in failure stress.
  • More predictable behavior (lower safety factors needed).
• Material with high Weibull modulus results in less spread of failure strengths, while low modulus results in greater spread.
• Example Interpretation:
  • Weibull modulus of 5 may show a range of failure strengths over hundreds of MPa, whereas 15 may limit the range to less than 100 MPa.

Practical Implications of Weibull Modulus Values

• Low Weibull modulus indicates unpredictable strengths requiring higher safety factors in designs.
• Typical values for different materials:
  • Whiteware ceramics: 3-5
  • Engineering ceramics: 10-20
• Examples from experience:
  • Silicon nitride (modulus ~14) vs. fused silica (modulus ~7) for missile components and their implications on safety factors.

Deterministic Properties

• If Weibull modulus reaches ~30, material properties can be considered deterministic, implying very predictable behavior.

Testing and Determining Weibull Modulus

• Methodology for testing engineering ceramics (e.g., aluminum oxide):
  • Use of four-point flexure coupons to measure failure stress.
• Example results:
  • List of failure strengths gives corresponding probabilities based on the order of strengths.
  • Probability calculations: For example, failure strength of 263 MPa has a probability of ~2%, whereas 452 MPa has ~99%.
• The slope of the linear fit of the logged data yields Weibull modulus ($m$).

Conclusion

• Understanding the Weibull distribution is crucial for predicting the strength of brittle materials and ensuring safe designs in engineering applications.