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).
- Weibull Modulus (m):
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.
- Probability of Survival:
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.
- Increasing Weibull modulus ($m$) leads to steeper CDF curves indicating:
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 ($m$).
- Methodology for testing engineering ceramics (e.g., aluminum oxide):
Conclusion
- Understanding the Weibull distribution is crucial for predicting the strength of brittle materials and ensuring safe designs in engineering applications.