Weibull Distribution (Cumulative & Density)
The standard continuous probability distribution for brittle fracture strength and component failure lifetime analysis.
Mathematical Formulation
Pf = 1 - exp(-(sigma / sigma0)^m)Parameters & Physical Meanings
Detailed parameter breakdown, standard units, valid mathematical constraints, and initial guess heuristic algorithms.
| Symbol | Name | Unit | Constraints | Physical Interpretation | Initial Guess Heuristic |
|---|---|---|---|---|---|
| $\sigma_0$ | Characteristic Strength / Scale Parameter | Stress (MPa) / Time (hours) | sigma0 > 0 | Scale parameter corresponding to the stress or time at which exactly 63.2% of specimens have failed | Estimated from the 63rd percentile of failure stress data |
| $m$ | Weibull Modulus / Shape Parameter | Dimensionless | m > 0 | Measures material defect homogeneity; higher m indicates narrower strength scatter and more uniform defect distribution | Estimated from linear regression slope of ln(ln(1/(1-Pf))) vs ln(sigma) |
When to Choose This Model
- Analyzing brittle tensile and flexural fracture strength scatter in advanced ceramics and glasses
- Semiconductor package time-to-failure (TTF) life data analysis
- Optical fiber tensile proof testing and fatigue lifetime modeling
- Wind turbine and mechanical bearing fatigue reliability forecasting
Typical Scientific Applications
- Ceramic fracture mechanics
- Electronic component reliability testing
- Fiber optic mechanical durability
- Composite material fatigue failure
Expected Fit Profile & Curve Morphology
Shape: distributionExample Dataset & Expected Fit Output
Synthetic 15-point ceramic tensile fracture strength test data with scale sigma0 = 345 MPa and Weibull modulus m = 8.4
Expected Converged Parameters
- sigma0345.200
- m8.420
Sample Experimental Vectors (14 points)
| # | x (Independent) | y (Observed) |
|---|---|---|
| 1 | 180.000 | 0.003 |
| 2 | 220.000 | 0.021 |
| 3 | 250.000 | 0.065 |
| 4 | 275.000 | 0.142 |
| 5 | 295.000 | 0.258 |
| 6 | 310.000 | 0.384 |
| 7 | 325.000 | 0.521 |
| 8 | 340.000 | 0.654 |
| + 6 more points available in AltaiPlot preset | ||
Python / SciPy Reference Implementation
Reference library: scipy.optimize (Trust-Region-Reflective).Exact parameterization parity verified.
import numpy as np
from scipy.optimize import curve_fit
def weibull_cdf_model(sigma, sigma0, m):
sigma_safe = np.maximum(sigma, 1e-12)
return 1.0 - np.exp(-(sigma_safe / sigma0)**m)
# Linearization heuristic: ln(-ln(1 - Pf)) = m * ln(sigma) - m * ln(sigma0)
valid = (Pf_data > 0) & (Pf_data < 1)
sigma_v = sigma_data[valid]
Pf_v = Pf_data[valid]
y_lin = np.log(-np.log(1.0 - Pf_v))
x_lin = np.log(sigma_v)
p_coeffs = np.polyfit(x_lin, y_lin, 1)
m_init = max(p_coeffs[0], 0.5)
sigma0_init = np.exp(-p_coeffs[1] / m_init)
p0 = [sigma0_init, m_init]
bounds = ([1e-6, 0.1], [np.inf, 50.0])
popt, pcov = curve_fit(weibull_cdf_model, sigma_data, Pf_data, p0=p0, bounds=bounds)Comparative & Alternative Models
Key decision trade-offs between this model and related functional alternatives:
Frequently Asked Questions (Weibull Distribution (Cumulative & Density))
What is the physical interpretation of a high Weibull modulus (m > 20)?
A high Weibull modulus indicates that the material fails within a very narrow stress range, reflecting highly uniform defect sizes. Ductile metals typically exhibit effective m > 30, while structural ceramics typically exhibit m between 5 and 15.
Scientific References & Citations
- Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293-297.[Source / DOI]