Weibull Modulus Calculator for Brittle Materials

Calculator Input

Material or batch name

Strength unit

Plotting position

Target stress for reliability

Failure percentile for Bx strength

Displayed decimals

Fracture strength samples

Use at least 5 positive strength values. The page sorts them automatically.

Example Data Table

Example fracture strength data for a brittle ceramic batch.

Specimen	Strength (MPa)
1	420
2	455
3	470
4	489
5	510
6	528
7	545
8	563
9	585
10	612

Using this sample with Benard ranking gives a modulus near 9.37 and characteristic strength near 544.03 MPa.

Formula Used

The two-parameter Weibull failure model is:

F(σ) = 1 − exp[−(σ / σ₀)^m]

Where:

F(σ) = failure probability at stress σ
m = Weibull modulus
σ₀ = characteristic strength

For linear regression, the calculator uses:

ln(ln(1 / (1 − F))) = m ln(σ) − m ln(σ₀)

So the fitted straight line becomes:

Y = mX + b

X = ln(σ)
Y = ln(ln(1 / (1 − F)))
m = slope
σ₀ = exp(−b / m)

Default ranking uses Benard’s median rank:

F_i = (i − 0.3) / (n + 0.4)

Reliability at a target stress is:

R(σ) = exp[−(σ / σ₀)^m]

Percentile strength for a chosen failure percentage p is:

σ_p = σ₀ [−ln(1 − p)]^1/m

How to Use This Calculator

Enter the material or batch name.
Enter the strength unit, such as MPa or ksi.
Paste fracture strengths into the sample box, one per line or comma separated.
Choose the plotting position method. Benard is a common default.
Optionally enter a target stress to estimate survival and failure probability.
Enter a failure percentile, such as 10 for B10 strength.
Press Calculate Weibull Modulus.
Review modulus, characteristic strength, curve fit, reliability, ranked table, and graphs.
Use the CSV or PDF buttons to export the results.

FAQs

1) What does the Weibull modulus represent?

It measures strength scatter in brittle materials. A higher modulus means test results are clustered tightly, while a lower modulus indicates wider variability and less predictable failure behavior.

2) What is characteristic strength?

Characteristic strength, σ0, is the stress where predicted failure probability reaches 63.2%. It is a scale parameter of the Weibull model and is not simply the average strength.

3) Why are the strengths sorted before fitting?

Weibull plotting uses ordered data. Each sorted value gets a rank-based failure estimate, which is then transformed into a straight-line fit for modulus and characteristic strength.

4) Which plotting position should I choose?

Benard’s median rank is a standard choice for many engineering datasets. Hazen and Mean Rank are also useful for comparison studies when you want to test sensitivity to ranking assumptions.

5) What does the R² value tell me?

R² shows how well the linearized Weibull model matches your transformed data. Values closer to 1 suggest a better straight-line fit and stronger support for the chosen Weibull model.

6) What is B10 strength?

B10 strength is the stress level where 10% failure probability is predicted, or 90% survival. It is often used as a conservative design strength for brittle components.

7) Can I use this for ductile metals?

It is mainly intended for brittle fracture datasets such as ceramics, glass, and some composites. Ductile metal failure often needs different statistical or physical models.

8) How many samples should I use?

More samples improve confidence in the fitted modulus. This calculator requires at least five values, but engineering decisions are stronger when based on larger, representative test batches.