This article is a second article on the Weibull Law which explains how to use Python to calculate the law’s parameters.

If you didn’t read the first article, you can read it here

## 1 How to determine the parameters of the Law

If we start from the Weibull Probability that we determined previously:

After a few simple mathematical operations (take the log of both sides), we can convert this expression into a linear expression, such as the following one:

This means that we can pose:

and

And the previous equation can be written:

## 2 Numerical Example with Python

Let’s import first the python modules we will need for the study:

- os is a classic module always useful to handle the link with files and the system
- numpy is here for the numerical calculations
- matplotlib will be useful to draw the graphs
- scipy will provide us with an useful function to do regression of the curve and fit the parameters
- pandas will facilitate the interaction with the data

import os import numpy as np import matplotlib.pyplot as pl from scipy import stats import pandas as pd

Let’s define now a function to calculate the Y=ln[-ln(1-P_{F})] value mentionned previously

def weibull_inv(p): return np.log(-np.log(1.0-p))

Now let’s suppose that we have the following data into a data.csv file

Sample N | failure load(N) | Sigma Failure(MPa) |
---|---|---|

1 | 1168 | 17.7 |

2 | 1175 | 17.8 |

3 | 1195 | 18.1 |

4 | 1287 | 19.5 |

5 | 1360 | 20.6 |

6 | 1366 | 20.7 |

7 | 1426 | 21.6 |

8 | 1445 | 21.9 |

9 | 1452 | 22 |

10 | 1491 | 22.6 |

11 | 1524 | 23.1 |

12 | 1537 | 23.3 |

13 | 1551 | 23.5 |

14 | 1564 | 23.7 |

15 | 1564 | 23.7 |

16 | 1577 | 23.9 |

17 | 1637 | 24.8 |

18 | 1656 | 25.1 |

19 | 1676 | 25.4 |

20 | 1682 | 25.5 |

Let’s use pandas to import those datas into a DataFrame

#importing the CSV file Samples_DF = pd.read_csv('data.csv') #Sorting the data according to stress values and re-indexing Samples_DF = Samples_DF.sort_values(by='Sigma Failure(MPa)') Samples_DF = Samples_DF.reset_index(drop=True)

Let’s plot that now

pl.plot(Samples_DF['Sigma Failure(MPa)'],'o') pl.grid() pl.xlabel('# Sample') pl.ylabel('Stress at fracture (Principal)') pl.title("Failure Stress results of experiment")

Let’s add to the dataframe table 2 new columns corresponding respectively to:

- The law of probability (an empirical distribution function)
- The inversed Weibull distribution law

Samples_DF['Proba'] = (Samples_DF.index-Samples_DF.index[0]+1) / (len((Samples_DF.index))+1) Samples_DF['Weibull'] = weibull_inv(Samples_DF['Proba'])

We obtain the following dataframe:

Sample N failure load(N) Sigma Failure(MPa) Proba Weibull 0 1 1168 17.7 0.047619 -3.020227 1 2 1175 17.8 0.095238 -2.301751 2 3 1195 18.1 0.142857 -1.869825 3 4 1287 19.5 0.190476 -1.554433 4 5 1360 20.6 0.238095 -1.302197 5 6 1366 20.7 0.285714 -1.089240 6 7 1426 21.6 0.333333 -0.902720 7 8 1445 21.9 0.380952 -0.734859 8 9 1452 22.0 0.428571 -0.580505 9 10 1491 22.6 0.476190 -0.435985 10 11 1524 23.1 0.523810 -0.298490 11 12 1537 23.3 0.571429 -0.165703 12 13 1551 23.5 0.619048 -0.035543 13 14 1564 23.7 0.666667 0.094048 14 15 1564 23.7 0.714286 0.225351 15 16 1577 23.9 0.761905 0.361224 16 17 1637 24.8 0.809524 0.505750 17 18 1656 25.1 0.857143 0.665730 18 19 1676 25.4 0.904762 0.855000 19 20 1682 25.5 0.952381 1.113344

In order to perform a linear regression, we have to extract w and lnsw as below:

w = Samples_DF['Weibull'] lnsw = np.log(Samples_DF['Sigma Failure(MPa)'])

Let’s calculate now the parameters of the linear regression:

with:

and

m, lnsm0, *t = stats.linregress(lnsw,w) sigma0 = np.exp(- lnsm0 / m) print('m=', m) print('sigma0=',sigma0)

Here’s what I get:

m= 9.23254658432

sigma0= 23.3758678631

m expresses the dispersion around the Weibull Law

sigma0 is the scale factor of the Weibull Law.

It has the same order than the stress at failure.

Let’s draw all of that now with mathplotlib:

```
pl.figure()
pl.plot(lnsw,w)
pl.plot(lnsw,w,'*')
x = np.arange(lnsw.iloc[0], lnsw.iloc[-1]+0.1, 0.1)
y = (lambda x : m * x + lnsm0)(x)
pl.plot(x, y)
pl.plot()
pl.grid()
pl.ylabel('log(-log(1 - Probability of fracture))')
pl.ylabel('-log(Stress at fracture)/m')
pl.title("Weibull Analysis of experiment data")
```

That’s all for today, hope you understand better now how to use the Weibull criteria to identify the failure stress of brittle material

Obviously, there is more to know about this topic… but that will be for next time!

–Cyprien

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