# Plate with edge cracks (Fatigue and Fracture)

This study is focussed on investigating the dispersion in the fatigue life of a structure using Monte Carlo simulation. Short cracks are initially present and the Paris equation is integrated using the Fatigue and Fracture toolbox of COSSAN-X.

## Definition of the mechanical model

In this study, the fatigue life of a plate with two edge cracks will be analysed (see picture below). It is assumed that the cracks do not interact. Hence the stress intensity factor at each tip is estimated considering its crack only. An analytical approximation of the stress intensity factor can be found at [1]:

$K_I = F_0 \cdot \sigma \cdot \sqrt{\pi \cdot a}$

where $\sigma$ is the applied stress, $a$ is the crack length and $F_0$ is a correction factor, determined by the gemetry:

$F_0 = sec(\beta ) \cdot \left( \frac{tan\beta }{\beta } \right)^{\frac{1}{2}}\cdot \left[0.752 + 2.02 \cdot\dfrac{a}{w} + 0.37 \cdot \left(1- sin \beta \right)^3 \right]$

$\beta = \frac{\pi\cdot a}{2 \cdot w}$

The structure undergoes alternating loading, the maximum applied stress is 100 MPa and the minimum applied stress is 20 MPa.

Geometry of the structure considered in this study.

## Definition of the probabilistic model

The initial crack length and the coefficients of Paris equation are modelled using random variables. The distributions used are summarized in the table below.

 Property Distribution Mean Coefficient of variation Initial crack length (2 variables) lognormal 0.001 m 0.50 Coefficient C of Paris equation lognormal $10^{-23}$ 0.10 Coefficient m of Paris equation deterministic 2.5 - Fracture toughness lognormal $7 \cdot 10^{7} \; Pa \sqrt{m}$ 0.10

The random variables are assumed to be uncorrelated.

## Creation of the COSSAN objects

A Mio object estimating the maximum value and the variation of the stress intensity factor over one cycle has been written. It is based on the approximation of the stress intensity factor developed in the first section.

A CrackGrowth object has been prepared. It used the outputs of the Mio described previously and estimates the variation of the crack length over one cycle using Paris equation.

A Fracture object has been prepared. It returns a number smaller than zero as long as stable crack occurs, and a number bigger than zero when fracture happens. Practically, the output of this object is the difference between the maximum value of stress intensity factor and the toughness of the material.

Finally, a FatigueFracture object is created. It takes as an input all the objects defined previously and handles the numerical integration of Paris equation.

## Results

The following figure shows the scatter in the fatigue life obtained using Monte Carlo simulation with 200 samples.

Histogram of the fatigue life of the structure.