# ENBIS: European Network for Business and Industrial Statistics

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## ENBIS-18 in Nancy

*2 – 25 September 2018; Ecoles des Mines, Nancy (France)*Abstract submission: 20 December 2017 – 4 June 2018

### Detection and Modelling of the Propagation Regimes in Fatigue Crack Propagation

*5 September 2018, 11:30 – 11:50*

#### Abstract

- Submitted by
- Florine Greciet
- Authors
- Florine Greciet (Safran Aircraft Engines, Université de Lorraine, IECL, Inria BIGS), Romain Azaïs (Université de Lorraine, IECL, Inria BIGS), Anne Gégout-Petit (Université de Lorraine, IECL, Inria BIGS)
- Abstract
- In aeronautics the risks of engine parts failure are accentuated by the extreme environment in which these structures must operate. Each environment stress can cause a flaw in structure and the stress accumulation repeated on tens of thousands of hours of flight leads to the propagation of the crack. In the long term a crack can cause the break of the structure which can be fatal during a flight. It is essential to control this phenomenon to size the engine parts accordingly. In terms of sizing, the parts are designed to withstand during a defined period the stresses applied to them.

Crack propagation modeling goes through the study of the fatigue crack growth rate law according to the stress intensity factor. The following three regimes are empirically observable through the propagation law. In regime I, referred to as the crack initiation region, crack propagation is a discontinuous process which is extremely slow at very low values of stress intensity factor. In regime II, a power-law relationship between crack growth rate and stress intensity factor range is observed. Regime III corresponds to a quick and unstable crack growth leading to rupture when the stress intensity factor tends to a critical value.

To model the propagation law we propose to use Piecewise Deterministic Markov Processes (PDMPs) which are good candidates to model the three regimes mentioned before because it is a process governed by punctual random jumps governed by an ordinary differential equation between the jump times. It is important to note that times of transition between regimes are non observable on propagation curve because of these two following reasons: the crack propagation phenomenon is continuous and data are usually noisy.

Therefore, the trajectory is hidden, that means times of transition between regimes, propagation parameters in each regime and the number of propagation regimes are unknown.

The two methodologies that we propose aim at estimating all these quantities under the following assumptions:

- In each regime the propagation is described by a polynomial function;

- Regime changes preserve the propagation continuity;

- Trajectories are observed through a Gaussian additive noise.

The first methodology consists in assuming that the number of regimes is k and maximizing the likelihood with k regimes in two steps: the models parameters are estimated by an EM algorithm and we use an exhaustive search to find the times of transition. The number of regimes is determined a posteriori by a BIC criterion. The second approach is recursive and consists in estimating only one time of transition in the observed trajectory at each step. To overcome the invalidity of the model we maximize a penalized version of the likelihood by an EM algorithm (parameters estimation) through a trichotomic algorithm (estimation of transition times). When a time of transition is estimated, this procedure is repeated on the observed data after this time of transition. The procedure stops when the last estimated time of transition improves no more the BIC.

Results of simulations obtained for these two methods will be presented.