# Fatigue crack growth in reliability analysis

### Description

It provides the lifetime distribution of metallic materials that fail due to the crack growth produced by mechanical fatigue efforts. The crack growth trends are fitted by linear or nonlinear mixed effects regression models in order to make predictions about the material lifetime. The lifetime is defined as the time passed before the material does not meet the specification requirements and it is conditioned by a critical crack length that induces the material failure. Three different methods can be applied to estimate the fatigue lifetime distribution: "SEP-lme_bkde" and "SEP-lme_kde" are nonparametric while "PB-nlme" corresponds to the parametric approach proposed by Pinheiro and Bates (2000).

### Usage

1 2 | ```
cracks.growth(x, aF, T_c, method = c("SEP-lme_bkde", "SEP-lme_kde",
"PB-nlme"), nMC = 5000, nBKDE = 5000, nKDE = 5000)
``` |

### Arguments

`x` |
Matrix or data frame composed of three columns: times or number of cycles, crack lengths and specimen number. |

`aF` |
Critical crack length for which the material failure is produced. |

`T_c` |
Censoring time or frequency. |

`method` |
A string of characters: "SEP-lme_bkde" (default methodology) indicates that a mixed effects linear regression model is applied to crack growth data and the lifetime density is estimated by bkde method. "SEP-lme_kde" indicates that a mixed effects linear regression model is applied to crack growth data and the lifetime distribution is estimated by kde method. "PB-nlme" indicates that a mixed effects nonlinear regression model is applied to crack growth data and the lifetime the parameters are estimated maximum likelihood methodologies, and the lifetime distribution by Monte Carlo. |

`nMC` |
Number of Monte Carlo estimates, by default 5000. |

`nBKDE` |
Number of bkde estimates, by default 5000. |

`nKDE` |
Number of kde estimates, by default 5000. |

### Details

This function provides a simultaneous fitting of crack growth data corresponding to different specimens when these are subjected to mechanical fatigue efforts. For this purpose, mixed effects linear models (lme) with B-spline smoothing are applied. Since the failure is defined at a specific critical crack length, predictions of material lifetime are obtained assuming the linearized Paris-Erdogan law and the material lifetime distribution is estimated. There are available three different techniques to estimate the lifetime distribution: the binned kernel density estimate (bkde), the kernel estimator for the distribution function (kde) computed by Quintela del Rio and Estevez-Perez (2012), and in addition the parametric method proposed by Pinheiro and Bates (2000) based on mixed effects nonlinear regression (nlme), maximum likelihood and Monte Carlo simulation.

### Value

Return a list with the following values:

`data` |
Data frame with the data corresponding to number of cycles, crack length, and sample. |

`a.F` |
Critical crack length. |

`Tc` |
Censoring time. |

`param` |
Data frame with the estimates of Paris law parameters: C and m |

`crack.est` |
Data frame with time, crack growth estimates, and corresponding sample or specimen. |

`sigma` |
Residual standard deviation. |

`residuals` |
Residuals resulting from the crack length fitting. |

`crack.pred` |
Data frame with time, crack growth predictions out of the experimental time interval, and corresponding sample or specimen. |

`F.emp` |
Data frame with the empirical lifetime distribution and the corresponding time: time, Fe. |

`bw` |
Bandwidth used in bkde and kde methods. |

`F.est` |
Data frame with the estimated lifetime distribution and the corresponding time: time, F. |

`nBKDE` |
Number of bkde estimates. |

`nKDE` |
Number of kde estimates. |

`nMC` |
Number of Monte Carlo estimates. |

### Author(s)

Antonio Meneses antoniomenesesfreire@hotmail.com, Salvador Naya salva@udc.es, Javier Tarrio-Saavedra jtarrio@udc.es, Ignacio Lopez-Ullibarri ilu@udc.es

### References

Meeker, W., Escobar, L. (1998) Statistical Methods for Reliability Data.
John Wiley & Sons, Inc. New York.

Pinheiro JC., Bates DM. (2000) Mixed-effects models in S ans S-plus.
Statistics and Computing. Springer-Verlang. New York.

Paris, P.C. and Erdogan, F. (1963) A critical analysis of crack
propagation laws. J. Basic Eng., 85, 528.

Quintela-del-Rio, A. and Estevez-Perez, G. (2012) Nonparametric Kernel
Distribution Function Estimation with kerdiest: An R Package for Bandwidth
Choice and Applications, Journal of Statistical Software 50(8), 1.
URL http://www.jstatsoft.org/v50/i08/.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
## Not run:
## Using the Alea.A dataset
data(Alea.A)
x <- Alea.A
## Critical crack length
aF <- 1.6
## Censoring time
T_c <- 0.12
## cracks.growth function applied to Alea.A data
cg <- cracks.growth (x, aF, T_c, method = c("SEP-lme_bkde", "SEP-lme_kde",
"PB-nlme"), nBKDE = 5000, nKDE = 5000, nMC = 5000)
## cracks.growth values using the "SEP-lme_bkde" by default method.
names(cg)
# [1] "data" "a.F" "Tc" "param" "crack.est"
# [6] "sigma" "residuals" "crack.pred" "F.emp" "bw"
#[11] "F.est" "nBKDE"
## End(Not run)
``` |