A gamma process model for the analysis of fatigue crack growth data Maurizio Guida a , Francesco Penta b,⇑ a Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, 84084 Fisciano, SA, Italy b Department of Industrial Engineering, University of Naples ‘‘Federico II’’, 80125 Naples, Italy article info Article history: Received 6 November 2014 Received in revised form 19 March 2015 Accepted 13 May 2015 Available online 22 May 2015 Keywords: Gamma process Stochastic fatigue crack growth Lifetime prediction R-ratio effect Reliability abstract In this paper, the time to reach any given crack size in fatigue testing is directly modeled as a stochastic process. In particular, a gamma process with non-stationary independent increments is assumed for each specimen, where the shape parameter is a suitable function of the crack length. Then, the variability across specimens is accounted for by assuming that the scale parameter is a gamma random variable, resulting in simple mathematical forms for the distribution of service time, its mean and variance. The correlation between the Paris law parameters C and m is also revisited and some useful results are given. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Although since a long time principles and methods of the Damage Tolerance Design Philosophy have been extensively applied in almost all fields of the industrial design, modeling the material fatigue crack growth properties is still a problem that raises a strong interest among researchers, since the outcomes of the fundamental tasks by which fatigue design is car- ried out (i.e. service failure risk analysis, inspection planning and, eventually, reliability updating based on inspection results) are strongly dependent on the accuracy of the probabilistic model adopted to describe the material behaviour. So far the problem has been faced by approaches essentially phenomenological, inspired by particular features emerged during material testing, with the aim of defining relatively simple models whose parameters could be evaluated by the usual statistical estimation methods. For example, since fatigue cracks are observed to grow intermittently and to go through active and dormant periods, it seemed reasonable to assume that the growth of a crack is due to a sequence of peaks of a random stress process or, equivalently, to a sequence of shocks occurring randomly in time as events of a point stochastic process. This leads to model the phenomenon by jump processes [1–3]: crack length is represented as a sum of non-negative random variables X i , characterising the magnitude of length increments, indexed by a counting process defining the number of increments in time. These models are specified by hypotheses posed on the counting process and the random variables X i . Intermingling characteristic of the families of crack paths experimentally acquired under the same loading conditions has been also modeled by continuous in time stochastic processes. Their governing equations were obtained randomising a deterministic phenomenological crack growth equation, i.e. the Paris law, by introducing (multiplicatively) into it a random process XðtÞ that represents the combined effect of unknown random factors such as temperature, internal stress, material http://dx.doi.org/10.1016/j.engfracmech.2015.05.027 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +39 081 7682451. E-mail address: penta@unina.it (F. Penta). Engineering Fracture Mechanics 142 (2015) 21–49 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech