A generalized model for static fatigue lifetime evaluation of optical fibers to reduce the model uncertainty

Highlights

• A generalized lifetime model is proposed to expand the model space and encompass reasonable possibilities.

• A suitable model is determined in the expanded model space to address model uncertainty.

• It is shown that the proposed framework has better performance compared with existing models on three datasets.

Abstract

The mechanical reliability of silica-based optical fibers declines especially under the combination of applied stresses and severe chemical environment in service. It is of great concern to assess the mechanical reliability and lifetime based on the short-term accelerate testing and the relevant mechanism. The long-term lifetime prediction of optical fibers is sensitive to the form of kinetic model, yet no fair and common agreement has been reached on the best model. This study proposes a generalized lifetime model to search the best model in an expanded model space and hence can reduce model uncertainty. Following that, a procedure using the maximum likelihood estimation and the likelihood ratio test is conducted to determine the model structure. Furthermore, three sets of static fatigue lifetime data are used to illustrate the validity and superiority of the proposed framework. The result demonstrates that the proposed framework can reduce the uncertainty in model choice and is readily applicable to evaluation of static fatigue lifetime of optical fibers.

Introduction

Optical fibers are applied in the field of both telecommunication and sensing for their excellent properties of capacity, anti-interference and flexibility [1,2]. The mechanical reliability and lifetime prediction of optical fibers is of great concern and has been a topic of research [3]. It is generally acknowledged that the surface flaws on silica-based optical fibers grow under the combination of mechanical stresses and chemical environment, also known as static fatigue, leading to the delayed fracture of an initially intact optical fiber [4].

Several kinetics models have been proposed to describe the growth process of flaws, including the empirical and mechanism based models, to describe the crack growth behavior of the flaws and then predict the lifetime accurately. Empirical power law model is the most commonly used model to describe the dependence of the crack growth velocity on stress intensity at the crack tip. The existing industry standard adopts this kind of form due to its mathematical concision [5]. The power law model is not only convenient for integral for both dynamic and static fatigue loads but also able to combine with Weibull distribution analytically to incorporate the statistical effect. However, the power law model has no physical implications and its assumption of constant pre-exponent is inconsistent with experimental results where the pre-exponent was found dependent on stress [6]. Therefore, several kinetics models based on chemical dynamics have been proposed in the literature. Wiederhorn and Bolz [7] proposed a model in exponential form by assuming that the tensile stress at the crack tip reduced the activation free energy and increased the reaction rate via an activation volume. Nevertheless, there were some imprecise deviations since the localized tensile pressure was incompatible with the assumption of the uniform pressure at the crack tip. A more rigorous model proposed by Lawn [8] considered both the breaking and forming processes of silicon-oxygen bonds during the fracture and the crack propagation modified the strain energy density thus affecting the activation energy. The models of Wiederhorn and Lawn in exponential form cannot be solved analytically for the fatigue equations, hence not favored compared with the power law model in general. There are also other alternative models which are modifications of the three mentioned models [9,10].

Considering that we do not have profound understanding on the degradation mechanisms and the kinetics, it is reasonable to employ any feasible models [11]. However, different models would give quite different results for data fitting and long-term lifetime prediction. On the one hand, the three models have different fitting to test data due to their respective descriptions of fracture mechanism and the explanations of model parameters. For example, the power law model best matches the static and dynamic fatigue test results for both bare and coated fiber in buffer solution whereas the Wiederhorn's model gives the best description for the same test in ambient air [12]. Similarly, the Lawn's model with the most physical meaning is favored by the observation of slow crack growth in fused silica whereas it gives a poor fit to fatigue data in ambient environment [13,14]. On the other hand, lifetime prediction by extrapolating from experimental data is sensitive to the model forms. The power law model gives the longest lifetime prediction, whereas the Lawn's model gives the shortest predicted lifetime under the same condition [11]. Therefore, the model uncertainty exists in long-term life prediction of optical fibers and may result in heavily biased inference if it is not carefully considered.

Many efforts have been made to reduce the model uncertainty, including the methods of alternate hypotheses (or model averaging) and adjustment factor [15]. The former one combines all the candidate models with the mixture of respective probabilities, and it has been fully extended to the Bayesian model averaging by integrating the model prior knowledge and the likelihood function of the experimental data [16]. The latter one updates one of the best model with information from the other models. One of the problems is that the methods assume the true process is among the list of candidate models [17]. However, it is difficult to interpret and controversial in practical application because the truth is typically thought to be more complex than any model under consideration [18]. Instead of averaging over a discrete set of models, it is recommended to reduce the model uncertainty within a single model encompassing all reasonable possibilities [19]. The candidate models are the subsets of the model space of a generalized model and it greatly increases the number of possible models. Then the way to reduce the model uncertainty is to search the best model in a continuous model space rather than adjusting the model weight and averaging among a finite number of discrete models. Therefore, in this work, a generalized lifetime model is proposed to incorporate the characteristics of different models and a corresponding procedure is put forward to simplify the proposed model for various conditions.

The remainder of the study is organized as follows: Section 2 reviews the mentioned models and gives a brief analysis. Section 3 develops the generalized model according to the form features of the three models. Meanwhile, the procedure to simplify the proposed model via the likelihood-ratio test is introduced according to the relationships among these models. Following that, three examples are given to verify the validity of the method in Section 4. Section 5 summaries the study with some conclusions.