Bayesian prediction for a jump diffusion process with application to crack growth in fatigue experiments

In many felds of technological developments, understanding and controlling material fatigue is an important point of interest. This article is concerned with statistical modeling of the damage process of prestressed concrete under low cyclic load. A crack width process is observed which exhibits jumps with increasing frequency. Firstly, these jumps are modeled using a Poisson process where two intensity functions are presented and compared. Secondly, based on the modeled jump process, a stochastic process for the crack width is considered through a stochastic differential equation (SDE). It turns out that this SDE has an explicit solution. For both modeling steps, a Bayesian estimation and prediction procedure is presented

Prediction of crack growth based on a hierarchical diffusion model

A general Bayesian approach for stochastic versions of deterministic growth models is presented to provide predictions for crack propagation in an early stage of the growth process. To improve the prediction, the information of other crack growth processes is used in a hierarchical (mixed-effects) model. Two stochastic versions of a deterministic growth model are considered. One is a nonlinear regression setup where the trajectory is assumed to be the solution of an ordinary differential equation with additive errors. The other is a diffusion model defined by a stochastic differential equation (SDE) where increments have additive errors. Six growth models in the two versions are compared with respect to their ability to predict the crack propagation in a large data example. Two of them are based on the classical Paris-Erdogan law for crack growth, and four are other widely used growth models. It turned out that the three-parameter Paris-Erdogan model and the Weibull model provide the best results followed by the logistic model. Suprisingly, the SDE approach has no advantage for the prediction compared with the nonlinear regression setup

BaPreStoPro: an R package for Bayesian prediction of stochastic processes

In many applications, stochastic processes are used for modeling. Bayesian analysis is a strong tool for inference as well as for prediction. We here present an R package for a large class of models, all based on the definition of a jump diffusion with a non-homogeneous Poisson process. Special cases, as the Poisson process itself, a general diffusion process or a hierarchical (mixed) diffusion model, are considered. The package is a general tool box, because it is based on the stochastic differential equation, approximated with the Euler scheme. Functions for simulation, estimation and prediction are provided for each considered model

Bayesian prediction for stochastic processes

In many fields of statistical analysis, one is not only interested in estimation of model parameters, but in a prediction for future observations. For stochastic processes, on the one hand, one can be interested in the prediction for the further development of the current, i.e. observed, series. On the other hand, prediction for a new series can be of interest. This work presents two Bayesian prediction procedures based on the transition density of the Euler approximation, that include estimation uncertainty as well as the model variance. In a first algorithm, the pointwise predictive distribution is calculated, in a second, trajectories will be drawn. Both methods will be compared and analyzed with respect to their advantages and drawbacks and set in contrast to two commonly used prediction approaches

Modelling wear degradation in cylinder liners

We present and discuss a stochastic model describing the wear process of cylinder liners in a marine diesel engine. The model is based on a stochastic differential equation and Bayesian inference is illustrated. Corrosive action and measurement error, both quite negligible, are modeled with a Wiener process whereas a jump process is used to describe the contribution of soot particles to the wear process. The model can be used to forecast the wear process and, consequently, plan condition based maintenance activities. In the paper, we provide a critical illustration of the mathematical and computational aspects of the model. We propose a strategy that, implemented for simulated and real data, allows for stable parameter estimation and forecasts

Resistance to fatigue and prediction of lifetime of wire tendons cast into concrete up to 10^8 cycles

Usually for verification of compliance, the fatigue resistance of prestressing steel is determined from tests of naked specimens at 2 million cycles. However, for design the fatigue resistance of tendons cast into concrete, is substantially lower. To verify the resistance of existing older prestressed concrete bridges and for the design of new bridges, S-N curves of prestressing steel in curved steel ducts embedded into concrete are needed. In bridges, the load cycles due to heavy vehicles may rise up to about 10E8 cycles or even more. Previous tests with curved tendons in steel ducts primarily cover a range of up to about 20 million cycles. Thereby no real endurance strength has been estimated jet. Hence the S-N curves given in Eurocode 2 and Model Code 2010 are defined hypothetically for a range from 10^6 up to 10^8 and are not based on test results. The reason is that experimental investigations in a range up to 10 8 cycles are very expensive and also demand a very long duration. Essential progress results from the development of an optimized test setup that allows a frequency of 10Hz for the applied load cycles. Therewith, the experimental investigations up to 10^8 cycles have been done by means of prestressed concrete beams with embedded curved tendons in steel ducts. Furthermore, procedures to also forecast the lifetime in the case of very low stress ranges respectively the remaining lifetime of a running test had been developed in conjunction with an interdisciplinary research project. The procedures are based on refined statistical analysis of the extensively measured data including increase of crack width, strains, sound emission etc. Additionally the analysis of the latter leads to some interesting new perceptions

A microstructure based fatigue life prediction framework and its validation

Fatigue crack initiation in polycrystalline materials can be attributed to various mechanistic and microstructural features acting in concert like the elastic stress anisotropy, plastic strain accumulation, resolved shear stress, normal stress, slip-system length, and grain boundary character. In nickel-base superalloys, fatigue cracks tend to initiate near twin boundaries. The factors causing fatigue crack initiation depend on the material’s microstructure, the variability of which results in the scatter observed in the fatigue life. In this work, a robust microstructure based fatigue framework is developed, which takes into account i) the statistical variability of the material\u27s microstructure, ii) the continuum scale complex heterogeneous 3D stress and strain states within the microstructure, and iii) the atomistic mechanisms such as slip-grain boundary (GB) interactions, extrusion formations, and shearing of the matrix and precipitates due to slip. The quantitative information from crystal plasticity simulations and molecular dynamics is applied to define the energy of persistent slip bands (PSB). The energy of a critical PSB and its associated stability with respect to the dislocation motion is used as the failure criterion for crack initiation. This unified framework helps us gain insights on why fatigue cracks tend to initiate at twin boundaries. In addition to that, the computational framework links variability in material’s microstructure to the scatter observed in fatigue life