Reliability Analysis of Processes with Moving Cracked Material

TL;DR

This paper analyzes the reliability of moving cracked elastic materials under tension, considering stochastic crack occurrence and tension models, providing probabilistic formulas and numerical examples for practical applications.

Contribution

It introduces a stochastic model for crack occurrence and tension variation, deriving explicit reliability formulas and employing Monte Carlo simulation for complex cases.

Findings

Reliability formulas for systems with constant and variable tension.

Explicit first passage time results for Ornstein-Uhlenbeck processes.

Numerical examples demonstrate applicability to printing presses and paper materials.

Abstract

The reliability of processes with moving elastic and isotropic material containing initial cracks is considered in terms of fracture. The material is modelled as a moving plate which is simply supported from two of its sides and subjected to homogeneous tension acting in the travelling direction. For tension, two models are studied: i) tension is constant with respect to time, and ii) tension varies temporally according to an Ornstein-Uhlenbeck process. Cracks of random length are assumed to occur in the material according to a stochastic counting process. For a general counting process, a representation of the nonfracture probability of the system is obtained that exploits conditional Monte Carlo simulation. Explicit formulae are derived for special cases. To study the reliability of the system with temporally varying tension, a known explicit result for the first passage time of an Ornstein-Uhlenbeck process to a constant boundary is utilized. Numerical examples are provided for printing presses and paper material.