Quantificação da incerteza do modelo randômico de McEvily via metodologia fast crack bounds - Monte Carlo

Abstract

The fracture mechanics studies the behavior of cracks in order to understand and predict their propagation until the fracture, in order to avoid catastrophic accidents, since about 80% of failures in industries occur due to fatigue that is caused by cracks. In general, the predictions made are based on mathematical models. The high complexity of structural analysis problems has encouraged engineers to resort to numerical methods such as finite element methods, finite differences or boundary elements to quantify the uncertainty, since normally some variables and conditions of the analyzed problem are unknown. The objective of this work is to quantify the uncertainty of crack propagation in the McEvily model via the Fast Crack Bounds (FCB) methodology proposed by Avila et al. (2016). The uncertainty quantification consists of to obtain quotas for the estimators of the statistical moments of the “Crack Size” stochastic process using the FCB method and Monte Carlo simulation. The FCB method focuses to obtain quotas lower and upper (functions) for the crack size function, these dimensions “envelop” the numerical solution of Runge-Kutta of order 4. The quotas are obtained by from the Initial Value Problem (IVP) of McEvily crack propagation, through the Taylor series retaining the second order term with Lagrange’s remainder. After to define the quotas, the MATLAB software is used to execute the implemented algorithms to quantify the uncertainty. The results generated in MATLAB are the estimates of the first and second statistical moments, as well as their deviations and the computational times between the quotas and the numerical solution RK4. These theoretical results are later presented and analyzed in the form of tables and graphs.