Uncertainty quantification of the Collipriest random model by the Fast Crack Bounds and Monte Carlo methodology

Resumo

Mechanical structures are subjected to cyclical stresses and collapse under fatigue conditions. There
are several mathematical models that describe the crack propagation. In general, crack propagation models are
classified by the type of loading, which can have constant stress amplitude or variable stress amplitude. In this
work, the constant stress amplitude model proposed by Collipriest will be explored. For many engineering
applications a reliable estimate of crack behavior is required. Therefore, this work presents theoretical results,
which consist of the quantification of uncertainty in the definition parameters in the model used, based on lower
and upper bounds, which envelop the estimators of the first and second order statistical moments of the function
size of crack, based on the Fast Crack Bounds method. These dimensions are polynomials, defined in the
variable number of cycles, which take into account the uncertainties in the parameters that describe the crack
propagation models. The Monte Carlo simulation method will be used to obtain the realizations of the crack size
function from a set of random samples of the characteristic parameters of the Collipriest model. These
achievements will be used to obtain the estimators of statistical moments of crack size. The efficiency of the
bounds for the estimators of the statistical moments of the crack size is evaluated through relative deviation
functions between the bounds and the approximate numerical solutions of the initial value problem that describes
the Collipriest model. In general, the solution of the initial value problem that the crack propagation models
describe is obtained through the use of numerical methods, such as the explicit fourth order Runge-Kutta
method. In this work, a mathematical software will be used for numerical analysis of the solutions of the crack
propagation model of the Collipriest model, therefore an analysis of the computational performance between the
Fast Crack Bounds and Runge-Kutta methods and presented by the computational time, graphs and tables.