On the application of numerical differentiation methods to the determination of the fatigue crack growth rate

Abstract

Using a sample of test results from 68 compact eccentric tensile specimens made of titanium alloys, nickel alloys and steel, the effect of the choice of numerical differentiation method (secant method and the method of differential polynomials on three, five and seven points) used to calculate the fatigue crack growth rate on characteristics of the linear section of the kinetic diagram of the fatigue failure. The purpose of the study is to determine the advantages, disadvantages and consistent patterns of the considered methods. The coefficient of determination R 2, integral criterion χ which characterizes the difference between the predicted and actual number of cycles corresponding to the section of stable crack growth, and correlation between the logarithms of the Paris constants for alloys of the same class were used as criteria for the correct choice of the method of numerical differentiation. The main results and conclusions of the study: the use of the method of differential polynomials over three points compared to the secant method slightly increases the correlation between the logarithms of the fatigue crack growth rate and the range of the stress intensity factor (an increase in R 2) and increases the difference between the calculated and experimental number of cycles corresponding to stable crack growth (an increase in χ). However, when determining the fatigue crack growth rate by the method of differential polynomials for five and seven points, a more significant smoothing of the experimental data is observed, accompanied by a significant increase in R 2 and a decrease in χ; proximity to zero of the integral accuracy parameter χ is a necessary but not sufficient criterion for good agreement between the test result and the mathematical model that describes it, while the combination of parameters χ and R 2 uniquely forms this criterion; the choice of the method of numerical differentiation does not affect the correlation of the logarithms of the constants of the Paris equation.