Application of Numerical Differentiation Methods to Determination of the Fatigue Crack Growth Rate

A sample of test results for 68 eccentric tensile specimens of titanium alloys, nickel alloys, and steel is used to assess the effect of the choice of the numerical differentiation method (the secant method and the method of differential polynomials with three, five, or seven data points) for calculating the fatigue crack growth rate on characteristics of the linear portion in the kinetic diagram of fatigue fracture. The purpose of this study is to understand the advantages, drawbacks, and general aspects of the methods in question. The criteria used to ascertain the correctness of the choice of the numerical differentiation method are the coefficient of determination, R²; the integral criterion χ characterizing the difference between the predicted and actual numbers of cycles corresponding to the region of stable crack growth; and correlation between the logarithms of Paris’ equation constants for alloys of a particular class. The results of this study demonstrate that, compared to the secant method, fitting to three data points by the method of differential polynomials slightly improves correlation between the logarithms of the crack growth rate and the stress intensity factor range (increases R²) and increases the difference between the calculated and experimentally determined numbers of cycles corresponding to stable crack growth (increases χ). At the same time, fitting to five and seven points in determining the fatigue crack growth rate by the method of differential polynomials leads to appreciable smoothing of the experimental data, accompanied by a noticeable increase in R² and decrease in χ. The integral accuracy parameter χ being almost zero is a necessary but not sufficient criterion for good agreement between the test results and the mathematical model used to describe them, whereas a combination of the parameters χ and R² unambiguously ensures such a criterion. The choice of the numerical differentiation method has no effect on correlation between the logarithms of Paris’ equation constants.