Application of Complex Variable Methods for Fatigue Sensitivity Analysis

Complex variable, numerical differentiation techniques have proven useful in many fields of engineering analysis. Complex Taylor series expansion and Fourier differentiation are two such complex variable methods. This paper adapts the use of both complex Taylor series expansion and Fourier differentiation for fatigue sensitivity analysis. The sensitivity of the number of cycles to failure to input parameters and initial conditions has been determined by traditional numerical differentiation techniques as well as through complex variable methods. Both complex Taylor series expansion and Fourier differentiation have been found to be more accurate and stable than traditional central differencing. Nomenclature h = step size or sampling radius of numerical differentiation an = Fourier series coefficients N = number of terms in truncated Fourier series co = initial crack length ao = initial crack depth Δc = step size of the differential equation solver Nf = number of cycles to failure C1 = Paris constant for region I C2 = Paris constant for region II m1 = Paris exponent for region I m2 = Paris exponent for region II da/dN = crack propagation rate ΔK = stress intensity factor i = square root of negative one I