A Simplified Conservative Runge–Kutta Method for Numerical Prediction of Fatigue Crack Growth

Accurate numerical prediction of fatigue crack growth under cyclic loading is crucial for ensuring the safety and structural integrity of engineering structures. However, classical numerical integration schemes, such as the fourth-order Runge–Kutta (RK4) method, offer high accuracy but can fail to produce conservative crack size predictions when larger integration step sizes are used; as a result, they may underestimate crack growth and compromise safety margins. To address this issue, modified RK4 schemes have been proposed to enforce conservatism, but these approaches often introduce additional computational overhead or involve complex correction terms. In this study, a simplified conservative RK4 algorithm is proposed to enhance computational efficiency without sacrificing the conservative bias (overestimation) in crack growth prediction. The method integrates Paris law for fatigue crack growth with a reduced-order correction mechanism applied at each stage of the standard RK4 process. The performance of the proposed scheme is evaluated by analysing convergence behaviour, relative error, and conservative prediction regions across a range of Paris law exponents (m = 2 to 5). Simulation results demonstrate that the proposed scheme consistently yields upper-bound crack growth predictions across various step sizes, thereby providing an effective balance between structural safety and computational efficiency. This approach is particularly well-suited for high-cycle fatigue analysis, where the large number of load cycles demands efficient computation and a high level of prediction reliability.