Closed‐form solution for the Fatemi‐Socie extended critical plane parameter in case of linear elasticity and proportional loading

Fatigue damage remains a significant issue for both metallic and non-metallic components, being the main cause of in-service failures. Among the different assessment methodologies, critical plane methods have gained significance as they enable identifying the critical location and the early crack propagation orientation. However, the standard plane scanning method used for calculating critical plane factors is computationally intensive, and as a result, it is usually applied only when the component's critical region is known in advance. In the presence of complex geometries, loads, or constraints, a more efficient method would be required. This work presents a closed-form solution to efficiently evaluate a critical plane factor based on the Fatemi-Socie criterion, in the case of isotropic linear-elastic material behavior and proportional loading conditions. The proposed algorithm, based on tensor invariants and coordinate transformation laws, was tested on different case studies under various loading conditions, showing a significant reduction in computation time compared to the standard plane scanning method.