Dual Bubble-Enriched SFEM Approaches for Locking-Free Analysis of Structures Under Cyclic Loads

Shakedown analysis, a robust technique for assessing fatigue failure in cyclically loaded structures, may encounter volumetric locking challenges in incompressible materials under plane strain conditions, particularly with low-order approximations. This study presents dual numerical approaches that pioneer integrating the bubble-enriched edge-based smoothed finite element method (bES-FEM) with conic programming within the shakedown analysis framework. The bubble enrichment enables locking-free performance across arbitrary mesh configurations while enhancing numerical accuracy. Two distinct formulations based on Koiter’s and Melan’s theorems are presented in conic form, with the associated kinematic and static constraints satisfied over the smoothed domains, minimizing problem size. These optimization problems are efficiently solved using the highly effective primal–dual interior-point solver, Mosek. The proposed methods are validated through various benchmark numerical examples featuring diverse geometries and loading combinations, achieving accurate shakedown limits with minimal computational effort. Insights into structural failure mechanisms are derived from plastic dissipation work distributions obtained from the kinematic formulation, corroborated by limit state stress fields from its dual form — the static approach. Furthermore, the ability to predict failure modes — whether due to incremental collapse or alternating plasticity — by analyzing failure mechanisms and interaction diagrams across multiple load ranges, along with identifying transition points between these modes, represents an intriguing finding of this study.