From Muskhelishvili potentials to Williams crack-tip expansions

The Williams expansion provides a fundamental asymptotic description of crack-tip stress fields in linear elastic fracture mechanics, where the leading singular term governs K-dominance and the non-singular and higher-order terms control crack-tip constraint and deviations from near-tip asymptotics. Despite their importance, a systematic analytical route for extracting the full Williams expansion directly from full-field solutions remains lacking. Here we establish a direct connection between Muskhelishvili complex potentials and the Williams expansion for two-dimensional elastic crack problems. By matching the asymptotic structure of complex potentials near crack tips, explicit expressions for T-stress are derived and recurrence relations for higher-order Williams coefficients are obtained. This formulation provides a systematic procedure for constructing complete crack-tip asymptotic fields from full-field elasticity solutions. The framework is validated using classical crack configurations and extended to kinked and branched cracks, demonstrating its generality and advantages over perturbative or fitting-based approaches. The resulting higher-order expansions enable quantitative assessment of the spatial extent of K-dominance and clarify when non-singular terms must be retained. Motivated by recent gap-test experiments highlighting the role of crack-parallel stress in fracture behavior, the method is further applied to relate macroscopic crack-parallel stress to local T-stress in crack-tip fields. By bridging Muskhelishvili potentials and the Williams expansion, the present work provides a systematic analytical basis for evaluating crack-tip constraint and higher-order effects in fracture problems involving complex crack geometries, deviations from K-dominance, and multiphysics coupling.