Derivation of complete crack-tip stress expansions from Westergaard–Sanford solutions
Résumé
The description of mechanical fields at the vicinity of a bi-dimensional crack-tip can be performed using the classic Williams series expansion. While its general structure is well known, complete expressions are rarely available for specific problems. This article describes and applies a methodology to express complete expansions for four given fracture configurations. The procedure relates Williams series coefficients to those of expanded Westergaard–Sanford complex potentials for modes I and II. Actual expansions of complex solutions for mode I cases are derived using classical complex analysis techniques. Complete closed-form results, four power series and one Laurent series, have been determined with this approach. The correctness of analytical results and series convergence behavior have been conclusively investigated through numerical tests comparing reference complex solutions with truncated series representations. The methodology can be applied straightforwardly to new fracture configurations where complex solutions are known. Complete closed-form expressions can be used to derive, test and improve numerical and experimental techniques involving higher order terms in crack-tip expansions.