The paper by Wang et al. [1] reports calculations of the stress intensity factors for a surface-breaking cylindrical crack in a half-space, subjected to Hertzian contact stresses. The stress intensity factors are computed as a result of solving two coupled Cauchy singular integral equations, formulated by modeling the cylindrical crack as a distribution of normal and shear ring dislocations [2,3].
Our original concern with this analysis arises from its reliance on the Somigliana ring dislocation solution derived by Korsunsky [2,3], to account for the radial displacement jump along the cylindrical surface. The authors may not have been aware that this particular dislocation solution actually corresponds to a radial displacement jump across an exterior disk (perpendicular to the axis of symmetry and bounded by the dislocation loop), and not across a cylindrical surface, as recently noted by Paynter et al. [4,5]. This issue illustrates the fundamental difference in the properties of Volterra and Somigliana dislocations that are characterized by constant and variable Burgers vectors, respectively. A Somigliana dislocation, unlike a Volterra dislocation, induces a stress field that depends on the orientation of the surface of discontinuity [6]. Thus, the kernels corresponding to the radial Somigliana ring dislocations in Eqs. ( 5) and ( 6) of [1] cannot be used for a cylindrical surface. The kernels for a surface-breaking cylindrical crack in a half-space are given in Eq. ( 57) in [5].
A secondary issue is the numerical technique used by the authors to solve the set of coupled Cauchy singular equations. This particular method assumes that the aperture gradient at the crack mouth has a square-root singularity, cf. Eq. ( 19) of [1], in contradiction with the nature of the solution at the free surface. Enforcing a non-existent singular behavior to the dislocation density may lead to some inaccuracy in the stress intensity factors computed at the other crack end and/or require an unnecessary number of Chebyshev polynomials when approximating the dislocation density functions to solve this problem accurately. For a general surface-breaking crack, the choice of a suitable behavior of the crack aperture and its gradient is discussed by Hills et al. [7], who also describe appropriate numerical techniques.
To assess the accuracy of the numerical results reported in [1], we re-analyzed the problem studied by Wang et al. by solving the coupled Cauchy singular equations using either the kernels from Eq. (57) in [5], corrected for a few typographical errors, (solution I) or the kernels from Eqs. ( 5) and ( 6) in [1] (solution II). The integral equations were solved by representing the unknown dislocation densities as the products of shape functions that vanish at the crack mouth but are characterized by a square-root singular behavior at the crack closed end, and smooth functions that are approximated by series of Chebyshev