Elliptic-hyperbolic coordinates for the numerical investigation of elliptical hole problems: A finite difference solution

An elliptical hole can cover several different defect shapes within a material, ranging from thin cracks to round holes. In this study, an elliptical-hyperbolic coordinate system was used to mesh a geometry with traction-free elliptical holes. A new transformation was applied to map the physical domain to a computational Cartesian domain, in which the finite difference technique was applied. A few test cases involving ellipses of various shapes, from an almost round hole to a crack, were analyzed numerically. They were subjected to unidirectional and bidirectional loads far away from the hole location. The numerical and exact analytical results showed a good agreement. The stress intensity factors of a central crack and an edge crack subjected to unidirectional loading were extracted by correlating the normal stress ahead of the crack tip and the vertical crack face displacement behind the crack tip using three terms of the asymptotic expansion. The accuracy of both methods was excellent, with an error <1%. The good agreement between the numerical and analytical results confirms the reliability of the proposed technique and paves the way for addressing more complex geometries and material models.