Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method

A solution method of an inverse problem is developed to extract cohesive-zone laws from elastic far-fields surrounding a crack-tip cohesive zone. The solution method is named the “field projection method (FPM).” In the process of developing the method a general form of cohesive-crack-tip fields is obtained and used for eigenfunction expansions of the plane elastic field in a complex variable representation. The closing tractions and the separation-gradients at the cohesive zone are expressed in terms of orthogonal polynomial series expansions of the general-form complex functions. The series expansion forms a set of cohesive-crack-tip eigenfunctions, which is complete and orthogonal in the sense of the interaction J-integral in the far field as well as at the cohesive-zone faces. The coefficients of the eigenfunctions in the J-orthogonal representation are extracted directly, using interaction J-integrals in the far field between the physical field of interest and auxiliary probing fields. The path-independence of the interaction J-integral enables us to identify the cohesive-zone variables, i.e. tractions and separations, and thus the cohesive-zone constitutive laws uniquely from the far-field data. A set of numerical algorithms is developed for the inversion method and the results from numerical experiments suggest that the proposed algorithms are well suited for extracting cohesive-zone laws from the far-field data. The set includes methods to find the position and size of a cohesive zone. Further included are discussions on error analysis and stability of the inversion scheme.