Computation of two dimensional mixed-mode stress intensity factor rates using a complex-variable interaction integral

The well-known interaction integral, also known as the M-integral or I-integral, is a method to compute the mixed-mode stress intensity factors (SIFs) for fracture mechanics problems. The capabilities of the M-integral are extended here to compute derivatives of the SIFs with respect to the crack extensions for an isotropic linear elastic material under static loading. These derivatives were calculated using the complex Taylor series expansion (CTSE) numerical differentiation method. The derivatives of the auxiliary fields are calculated by applying CTSE directly to the M-integral formulation. The derivatives of the actual displacement fields are computed using a hypercomplex-variable finite element method (ZFEM) that also employs CTSE. SIF rates with respect to all crack tips are computed in a single analysis. The method is general and can be easily extended to other two-dimensional loading scenarios and different material models in a straightforward manner through the use of the appropriate auxiliary fields. The complex-variable M-integral was implemented within the commercial finite element software Abaqus through a user-defined element subroutine (UEL). Numerical examples demonstrate the high accuracy of the method.