Mixed-mode stress intensity factors computation in functionally graded materials using a hypercomplex-variable finite element formulation

The hypercomplex-variable finite element method, ZFEM, is extended to compute the mode I and mode II energy release rates (ERR) for functionally graded materials. The ERR is computed using an efficient local stiffness derivative approach, L-ZFEM, that computes the derivative of the stiffness matrix at the element level using the highly accurate complex-variable sensitivity method. Mode I and II values are computed using the appropriate perturbation of the surrounding crack tip elements, i.e., perturbations in the self-similar (mode I) and perpendicular (mode II) directions. The energy release rate values are as accurate as the J-integral results. The advantage of this approach is that the derivatives are only required for a small number of elements surrounding the crack tip and no energy conservation integrals are required. In addition, derivatives of the ERR with respect to the FGM material properties are computed by combining the local stiffness derivative approach with a global complex variable formulation, G-ZFEM. This methodology was implemented into the commercial finite element software Abaqus through a combination of Abaqus intrinsic elements and a complex variable user element subroutine (UEL). Numerical results are compared against analytical solutions and other numerical approaches and demonstrate excellent accuracy.