Consistent Formulations of the Interaction Integral Method for Fracture of Functionally Graded Materials

The interaction integral method provides a unified framework for evaluating fracture parameters (e.g., stress intensity factors and $T$ stress) in functionally graded materials. The method is based on a conservation integral involving auxiliary fields. In fracture of nonhomogeneous materials, the use of auxiliary fields developed for homogeneous materials results in violation of one of the basic relations of mechanics, i.e., equilibrium, compatibility or constitutive, which naturally leads to three independent formulations: “nonequilibrium,” “incompatibility,” and “constant-constitutive-tensor.” Each formulation leads to a consistent form of the interaction integral in the sense that extra terms are added to compensate for the difference in response between homogeneous and nonhomogeneous materials. The extra terms play a key role in ensuring path independence of the interaction integral. This paper presents a critical comparison of the three consistent formulations and addresses their advantages and drawbacks. Such comparison is made both from a theoretical point of view and also by means of numerical examples. The numerical implementation is based on finite elements which account for the spatial gradation of material properties at the element level (graded elements).