Hypersingular integrodifferential equations and applications to fracture mechanics of homogeneous and functionally graded materials with strain-gradient effects

The focus of this work is to solve crack problems in functionally graded materials (FGMs) with strain-gradient effect. The method used and developed is called hypersingular integral equation method in which the integral is interpreted as a finite part integral, and it can be considered as a generalization of the well-known singular integral equation method. In developing the method, we have derived the exact formulas for evaluating the hypersingular integrals and used Mellin transform to study the crack-tip asymptotics; we have detailed the numerical approximation procedures; also, we have generalized the definition of stress intensity factors (SIFs) under strain-gradient theory and provided formulas for computing SIFs.; Different types of crack problems have been solved: Conventional classical linear elastic fracture mechanics (LEFM) vs. strain-gradient theory; scalar problems (Mode III fracture) vs. vector ones (Mode I fracture); homogeneous materials vs. FGMs; different geometric setting of crack location and material gradation. In particular, we obtain a closed form solution for the crack profile in one simple case—Mode III crack problems in homogeneous materials with the characteristic length ℓ′ responsible for surface strain-gradient term being zero.