Multiresolution analysis of a micromechanical model

Thin-film interconnects subjected to combined thermomechanical fatigue (TMF) and corrosion are considered. Thin-films are composed of polycrystalline materials, and grain boundaries (GBs) have a critical role in the deformation and failure of such materials. Intergranular void formation in a thin metal conductor occurs as a result of both electro-migration and stress-induced diffusion. In our theoretical model, the presence of voids in the interconnect serve as a potential site for corrosion fatigue cracking (CFC), where hydrogen diffusion occurs under a hydrostatic stress field near crack tips or notch roots and hydrogen concentration reaches a saturated value at notch tips. In effect the presence of hydrogen atoms decreases the surface energy of the film, which in turn causes crack initiation in front of the notch root. We consider that the formation of persistent slip bands (PSBs), which are narrow bands of highly localized cyclic strain is the mechanism of fatigue crack initiation along the GBs. We give a new expression for the Gibbs free energy change, which accounts for the reduction of the surface energy of metal film with cumulative increment of hydrogen intruded into the film. The micromechanically based governing equations consist of an advection-diffusion equation, which explains the time evolution of hydrogen concentration is coupled with the singular integral equation for the dislocation density distribution. The multiple scale reproducing kernel particle method (RKPM) in conjunction with multiresolution analysis provide us with a powerful tool for solving the governing evolutionary partial differential equation (PDE). The integral equation is a Fredholm type of the first kind with generalized Cauchy kernel and a bounded Fredholm kernel. We will solve the coupled advection-diffusion equation and the singular integral equation by simultaneous application of the RKPM and a Gauss-Jacobi integration technique, respectively.