Peridynamic model of vibrations in a two-dimensional periodic structure

Unlike classical continuum mechanics, where the linearized model is described by partial differential equations, the peridynamic model leads to an integro-differential equation with a non integrable kernel. The proposed method belongs to the category of nonlocal models, as particles separated by a finite distance can interact with each other. This allows the description of processes occurring in structures with cracks and discontinuities. Fracture is considered a natural result of deformation arising from the equation of motion and the constitutive model. Consequently, modeling crack growth in the peridynamic framework does not require additional data or equations that would be necessary in traditional fracture mechanics to determine crack initiation. The study examines a peridynamic model on a two-dimensional periodic structure related to graphene -a two dimensional allotropic form of carbon. It can be thought of as a single plane of layered graphite separated from the bulk crystal. Estimates suggest that graphene possesses high mechanical stiffness and record-breaking thermal conductivity. Its exceptionally high charge carrier mobility, which is the highest among all known materials (for the same thickness), makes it a promising material for various applications, particularly as a future foundation for nanoelectronics. The work investigates a hypersingular integro-differential equation describing oscillations in a two-dimensional periodic structure. A transformation has been found that allows the regularization of the singular integral operator involved in the equation. This made it possible to obtain a unique solution to the problem in the introduced Sobolev space.