Peridynamic Method

Continuum mechanics has been very effective in solving many mechanical problems and is widely used in the field. However, the classical continuum mechanics theory faces challenges when dealing with fracture problems, where the displacement field is discontinuous. Peridynamics is a new theory of continuum mechanics, which addressed this fundamental issue. In peridynamic theory in contrast to the classical continuum mechanics theory, the spatial derivative of the displacement field is replaced with an integral of forces exchanged between particles inside an influence domain, called horizon. In this article, we introduce three versions of the peridynamic theory, the bond-based, the ordinary state-based and the non-ordinary state-based. We will also explain the concept of dual horizon, which extends the capabilities of the computational implementation of the theory. We will then present some challenging fracture problems and their solutions with the peridynamic theory. These include predicting fracture in isotropic and anisotropic materials under static and dynamic loading conditions. We show that the peridynamic theory can accurately predict the fracture force and patterns in a wide range of materials, but we also show some limitations of the theory, which need to be addressed in future work.