Continuous and Discontinuous Finite Element Methods for a Peridynamics Model of Mechanics

Peridynamics is a recently developed theory of solid mechanics that replaces the partial differential equations (PDE) of classical continuum theories with integro-differential equations (IDE). We apply Finite Element Methods (FEM) as well as Discontinuous Galerkin Methods (DGM) to implement the peridynamic model. Since the integro-differential equations remain valid in the presence of discontinuities such as cracks, it has the potential to model fracture and damage with great generality. We use piecewise constant and discontinuous piecewise linear functions in regions where discontinuities may appear and continuous piecewise linear function in areas where the solutions is smooth and investigate how to combine these two methods. We are also interested in the choice of the horizon radius to implement the peridynamic model more accurately; cases when radius is fixed as a constant or as a function of grid distance are tested. Some theoretical analysis, i.e. existence and uniqueness of the solution and error estimation as well as numerical results for different cases are given.