A FEM-PD coupling based on Arlequin approach to impose boundary conditions in peridynamics

Abstract

The peridynamic approach (PD) is a continuous theory that is well suited for solving damage problems. Because of the nonlocal formulation, PD can predict the response of a material and fracture patterns with high probability in high dynamic processes. In PD, some parameters differ from the continuum formulation and have some deviation in discretized PD systems, such as a horizon. A material constant becomes a parameter dependent on the mesh size. A sticking point, which has to be considered, is that an incomplete horizon at the boundaries results in an unphysical variation of the material's stiffness in these regions. Material points at the boundaries do not have an entire nonlocal neighborhood, meaning the points have fewer bonds and are softer than points within the domain. This leads to the so-called surface effect. The difficulties in applying the classical local initial and boundary conditions happen because of the nonlocal character of the PD. To overcome this problem, several correction techniques have been developed. Nevertheless, a standard method to describe them is not available yet. An alternative approach is the application of the earlier proposed FEM-PD coupling, which can be seen as a local-nonlocal coupling method. The damage-free zones are analyzed by the FEM as classical local theory, while the domain where the fracture is expected is modeled with the PD as a nonlocal theory. Consequently, the reduction of the computational effort as well as the imposing of the conventional local boundary conditions, is achieved. The coupling method is based on the Arlequin method-an energy-based procedure where the energy of a system is found as a weighted average of both systems. The mechanical compatibility in the overlapping zone of both domains is reached by implementing constraints with the help of the penalty method. In the paper at hand, the focus is on imposing BCs. The proposed method is applied to both static and dynamic applications. The accuracy and convergence behavior is evaluated by analyzing test examples.