# Applications of Neumann-type nonlocal boundary condition on a nonlocal spring model

Project leader: Professor Logan Stokols
Project manager: Karim Shikh Khalil
Team members: Billy Cao, Alvaro Gonzalez, Erik Novak, Ra Pryor

In our project, we explored the applications of nonlocal operators in peridynamics. Peridynamics gives a non-local description of the behavior of particles and the deformation of bonds between particles in a solid. So, to understand the behavior of a particle, our model takes the positions of nearby particles around it into consideration. The goal of our project was to create an equation that can model the behavior of particles and the bonds between them, thought of as springs, when a fixed force is applied at the edge of our object. We also aimed to use numerical processes on our model so that we could corroborate our equation with known real world behavior. In our project we were able to derive a nonlocal operator, define and apply the inclusion of stress forces on the boundary of an object, and create a digital simulation of them for specific peridynamic spring model behavior. For the simulation, we approximate the object and its boundary through an evenly spaced lattice, and computed time-steps of effect on each lattice point over a period of time.

Peridynamics uses an integral operator to model interparticle bond forces, assuming that all particles are initially bonded to all other particles in a fixed radius called the horizon. We used a standard Hooke’s Law model for internal bonds, and modeled bonds as permanently breaking when the force exceeds an empirically-measured constant threshold. We created an integral equation for a nonlocal spring model by deriving a nonlocal integral operator which implements a nonlocal Neumann boundary condition that is linear in the normal direction, and used integral additivity to the standard Hooke’s Law model while approximating the displacement function outside the domain to derive a nonlocal operator that incorporates the nonlocal Neumann boundary condition. We then created a digitized model to represent the position of particles and their bonds by using an evenly-spaced mesh-based discretization of any convex object and then running an Euler numerical ODE algorithm on each mesh node to model the object’s macro behavior.