The Bond-based Peridynamic System with Dirichlet-type Volume Constraint

In this paper, the bond-based peridynamic system is analyzed as a nonlocal boundary value problem with volume constraint. The study extends earlier works in the literature on nonlocal diffusion and nonlocal peridynamic models to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some nonlocal Poincaré type inequalities and compactness of the associated nonlocal operators. It also offers careful characterizations of the associated solution spaces such as compact embedding, separability and completeness. In the limit of vanishing nonlocality, the convergence of the peridynamic system to the classical Navier equations of elasticity is demonstrated.