Extension of nonlocal problems with local boundary conditions from 1 dimension to 2 and 3 dimensions

Peridynamics (PD), a nonlocal extension of continuum mechanics, employs an integral based convolution as the governing operator. In Rn, Beyer et al. [2] showed that the PD governing operator is a bounded function of the classical (local) operator. In 1D, Aksoylu et al. [1] generalized the PD formulation to a bounded domain using a convolution operator based on Hilbert bases, which gives rise to a infinite sum. This way, local Boundary Conditions (BC) are incorporated to nonlocal theories through Hilbert bases of the classical operator with the chosen BC. An integral representation of the infinite sum is very useful, as it allows for a convenient numerical implementation. We extend the results in [1] to 2D and 3D and provide integral representations of the governing operators employing antiperiodic and periodic BC. A direct integral representation of the Neumann BC and Dirichlet BC are involved. Instead, in 1D, a construction, called as simple convolutions, was given [1] to obtain Neumann BC and Dirichlet BC, using antiperiodic and periodic BC. We also extend this construction to 2D and 3D. In addition, we provide explicit expressions of the corresponding regulating functions.