Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

The bond-based peridynamic system with Dirichlet-type volume constraint

Tadele Mengeshaa1 and Qiang Dua1

a1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (;


In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local 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 non-local Poincaré-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

(Received August 16 2012)

(Accepted January 11 2013)