No abstract

The peridynamic theory is an extension of the classical continuum mechanics theory. The peridynamic governing equations involve integrals of interaction forces between near particles separated by finite distances. These forces depend upon the relative displacements between material points within a body. On the other hand, the classical governing equations involve the divergence of a tensor field, which depends upon the spatial derivatives of displacements. Thus, the peridynamic governing equations are valid not only in the interior of a body, but also on its boundary, which may include a Griffith crack, and on interfaces between two bodies with different mechanical properties. Near the boundary, the solution of a peridynamic problem may be very different from the classical solution. In this work, we investigate the behavior of the displacement field of a unidimensional linearly elastic bar of length L near its ends in the context of the peridynamic theory. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young's modulus E in the classical theory and is given by different expressions found in the literature. We find that, depending on the expression of C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in the classical linear elasticity. In spite of the above, we show that the peridynamic displacement field converges to its classical counterpart as a length scale, called peridynamic horizon, tends to zero.

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

No abstract

No abstract