An improved asymptotic peridynamic plate theory

The peridynamic theory extends the classical continuum theory by allowing material points to interact with other material points separated by finite distances. This feature enables direct application to discontinuities, such as cracks, making the peridynamic theory particularly valuable in fracture mechanics. This work improves the bond-based peridynamic plate theory of [1], which employs a through-thickness expansion of the displacement field using the lower lateral surface as the reference plane. Whereas in the original formulation the governing equations are derived by evaluating the equation of motion at the reference and top surfaces, in this work arbitrary positions are considered along the thickness, yielding both zeroth- and first-order approximations of the displacement field. Computational results obtained from these two theories reveal numerical instabilities, motivating a further improvement by considering the midsurface of the plate as the reference plane. Because the integration of odd terms of the expansion vanishes, a second-order approximation of the displacement field is also considered. The theories together with a prototype microelastic brittle material model are then applied to formulate two fracture mechanics problems. In the first problem, the three brittle plate models yield similar results for small plate thicknesses, as expected from the peridynamic theory of membranes. For large enough thickness, however, the first two models predict very large displacements after a relatively small simulation time, and the third model predicts crack patterns that are consistent with those found for small thicknesses. In the second problem, the third brittle plate model predicts numerically stable results that are physically plausible.