Two-Phase Peridynamic Elasticity with Exponential Kernels. II: Bending, Buckling, and Vibration of Beams

This paper is devoted to the static and vibration behavior of a two-phase peridynamic (or relative rotation-based integral approaches) Euler–Bernoulli beam with exponential kernels. This two-phase peridynamic beam theory combines a local and a purely peridynamic phase and coincides with the so-called physically based nonlocal beam approach. For the considered exponential kernel, the two-phase peridynamic Euler–Bernoulli beam problem is reformulated as a two-length-scale differential model. The bending, buckling, and vibrations of the two-phase peridynamic Euler–Bernoulli beam are studied in closed-form solutions by solving an equivalent sixth-order differential eigenvalue problem. Results are also presented for the bending wave dispersive behavior in the infinite two-phase peridynamic Euler–Bernoulli beam. The two-phase peridynamic model is associated with the softening behavior of the small length-scale effects, both for static and dynamic analyses. It is also shown that the peridynamic problem, both for the finite and the infinite beam, can be reformulated as a two-phase curvature-driven nonlocal model for some specified chosen kernels. The exact analytical solutions derived, both in statics and in dynamics for the finite peridynamic beam, are corroborated by some complementary numerical investigations, based on the discretization of the peridynamic energy functional, for a consistent derivation of the associated nonlocal stiffness matrix.