Two-Phase Peridynamic Elasticity with Exponential Kernels. I: Statics and Vibrations of Axial Rods

This paper explores the static and vibration behavior of a two-phase peridynamic rod (or relative displacement-based integral rod model) with exponential kernels. The two-phase peridynamic theory combines a local and a purely peridynamic phase and coincides with the so-called physically based nonlocal approach. For the considered exponential kernel, the two-phase peridynamic problem is reformulated as a two-length-scale differential model. Exact solutions are derived for the behavior of a finite two-phase peristatic rod in pure tension, which highlights a boundary layer phenomenon. The free vibration of the two-phase fixed-fixed peridynamic rod is solved from an equivalent fourth-order differential eigenvalue problem. The axial wave dispersive behavior is also analytically studied for the infinite two-phase peridynamic rod, equivalent to a two-phase Eringen’s nonlocal strain-driven model. The two-phase peridynamic rod model is associated with the softening behavior of small length-scale effects, both for static or dynamic analyses. It is also shown that the peridynamic problem can be reformulated as a two-phase strain-driven nonlocal model for some specified chosen kernels. The exact analytical solutions derived, both in statics and in dynamics for the finite peridynamic rod, are corroborated by complementary numerical investigations, based on the discretization of the peridynamic energy functional, for a consistent derivation of the associated nonlocal stiffness matrix.