Effective elastic modulus of heterogeneous peristaic bar of periodic structure

The basic feature of the peridynamic model (introduced by Silling: J. Mech. Phys. Solids, 2000; 48: 175-209) considered is a continuum description of a material behavior as the integrated nonlocal force interactions between infinitesimal material points. A heterogeneous bar of periodic structure of constituents with the peristatic mechanical properties is analysed. One introduces the new volumetric periodic boundary conditions (PBCs) at the interaction boundary of a representative unit cell (UC) whose local limit implies the known locally elastic PBCs. The discretization of the equilibrium equation for peristatic composite materials (CMs) acts as a macro-to-micro transition of the deformation-driven type, where the overall deformation is controlled. Determination of the microstructural displacements in an accompany with the volumetric PBC allows one to estimate the peristatic traction at the geometrical UC's boundary which is exploited for estimation of the macroscopic stresses with subsequent evaluation of the effective moduli. Introduction of the volumetric PBCs opens the opportunities for systematic generalization of the classical computational homogenization approaches for CMs with the local constitutive laws for the different dimensions and physical phenomena to their peristatic counterparts. In particular, a convergence of effective modulus estimations is demonstrated for both the peristatic composite bar and locally elastic bar.