Analytical solutions of peristatics and peridynamics for 3D isotropic materials

Abstract

In a recent paper, Mikata (2019) has established two explicit analytical governing equations of peridynamics for 3D isotropic and orthotropic materials. In the present paper, general analytical solutions of peristatics and peridynamics for 3D isotropic peridynamic materials are obtained in infinite media for an arbitrary body force density, and arbitrary initial conditions in the case of peridynamics. In addition, peristatic and peridynamic Green's functions corresponding to a point load are also obtained. Classical limit of the peristatic and peridynamic Green's functions in the limit of the generalized material horizon δ going to 0 (δ→0) is analytically discussed in detail for one of the most commonly used nonlocality functions with a finite material horizon with a sharp cut-off. It is analytically and explicitly shown that the peristatic and peridynamic Green's functions converge to the elastostatic and elastodynamic Green's functions, respectively, as the generalized material horizon δ goes to 0 (δ→0). Numerical results are obtained and discussed for the peristatic Green's function and a 3D peristatic displacement field under a 1D-like loading. The governing equation investigated in the present paper is the first explicit analytical governing equation of peridynamics for 3D isotropic materials, which is comparable to Navier-Cauchy's equation. Thus, this equation opens up various ways to investigate peridynamics in 3D isotropic materials in an exact analytical fashion, which was not possible before.