Homogenization of the 1D Peri-static/dynamic Bar with Constant Micromodulus

Abstract

Because of the nonlocal interparticle forces inherent in peridynamics, surface, boundary, and end effects appear in 3D, 2D and 1D body problems, respectively. In certain situations, the effect is seen as a disturbance, and various efforts, mostly centering on 2D and 1D problems, have been made to reduce it. A simple method has been derived to remove the end effects in a 1D body by homogenizing the body. When a certain body type, common in practice, is homogenized, its linear elastic behavior, independent of the interparticle force range and with a finite number of material points, in the limit infinite, is identical to that of a corresponding classical continuum mechanics body.

Homogenization of the 1D Peri-static/dynamic Bar with Triangular Micromodulus

“…However, the improvements at edges and corners were smaller than for other existing methods, as seen in Le and Bobaru [14]. Even though the method can be applied in 1D, the direct calibration of the equilibrium equations presented in Eriksson and Stenström [1], and in this work, is simpler and more straightforward.…”

“…where u i is the only component of the displacement vector u of the material point situated at x i . Certain characteristic properties of the 1D peristatic bar are discussed in Eriksson & Stenström [1]. Like the previous case, a 1D bar is loaded in uniaxial tension by equal and opposite forces applied at the end points of the bar.…”

“…[3][4][5]. In a previous work [1], we considered the constant micromodulus. In the present work, we apply the linear, or "triangular," micromodulus.…”

“…For 1D bodies, with the micromodulus assumed to be constant, c(ξ ) = c 1 = 2E/(Aδ 2 ), and with the micromodulus assumed to be triangular, c 1 = 6E/(Aδ 2 ), which have been obtained by Bobaru et al [4] for a continuous bar, i.e., for an infinite number of points N H x inside the region H x . The constant micromodulus c is independent of N H x only in two special cases: a) for N H x infinite and b) for N H x finite, only in the case where the horizon divides an element in two equal parts and the element weight is halved [1]. This feature does however not hold for the triangular micromodulus, which is independent of N H x only if N H x is infinite, but is always a function of N H x when N H x is finite.…”

“…In a previous work [1], we derived the parameters necessary to remove the end effects in 1D bars with a constant micromodulus c, or to homogenize such bars. The micromodulus is the elastic stiffness of the bond between two material points.…”