Untitled

Abstract

A rod or beam is one of the most widely used members in engineering construction. Such members must be properly designed to resist the applied loads. When subjected to anti-plane (longitudinal) shear and torsional loading, homogeneous, isotropic, and elastic materials are governed by the Laplace equation in two dimensions under the assumptions of classical continuum mechanics, and are considerably easier to solve than their three-dimensional counterparts. However, when using the finite element method in conjunction with linear elastic fracture mechanics, crack initiation and its growth still pose computational challenges, even under such simple loading conditions. This difficulty is mainly due to the mathematical structure of its governing equations, which are based on the local classical continuum theory. However, the nonlocal peridynamic theory is free of these challenges because its governing equations do not contain any spatial derivatives of the displacement components, and thus, are valid everywhere in the material.This study presents the peridynamic equation of motion for anti-plane shear and torsional deformations, as well as the peridynamic material parameters. After establishing the validity of this equation, solutions for specific components that are weakened by deep edge cracks and internal cracks are presented.

Topology optimization of cracked structures using peridynamics

“…Therefore, direct solution approach presented by Bobaru et al [81] has been utilized in this study by directly equating the inertia term in Eq. 1 In order to introduce the resulting final set of equations for the direct solution [57], the PD force density given in Eq. (2) can be linearized for material point i by considering the work done by Silling [82] as:…”

“…PD has been also used for describing growth phenomena in bones in recent years [49][50][51], where nonlocal biological interaction has the same mathematical form as mechanical nonlocal interactions. The theory has been successfully used to simulate crack growth [52], failure in multi-scale models including nanostructures [53], study damage and failure in concrete columns [54], model fracture in functionally graded materials [55], fatigue [56], torsion [57], buckling [58] and corrosion [59] failures, simulate hydraulic fracturing process "fracking" [60], and predict failure in composite materials [61,62]. PD improves the simulation of the crack growth compared to the weakly nonlocal approaches reported in the literature [63][64][65][66].…”

Modelling of Granular Fracture in Polycrystalline Materials Using Ordinary State-Based Peridynamics

“…Hence, these equations are always applicable regardless of discontinuities such as cracks. Peridynamics has been used for the fracture analysis of many different types of materials and material behaviours [12,13,14,15,16,17,18,19]. It has also been applied for the analysis of polycrystalline materials [20,21,22].…”