Determination of horizon size in state-based peridynamics

Abstract

Peridynamics is based on integro-differential equations and has a length scale parameter called horizon which gives peridynamics a non-local character. Currently, there are three main peridynamic formulations available in the literature including bond-based peridynamics, ordinary state-based peridynamics and non-ordinary state-based peridynamics. In this study, the optimum horizon size is determined for ordinary state-based peridynamics and non-ordinary state-based peridynamics formulations by using uniform and non-uniform discretisation under dynamic and static conditions. It is shown that the horizon sizes selected as optimum sizes for uniform discretisation can also be used for non-uniform discretisation without introducing significant error to the system. Moreover, a smaller horizon size can be selected for non-ordinary state-based formulation which can yield significant computational advantage. It is also shown that same horizon size can be used for both static and dynamic problems. Keywords Peridynamics • Horizon • State-based • Ordinary state-based • Non-ordinary state-based 1 Introduction Solid mechanics is an important area of engineering dealing with deformations of materials and structures under external loading conditions. Continuum mechanics has been widely used for this purpose for the last two hundred years, and there are currently different continuum mechanics formulations available in the literature with different advantages and limitations. The most common continuum mechanics formulation was developed by Cauchy, and the equation of motion of the fundamental object of continuum mechanics, i.e. "material point", is expressed in the form of a partial differential equation. Since the analytical solution of this equation is limited to particular geometries, boundary conditions and material systems, different numerical techniques including finite element method have been developed to obtain solution for numerous problems of interest. However, due to the spatial derivatives in the partial differential equations, the standard solution procedures become invalid if discontinuities such as cracks exist in the solution domain. In such cases, additional steps should be taken to get around the discontinuity problem. As an alternative approach, Silling [1] developed a new continuum mechanics formulation and named it as "peridynamics". There are several fundamental differences between peridynamics and Cauchy's continuum mechanics (CCM) formulation. First of all, the equation of motion of a material point in peridynamics is in integral form and does not contain any spatial derivatives. Therefore, it does not suffer from the discontinuity problem mentioned earlier. Moreover, peridynamics [2-20] is a non-local continuum mechanics formulation Communicated by Luca Placidi.

Peridynamic Approach to Digital Image Correlation Strain Calculation Algorithm

“…The horizon size δ is, usually expressed as a radius equal to a multiple of the PD grid spacing δ = m∆x (Figure 4). According to studies on the parameter m [44], a PD horizon of about 3•∆x ensures the PD model stability and convergence in most cases. The PD model grid size ∆x should be selected to evaluate the local effects with suitable accuracy, and can be found based on results of convergence analysis.…”

An In-depth Investigation of Bimaterial Interface Modeling Using Ordinary State-based Peridynamics

“…The plate is discretized with the nodal spacing .dx = 0.0025 m. In the PD model, displacement boundary conditions (BCs) are applied through a fictitious boundary layer with depth equal to the horizon size 8 [57], which is chosen to be 3.015.dx. The horizon size, 8 "" 3.0.dx and specifically 8 = 3.015.dx, is commonly used in PD literature, and it is an optimal value for the majority of problems [57,58]. The traction and shear loading are converted into the body force to apply to the PD material points.…”

“…Actually, it can work for one dimensional cases, but for 2D or 3D dimensional problems, too small horizon sizes have undesirable effects on overall solutions. Discussion about horizon size can be found in [57,58]. Recently, a few attempts were made to develop nonlocal models which are able to converge to classical models as the horizon size vanishes.…”

Prediction of fracture toughness of metallic materials

“…The global response of a system, e.g., toughness, strength, is often governed by the material behaviour at smaller length scales [28]. Therefore, length scale is an essential parameter in order to capture the non-classical material behaviour which usually appears at micro-scale [29,30].…”