A non-ordinary state-based peridynamics framework for anisotropic materials

Abstract

Peridynamics (PD) represents a new approach for modelling fracture mechanics, where a continuum domain is modelled through particles connected via physical bonds. This formulation allows us to model crack initiation, propagation, branching and coalescence without special assumptions. Up to date, anisotropic materials were modelled in the PD framework as different isotropic materials (for instance, fibre and matrix of a composite laminate), where the stiffness of the bond depends on its orientation. A non-ordinary state-based formulation will enable the modelling of generally anisotropic materials, where the material properties are directly embedded in the formulation. Other material models include rocks, concrete and biomaterials such as bones. In this paper, we implemented this model and validated it for anisotropic composite materials. A composite damage criterion has been employed to model the crack propagation behaviour. Several numerical examples have been used to validate the approach, and compared to other benchmark solution from the finite element method (FEM) and experimental results when available.Fracture mechanics have been studied for nearly a century, from the first work of Griffith [15] for brittle materials. Over the years a number of researchers have modelled fracture mechanics analytically [28,36] for simple problems and geometries, and more commonly using numerical frameworks, such as the extended finite element method (XFEM) [4,27] and the extended boundary element method (XBEM) [16], among many others. Nevertheless, these methods suffer when crack branching or coalescence are involved.The phase-field method has been shown to model crack branching behaviour [1,18]. The method consists in describing the crack as an interface directly in the formulation and is used conjointly to the finite element method (FEM) [35]. Nevertheless, the method requires a fine mesh around the crack to model the interface correctly. Another drawback of the method is that it can provide unrealistic results. A novel numerical method entitled peridynamics (PD) [40] has been recently developed, and has shown great potential in fracture mechanics problems involving initiating, propagating, branching and coalescing cracks.The original peridynamics (PD) formulation was proposed by Silling [40], where he redefined the classical approach for continuum mechanics using an integral framework instead of partial derivatives. The main reason for using this approach is that the partial derivatives pose a challenge when dealing with fracture mechanics problems, since the governing partial differential equations in elasticity imply that singularities will appear due to the presence of discontinuities, which is not desirable. Due to the integral form of the formulation, no special assumptions are needed to deal with singularities, such as a crack in the domain.The first PD formulation described a continuum medium through discrete particles, interacting between each other through physical connections entitled bonds. Each bond has a stif...

A Review on the Developments of Peridynamics for Reinforced Concrete Structures

“…Bobaru et al [17] have investigated different aspects on convergence. Rossi Cabral et al [125] is one of the few studies to establish a physical relation to the horizon size, which was also mentioned by [58] for anisotropic materials.…”

“…Queiruga and Moridis [120] have investigated how different weight functions affect the error and convergence in PD analysis for 2D problems. The shape tensor becomes a singular matrix if a particle has less than 2 bonds in 2D problems and 3 bonds in case of a 3D problem [58,148]. Silling [133] highlighted that the shape tensor calculation may suffer in problems with large deformation or when extensive damage is present.…”

A stable non‐ordinary state‐based peridynamic model for laminated composite materials

“…In References 26, 30, a stabilized NOSBPD model is derived by the linearized BBPD, and the model does not require any control coefficient, but which is not used for composites materials. PD has been used in composite materials 10,19,31‐40 . The bonds in composite materials are divided into the fiber bond and the matrix bond in References 10, 31, 32, in which the fiber bond is used in fiber direction, and the matrix bond is used in other directions.…”

“…However, there only includes two material constants in References 35, 36, while composites have four independent material properties in continuum mechanics. Furthermore, a NOSBPD formulation was presented to model general anisotropic materials in Reference 37, while zero‐energy modes were not controlled in this model. Thus, it can be stated that there is still no effective method to control zero‐energy modes in NOSBPD for composite materials.…”

“…PD has been used in composite materials. 10,19,[31][32][33][34][35][36][37][38][39][40] The bonds in composite materials are divided into the fiber bond and the matrix bond in References 10,31,32, in which the fiber bond is used in fiber direction, and the matrix bond is used in other directions. Similarly, in References 19,33, transverse bonds were added in the transverse direction furthermore.…”