ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

A Fracture Multiscale Model for Peridynamic enrichment within the Partition of Unity Method

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic (PD) model is used in regions of fracture and smooth PUM is used in the surrounding linear elastic media. The method is a so-called global-local enrichment strategy. The elastic fields of the undamaged media provide appropriate boundary data for the localized PD simulations. The first steps for a combined PD/PUM simulator are presented. In part I of this series, we show that the local PD approximation can be utilized to enrich the global PUM approximation to capture the true material response with high accuracy efficiently. Test problems are provided demonstrating the validity and potential of this numerical approach

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized. © 2022, The Author(s)

Peridynamics review

Peridynamics (PD) is a novel continuum mechanics theory established by Stewart Silling in 2000 [1]. The roots of PD can be traced back to the early works of Gabrio Piola according to dell'Isola et al. [2]. PD has been attractive to researchers as it is a nonlocal formulation in an integral form; unlike the local differential form of classical continuum mechanics. Although the method is still in its infancy, the literature on PD is fairly rich and extensive. The prolific growth in PD applications has led to a tremendous number of contributions in various disciplines. This manuscript aims to provide a concise description of the peridynamic theory together with a review of its major applications and related studies in different fields to date. Moreover, we succinctly highlight some lines of research that are yet to be investigated

Peridynamic-based multiscale frameworks for continuous and discontinuous material response

This PhD thesis aimed to develop two broad classes of multiscale frameworks for peridynamic theory to address two pressing needs: first is increased computational efficiency and the second is characterisation of heterogeneous media. To achieve these goals, two multiscale frameworks were proposed: model order reduction methodologies and homogenization frameworks. The model order reduction schemes were designed to improve computational efficiency, while the homogenization methodology aimed to provide frameworks for characterisation of heterogeneous materials within the peridynamic theory. Two specific model order reduction schemes were proposed, including a coarsening methodology and a model order reduction method based on static condensation. These schemes were applied to benchmark problems and shown to be effective in reducing the computational requirement of peridynamic models without compromising the fidelity of the simulation results. Additionally, a first-order nonlocal computational homogenization framework was proposed to characterise heterogeneous systems in the framework of peridynamics. This framework was utilised to characterise the behaviour of elastic and viscoelastic materials and materials with evolving microstructures. The results from these studies agreed with published results. The thesis achieved the goal of contributing to the development of efficient and accurate multiscale frameworks for peridynamic theory, which have potential applications in a wide range of fields, including materials science and engineering.This PhD thesis aimed to develop two broad classes of multiscale frameworks for peridynamic theory to address two pressing needs: first is increased computational efficiency and the second is characterisation of heterogeneous media. To achieve these goals, two multiscale frameworks were proposed: model order reduction methodologies and homogenization frameworks. The model order reduction schemes were designed to improve computational efficiency, while the homogenization methodology aimed to provide frameworks for characterisation of heterogeneous materials within the peridynamic theory. Two specific model order reduction schemes were proposed, including a coarsening methodology and a model order reduction method based on static condensation. These schemes were applied to benchmark problems and shown to be effective in reducing the computational requirement of peridynamic models without compromising the fidelity of the simulation results. Additionally, a first-order nonlocal computational homogenization framework was proposed to characterise heterogeneous systems in the framework of peridynamics. This framework was utilised to characterise the behaviour of elastic and viscoelastic materials and materials with evolving microstructures. The results from these studies agreed with published results. The thesis achieved the goal of contributing to the development of efficient and accurate multiscale frameworks for peridynamic theory, which have potential applications in a wide range of fields, including materials science and engineering

Construction of a peridynamic model for viscous flow

We derive the Eulerian formulation for a peridynamic (PD) model of Newtonian viscous flow starting from fundamental principles: conservation of mass and momentum. This formulation is different from models for viscous flow that utilize the so-called “peridynamic differential operator” with the classical Navier- Stokes equations. We show that the classical continuity equation is a limiting case of the PD one, assuming certain smoothness conditions. The PD model for viscous flow is calibrated to the classical Navier-Stokes equations by enforcing linear consistency for the viscous stress term. Couette and Poiseuille flows, and incompressible fluid flow past a regular lattice of cylinders are used to verify the new formulation, at least at low Reynolds numbers. The constructive approach in deriving the model allows for a seamless coupling with peridynamic models for corrosion or fracture for simulating complex fluid-structure interaction problems in which solid degradation takes place, such as in erosion-corrosion, hydraulic fracture, etc. Moreover, the new formulation sheds light on the relationships between local and nonlocal models