Stewart A. Silling

Peridynamic Theory of Solid Mechanics

Publisher Summary The classical theory of solid mechanics is based on the assumption of a continuous distribution of mass within a body and all internal forces are contact forces that act across zero distance. The mathematical description of a solid that follows from these assumptions relies on PDEs that additionally assume sufficient smoothness of the deformation for the PDEs to make sense in their either strong or weak forms. The classical theory has been demonstrated to provide a good approximation to the response of real materials down to small length scales, particularly in single crystals, provided these assumptions are met. Nevertheless, technology increasingly involves the design and fabrication of devices at smaller and smaller length scales, even interatomic dimensions.

Deformation of a Peridynamic Bar

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venants principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Greens function for general loading problems.

Implementing peridynamics within a molecular dynamics code

Peridynamics (PD) is a continuum theory that employs a nonlocal model to describe material properties. In this context, nonlocal means that continuum points separated by a finite distance may exert force upon each other. A meshless method results when PD is discretized with material behavior approximated as a collection of interacting particles. This paper describes how PD can be implemented within a molecular dynamics (MD) framework, and provides details of an efficient implementation. This adds a computational mechanics capability to an MD code, enabling simulations at mesoscopic or even macroscopic length and time scales.

Peridynamics for multiscale materials modeling

The paper presents an overview of peridynamics, a continuum theory that employs a nonlocal model of force interaction. Specifically, the stress/strain relationship of classical elasticity is replaced by an integral operator that sums internal forces separated by a finite distance. This integral operator is not a function of the deformation gradient, allowing for a more general notion of deformation than in classical elasticity that is well aligned with the kinematic assumptions of molecular dynamics. Peridynamics effectiveness has been demonstrated in several applications, including fracture and failure of composites, nanofiber networks, and polycrystal fracture. These suggest that peridynamics is a viable multiscale material model for length scales ranging from molecular dynamics to those of classical elasticity.

Dynamic fracture modeling with a meshfree peridynamic code

Publisher Summary The peridynamic model is an alternate theory of continuum mechanics that is specifically oriented toward modeling problems, in which cracks or other discontinuities emerge spontaneously as a body deforms under load. In this study, a code that implements this theory is applied to the Kalthoff-Winkler dynamic single-fracture experiment in a tough steel specimen. Many problems of fundamental importance in mechanics involve the spontaneous emergence of discontinuities, such as cracks, in the interior of a body. The classical theory of continuum mechanics is in some ways suited to modeling this type of problem because the theory uses partial differential equations as a mathematical description. Although much work has been devoted to special techniques aimed at working around this problem—particularly in the theory of fracture mechanics—these techniques are not fully satisfactory either in principle or in practice as general descriptions of fracture. This difficulty is inherited by numerical methods that implement the classical theory, including almost all finite-element and finite-difference codes in common usage.

Peridynamic analysis of damage and failure in composites.

*† ‡ Nearly all finite element codes and similar methods for the analysis of deformation in structures attempt to solve the partial differential equations of the classical theory of continuum mechanics. Yet these equations, because they require the partial derivatives of displacement to be known throughout the region modeled, are in some ways unsuitable for the modeling of cracks and other discontinuities, in which these derivatives fail to exist. As a means of avoiding this limitation, the peridynamic model of solid mechanics has been developed for applications involving discontinuities. The objective of this method is to treat crack and fracture as just another type of deformation, rather than as a pathology that requires special mathematical treatment. The peridynamic theory is based on integral equations, rather than differential equations, so there is no problem in applying the equations directly on a crack tip or crack surface. In the peridynamic model, displacements and internal forces are permitted to have discontinuities and other singularities. Particles interact with each other directly across finite distances through central forces known as “bonds”. Damage is introduced into the peridynamic model by permitting these bonds to break irreversibly. Breakage occurs when a bond is stretched in tension (or possibly compression) beyond some prescribed critical amount. After a bond breaks, it sustains no force. A distinguishing feature of this approach is its ability to treat the spontaneous formation of cracks together with their mutual interaction and dynamic growth in a consistent framework. A three-dimensional code called EMU implements the peridynamic model on parallel computers. The peridynamic method has been applied successfully to the analysis of material and structural failure in aerospace composites, particularly in graphiteepoxy laminates. For example, the method has been applied to the prediction of failure mode and crack direction in large-notch composite panels under tension loads with different layups and stacking sequences. The results have reproduced the experimentally observed dependence of crack growth direction on the relative percentage of fibers in different directions. The authors also have analyzed the damage occurring in a composite panel due to low velocity impact. The method predicts in detail the delamination and matrix damage process. Although the numerical method in EMU lends itself to parallelization, threedimensional analysis of large problems is computationally intensive. The applications reported here were run on the Columbia supercomputer at NASA Advanced Supercomputing (NAS) division. The Columbia supercomputer is proving to be invaluable in high-resolution modeling of the failure of composite materials.

Peridynamic modeling of impact damage.

The peridynamic theory is an alternative formulation of continuum mechanics oriented toward modeling discontinuites such as cracks. It differs from the classical theory and most nonlocal theories in that it does not involve spatial derivatives of the displacement field. Instead, it is formulated in terms of integral equations, whose validity is not affected by the presence of discontinuities such as cracks. It may be thought of as a “continuum version of molecular dynamics” in that particles interact directly with each other across a finite distance. This paper outlines the basis of the peridynamic theory and its numerical implementation in a three-dimensional code called EMU. Examples include simulations of a Charpy V-notch test, accumulated damage in concrete due to multiple impacts, and crack fragmentation of a glass plate.Copyright

The formulation and computation of the nonlocal J-integral in bond-based peridynamics

This work presents a rigorous derivation for the formulation of the J-integral in bond-based peridynamics using the crack infinitesimal virtual extension approach. We give a detailed description of an algorithm for computing this nonlocal version of the J-integral. We present convergence studies (m-convergence and δ-convergence) for two different geometries: a single edge-notch configuration and a double edge-notch sample. We compare the results with results based on the classical J-integral and obtained from FEM calculations that employ special elements near the crack tip. We identify the size of the nonlocal region for which the peridynamic J-integral value is near the classical FEM solutions. We discuss how the boundary conditions and the peridynamic “skin effect” may influence the peridynamic J-integral value. We also observe, computationally, the path-independence of the peridynamic J-integral.

Peridynamics with LAMMPS : a user guide.

Peridynamics is a nonlocal formulation of continuum mechanics. The discrete peridynamic model has the same computational structure as a molecular dynamic model. This document details the implementation of a discrete peridynamic model within the LAMMPS molecular dynamic code. This document provides a brief overview of the peridynamic model of a continuum, then discusses how the peridynamic model is discretized, and overviews the LAMMPS implementation. A nontrivial example problem is also included.

Peridynamic 3D models of nanofiber networks and carbon nanotube‐reinforced composites

Here we employ a reformulation of the continuum mechanics theory, the peridynamic formulation (PF) in an integral form that, at the discretized level, resembles molecular dynamics (MD). The peridynamic theory is based on a continuum formulation and can capture nucleation and propagation of defects and discontinuities without ad‐hoc assumptions or special treatments needed by classical continuum theory. We analyze nanofiber networks and CNT‐reinforced polymer composites. We treat all crossovers contacts between fibers as perfect bonds. The use of repulsive short‐range forces eliminates the need for complex contact detection algorithms. We generate the fibers as 3D curves with random orientation, with or without preferred directionality. We use an object‐oriented code written in Fortran 90/95 to define the geometrical entities. The PF can capture the deformation and complex fracture behavior in fully 3D dynamic simulations. van der Waals forces are included in these calculations. The strength of the bonds b...