Publications

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Peridynamics is a nonlocal extension of classical continuum mechanics. The discrete peridynamic model has the same computational structure as a molecular dynamics model. This document provides a brief overview of the peridynamic model of a continuum, then discusses how the peridynamic model is discretized within LAMMPS. An example problem is also included.

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Fracture or tearing of ductile metals is a pervasive engineering concern, yet accurate prediction of the critical conditions of fracture remains elusive. Sandia National Laboratories has been developing and implementing several new modeling methodologies to address problems in fracture, including both new physical models and new numerical schemes. The present study provides a double-blind quantitative assessment of several computational capabilities including tearing parameters embedded in a conventional finite element code, localization elements, extended finite elements (XFEM), and peridynamics. For this assessment, each of four teams reported blind predictions for three challenge problems spanning crack initiation and crack propagation. After predictions had been reported, the predictions were compared to experimentally observed behavior. The metal alloys for these three problems were aluminum alloy 2024-T3 and precipitation hardened stainless steel PH13-8Mo H950. The predictive accuracies of the various methods are demonstrated, and the potential sources of error are discussed.

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The peridynamic theory is an extension of traditional solid mechanics that treats discontinuous media, including the evolution of discontinuities due to fracture, on the same mathematical basis as classically smooth media. A recent advance in the linearized peridynamic theory permits the reduction of the number of degrees of freedom modeled within a body. Under equilibrium conditions, this coarse graining method exactly reproduces the internal forces on the coarsened degrees of freedom, including the effect of the omitted material that is no longer explicitly modeled. The method applies to heterogeneous as well as homogeneous media and accounts for defects in the material. The coarse graining procedure can be repeated over and over, resulting in a hierarchically coarsened description that, at each stage, continues to reproduce the exact internal forces present in the original, detailed model. Each coarsening step results in reduced computational cost. This talk will describe the new peridynamic coarsening method and show computational examples.

The peridynamic model of solid mechanics treats internal forces within a continuum through interactions across finite distances. These forces are determined through a constitutive model that, in the case of an elastic material, permits the strain energy density at a point to depend on the collective deformation of all the material within some finite distance of it. The forces between points are evaluated from the Frechet derivative of this strain energy density with respect to the deformation map. The resulting equation of motion is an integro-differential equation written in terms of these interparticle forces, rather than the traditional stress tensor field. Recent work on peridynamics has elucidated the energy balance in the presence of these long-range forces. We have derived the appropriate analogue of stress power, called absorbed power, that leads to a satisfactory definition of internal energy. This internal energy is additive, allowing us to meaningfully define an internal energy density field in the body. An expression for the local first law of thermodynamics within peridynamics combines this mechanical component, the absorbed power, with heat transport. The global statement of the energy balance over a subregion can be expressed in a form in which the mechanical and thermal terms contain only interactions between the interior of the subregion and the exterior, in a form anticipated by Noll in 1955. The local form of this first law within peridynamics, coupled with the second law as expressed in the Clausius-Duhem inequality, is amenable to the Coleman-Noll procedure for deriving restrictions on the constitutive model for thermomechanical response. Using an idea suggested by Fried in the context of systems of discrete particles, this procedure leads to a dissipation inequality for peridynamics that has a surprising form. It also leads to a thermodynamically consistent way to treat damage within the theory, shedding light on how damage, including the nucleation and advance of cracks, should be incorporated into a constitutive model.

The peridynamic model of solid mechanics is a mathematical theory designed to provide consistent mathematical treatment of deformations involving discontinuities, especially cracks. Unlike the partial differential equations (PDEs) of the standard theory, the fundamental equations of the peridynamic theory remain applicable on singularities such as crack surfaces and tips. These basic relations are integro-differential equations that do not require the existence of spatial derivatives of the deformation, or even continuity of the deformation. In the peridynamic theory, material points in a continuous body separated from each other by finite distances can interact directly through force densities. The interaction between each pair of points is called a bond. The dependence of the force density in a bond on the deformation provides the constitutive model for a material. By allowing the force density in a bond to depend on the deformation of other nearby bonds, as well as its own deformation, a wide spectrum of material response can be modelled. Damage is included in the constitutive model through the irreversible breakage of bonds according to some criterion. This criterion determines the critical energy release rate for a peridynamic material. In this talk, we present a general discussion of the peridynamic method and recent progress in its application to penetration and fracture in nonlinearly elastic solids. Constitutive models are presented for rubbery materials, including damage evolution laws. The deformation near a crack tip is discussed and compared with results from the standard theory. Examples demonstrating the spontaneous nucleation and growth of cracks are presented. It is also shown how the method can be applied to anisotropic media, including fiber reinforced composites. Examples show prediction of impact damage in composites and comparison against experimental measurements of damage and delamination.

This report summarizes activities undertaken during FY08-FY10 for the LDRD Peridynamics as a Rigorous Coarse-Graining of Atomistics for Multiscale Materials Design. The goal of our project was to develop a coarse-graining of finite temperature molecular dynamics (MD) that successfully transitions from statistical mechanics to continuum mechanics. The goal of our project is to develop a coarse-graining of finite temperature molecular dynamics (MD) that successfully transitions from statistical mechanics to continuum mechanics. Our coarse-graining overcomes the intrinsic limitation of coupling atomistics with classical continuum mechanics via the FEM (finite element method), SPH (smoothed particle hydrodynamics), or MPM (material point method); namely, that classical continuum mechanics assumes a local force interaction that is incompatible with the nonlocal force model of atomistic methods. Therefore FEM, SPH, and MPM inherit this limitation. This seemingly innocuous dichotomy has far reaching consequences; for example, classical continuum mechanics cannot resolve the short wavelength behavior associated with atomistics. Other consequences include spurious forces, invalid phonon dispersion relationships, and irreconcilable descriptions/treatments of temperature. We propose a statistically based coarse-graining of atomistics via peridynamics and so develop a first of a kind mesoscopic capability to enable consistent, thermodynamically sound, atomistic-to-continuum (AtC) multiscale material simulation. Peridynamics (PD) is a microcontinuum theory that assumes nonlocal forces for describing long-range material interaction. The force interactions occurring at finite distances are naturally accounted for in PD. Moreover, PDs nonlocal force model is entirely consistent with those used by atomistics methods, in stark contrast to classical continuum mechanics. Hence, PD can be employed for mesoscopic phenomena that are beyond the realms of classical continuum mechanics and atomistic simulations, e.g., molecular dynamics and density functional theory (DFT). The latter two atomistic techniques are handicapped by the onerous length and time scales associated with simulating mesoscopic materials. Simulating such mesoscopic materials is likely to require, and greatly benefit from multiscale simulations coupling DFT, MD, PD, and explicit transient dynamic finite element methods FEM (e.g., Presto). The proposed work fills the gap needed to enable multiscale materials simulations.

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