Statistical coarse-graining of molecular dynamics into peridynamics.

This paper describes an elegant statistical coarse-graining of molecular dynamics at finite temperature into peridynamics, a continuum theory. Peridynamics is an efficient alternative to molecular dynamics enabling dynamics at larger length and time scales. In direct analogy with molecular dynamics, peridynamics uses a nonlocal model of force and does not employ stress/strain relationships germane to classical continuum mechanics. In contrast with classical continuum mechanics, the peridynamic representation of a system of linear springs and masses is shown to have the same dispersion relation as the original spring-mass system

Peridigm users' guide. V1.0.0.

Abstract Not Provide

A mathematical framework for multiscale science and engineering : the variational multiscale method and interscale transfer operators.

This report is a collection of documents written as part of the Laboratory Directed Research and Development (LDRD) project A Mathematical Framework for Multiscale Science and Engineering: The Variational Multiscale Method and Interscale Transfer Operators. We present developments in two categories of multiscale mathematics and analysis. The first, continuum-to-continuum (CtC) multiscale, includes problems that allow application of the same continuum model at all scales with the primary barrier to simulation being computing resources. The second, atomistic-to-continuum (AtC) multiscale, represents applications where detailed physics at the atomistic or molecular level must be simulated to resolve the small scales, but the effect on and coupling to the continuum level is frequently unclear

Singularities and Phase Transitions in Elastic Solids: Numerical Studies and Stability Analysis

Numerical studies of the deformation near the tip of a crack are presented for a family of incompressible solids in the context of the theory of finite anti-plane shear of an elastic material. The numerical model computes the near-field and far-field solutions simultaneously, enabling observations of both small-scale and large-scale nonlinearity. The computed near-field solution is compared with a lowest-order asymptotic solution. An approximation for the J-integral under conditions of very large loads is discussed and compared with numerical results. The size of the region over which the lowest-order solution applies is observed. Numerical solutions are presented for the same crack problem with materials for which the equilibrium equation changes in type from elliptic to hyperbolic as a result of deformation. These results show the emergence of surfaces of discontinuity in the displacement field in some cases. In other cases they show a chaotic mixture of elliptic phases near the crack tip. Analysis of the stability of such coexistent phases is carried out for a specific material, the trilinear material. It is shown that the Maxwell relation, and therefore local stability, cannot in general be satisfied exactly for an arbitrary boundary value problem with this material. However, in those cases where it cannot be satisfied exactly, it may be satisfied in the sense of a limit of a certain sequence of deformations. This sequence produces a progressively chaotic pattern of two coexistent elliptic phases, as was observed numerically. The phases mix over a definite region in a given boundary value problem. This region may be computed using a constitutive relation which characterizes the mixture in the limit of the sequence.</p

A coarsening method for linear peridynamics.

A method is obtained for deriving peridynamic material models for a sequence of increasingly coarsened descriptions of a body. The starting point is a known detailed, small scale linearized state-based description. Each successively coarsened model excludes some of the aterial present in the previous model, and the length scale increases accordingly. This excluded material, while not present explicitly in the coarsened model, is nevertheless taken into account implicitly through its effect on the forces in the coarsened material. Numerical examples emonstrate that the method accurately reproduces the effective elastic properties of a composite as well as the effect of a small defect in a homogeneous medium

Peridynamics as a rigorous coarse-graining of atomistics for multiscale materials design.

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

Peridynamics as a rigorous coarse-graining of atomistics for multiscale materials design.

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

Peridynamics with LAMMPS : a user guide.

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