Discrete and continuous nonlocal material models and their coupling

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The rational design of materials, the development of accurate and efficient material simulation, design, and control algorithms, and the determination of the response of materials to environments and loads occurring in practice all require an understanding of mechanics at disparate spatial and temporal scales. For this reason, there has been very considerable interest in the development of multiscale material models. A common approach for this purpose is to couple atomistic and continuum models, the first used to accurately resolve defects at small scales, the second to efficiently treat regions lacking defects. For example, many have tried to couple nonlocal molecular dynamics (MD) with local classical continuum elasticity (CE) models with limited success because, for all but the smallest samples, there remains a gap between the scales for which MD is tractable and CE is valid and also because one has to overcome problems arising from the coupling a nonlocal model (MD) to a local one (CE). The project addresses these difficulties by replacing MD with a newly developed variant (QC-QR) of the quasicontinuum (QC) method and CE by the nonlocal peridynamics (PD) continuum model. The QC-QR method approximates the well-known QC method by replacing the sums that determine the force on each active particle in the QC method by shorter sums defined using a ?quadrature? rule. The PD method does not involve spatial derivatives so that it can accurately account for defects at relatively small scales. The gains in efficiency effected by the QC-QR method relative to MD and QC and the gains in the range of validity effected by PD relative to CE, added to the fact that both QC-QR and PD are nonlocal models, means that a coupled QC-QR/PD model has the potential of overcoming the difficulties encountered for coupled MD/CE models that were alluded to above. In fact, QC-QR and PD are themselves multiscale material models, so that one significant aspect of the project is to explore the limits of their use as multiscale mono-models for materials. The project also considers the multiscale composite QC-QR/PD model whose efficacy is determined through computational and analytical studies. Likewise, the use of the QC-QR/PD coupled model as a bridge between MD and CE is considered.The rational design of new materials and their use in applications require an understanding of mechanics at disparate spatial and temporal scales ranging from that of atoms to that of the size of aircraft and bridges. For this reason, there has been very considerable interest in the development of multiscale material models that are valid over all that range of scales. Previous attempts at coupling models that are valid over limited scales so as to produce a composite model that is valid at all scales have not met with complete success because of several reasons, including the fact that a gap exists between the range of validity of some models and the range of tractability of others. Our goal is to produce a model for the mechanics of materials that is valid and tractable over a wider range of scales than can be handled by models in current use. We have participated in the development of new models, one that extends the range of validity of models that can operate at the large-end of the scales and one that improves the efficiency of models that operate at the atomistic scale. We make further studies of these models to determine more precisely their range of validity and tractability. We then study, through mathematical and computational means, how best to couple the two models and to quantify the resulting improvements over existing approaches. Finally, we test the new composite model by applying it to the solution of a series of test problems.