Discrete and continuous nonlocal material models and their coupling

材料的合理设计，准确和有效的材料模拟，设计和控制算法的发展，以及材料对环境和实际发生的负载的响应的确定，都需要在不同的空间和时间尺度上理解力学。由于这个原因，人们对多尺度材料模型的发展产生了相当大的兴趣。为此目的的一个常见的方法是耦合原子和连续模型，第一个用于精确地解决小尺度的缺陷，第二个有效地处理缺乏缺陷的区域。例如，许多人试图将非局部分子动力学（MD）与局部经典连续弹性（CE）模型结合起来，但成功有限，因为除了最小的样本外，MD易于处理的尺度与CE有效的尺度之间仍然存在差距，还因为必须克服将非局部模型（MD）与局部模型（CE）结合起来所产生的问题。该项目解决了这些困难，取代MD与新开发的变种（QC-QR）的准连续体（QC）方法和CE的非局部周波（PD）连续体模型。QC-QR方法近似于众所周知的QC方法，通过替换确定在QC方法中的每个活性颗粒上的力的总和，通过使用？求积？统治PD方法不涉及空间导数，因此它可以准确地解释相对较小尺度的缺陷。QC-QR方法相对于MD和QC的效率增益以及PD相对于CE的有效性范围增益，加上QC-QR和PD都是非局部模型的事实，意味着耦合QC-QR/PD模型具有克服上面提到的耦合MD/CE模型遇到的困难的潜力。事实上，QC-QR和PD本身就是多尺度材料模型，因此该项目的一个重要方面是探索它们作为材料的多尺度单模型使用的限制。该项目还考虑了多尺度复合QC-QR/PD模型，其有效性通过计算和分析研究确定。同样，使用的QC-QR/PD耦合模型之间的MD和CE.The新材料的合理设计和它们在应用中的使用需要在不同的空间和时间尺度的力学的理解之间的桥梁从原子的大小的飞机和桥梁。由于这个原因，人们对开发在所有尺度范围内都有效的多尺度材料模型非常感兴趣。以前曾试图将在有限尺度上有效的模型结合起来，以产生一个在所有尺度上都有效的复合模型，但由于几个原因，包括某些模型的有效性范围与其他模型的易处理性范围之间存在差距，这种尝试没有取得完全成功。我们的目标是产生一个模型的材料力学，是有效的，在更广泛的范围内比目前使用的模型可以处理的尺度易处理。我们参与了新模型的开发，其中一个扩展了可以在大尺度上运行的模型的有效性范围，另一个提高了在原子尺度上运行的模型的效率。我们对这些模型进行了进一步的研究，以更精确地确定它们的有效性和易处理性范围。然后，我们研究，通过数学和计算手段，如何最好地耦合这两个模型，并量化所产生的改进现有的方法。最后，我们测试了新的复合模型，将其应用到一系列测试问题的解决方案。

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.