CAREER: Adapting the Fluid Projection Method to Model Elasto-plastic Materials

材料在外力作用下变形主要有两种方式。变形可以是弹性的，因此当力被移除时，材料恢复其原始形状。可替代地，变形可以是塑性的，由此材料经历随后可能导致断裂的不可逆变化。许多具有重要技术意义的材料根据所施加的力表现出这两种类型的变形的组合，并且被称为弹塑性。一个例子是块体金属玻璃（BMG），其是与大多数金属相比具有无定形原子排列的合金。BMG具有理想的性能，例如能够像塑料一样加工，由于制造效率的显著提高，使它们成为许多应用（例如下一代智能手机外壳）的有吸引力的候选者。然而，实验测量的BMG断裂性能显示出很大的变化，限制了它们的使用。为了克服这些局限性，有必要发展预测的理论和计算模型的BMG弹塑性。这个项目是基于弹塑性材料方程和不可压缩流体方程之间惊人的相似性。利用这种相似性，最初为流体流动开发的计算方法将被转化为弹塑性。这些计算方法将用于与理论家和实验学家合作，研究BMG的断裂特性。最终目的是提供一个实用的工程工具，用于预测弹塑性材料何时会断裂，以及如何最好地设计使用它们的结构。这项工作将作为研究、教学和指导的综合计划的一部分进行，并将涉及新英格兰的外展活动，包括当地图书馆的系列讲座。Chorin（1968）的投影方法是一种成熟的方法，用于模拟流体流动的不可压缩Navier-Stokes方程。这个建议是基于一个令人惊讶的数学之间的对应关系的流体在不可压缩的限制和弹塑性固体在准静态限制（当惯性可以忽略不计）。在这个建议中，这种对应关系是利用翻译几个现代数值方法来自Chorin的投影方法准静态弹塑性，从而在一个实用的和强大的一套新的模拟工具，为不同类别的物理问题。与现有的技术相比，由此产生的数值方法可能是特别适合于涉及大塑性变形的问题。弹塑性材料的一个示例类型是块体金属玻璃（BMG），其是具有许多有利特性（例如优异的强度和耐磨性）的合金。这里开发的数值方法将用于与理论家和实验学家合作，研究BMG的断裂韧性特性，目的是在广泛的实验条件下预测BMG韧性。PI计划扩大数值方法的研究生课程，以满足这一领域的迫切需要。开源软件将作为该项目的一部分发布，PI将培训学生最佳实践，使软件可供广大受众使用。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

There are two main ways that materials deform under an applied force. The deformation can be elastic, so that when the force is removed the material recovers its original shape. Alternatively, the deformation can be plastic, whereby the material undergoes irreversible changes that may subsequently lead to breakage. Many materials of technological importance exhibit a combination of these two types of deformation depending on the applied force, and are called elasto-plastic. One example are bulk metallic glasses (BMGs), which are alloys that have an amorphous atomic arrangement in contrast to most metals. BMGs have desirable properties, such as the ability to be processed like plastics, making them attractive candidates for many applications (e.g. next-generation smartphone cases) due to considerable improvements in manufacturing efficiency. However, experimental measurements of BMG breakage properties show wide variations, limiting their usage. To overcome these limitations, it is essential to develop predictive theoretical and computational models of BMG elasto-plasticity. This project is based on a surprising similarity between the equations for elasto-plastic materials and the equations for incompressible fluids. Using this similarity, computational approaches that were originally developed for fluid flow will be translated to elasto-plasticity. These computational methods will be used in collaboration with theorists and experimentalists to study the fracture properties of BMGs. The ultimate aim is to provide a practical engineering tool for predicting when elasto-plastic materials will break, and how to best design structures using them. This work will be undertaken as part of an integrated program of research, teaching, and mentorship, and will involve outreach activities in New England, including a local library lecture series.The projection method of Chorin (1968) is a well-established approach for simulating the incompressible Navier-Stokes equations for fluid flow. This proposal is based on a surprising mathematical correspondence between fluids in the incompressible limit and elasto-plastic solids in the quasi-static limit (when inertia can be neglected). In this proposal, this correspondence is harnessed to translate several modern numerical approaches derived from Chorin's projection method to quasi-static elasto-plasticity, resulting in a practical and powerful set of new simulation tools for a different class of physical problem. Compared to existing techniques, the resultant numerical methods are likely to be especially well-suited to problems involving large plastic deformations. An example type of elasto-plastic material are the bulk metallic glasses (BMGs), which are alloys with many favorable properties such as excellent strength and wear resistance. The numerical methods developed here will be used in a collaboration with theorists and experimentalists to study the fracture toughness properties of BMGs, with the aim of predicting BMG toughness over a wide range of experimental conditions. The PI plans to expand the graduate curriculum in numerical methods to address a pressing need in this area. Open source software will be released as part of this project, and the PI will train students in best practices to make software accessible to a broad audience.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.