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

材料在外力作用下变形的主要方式有两种。变形可以是弹性的，这样当力被移除时，材料就会恢复到原来的形状。或者，变形可以是塑性的，因此材料经历了不可逆转的变化，随后可能导致断裂。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变形的组合，被称为弹塑性。一个例子是大块金属玻璃(BMG)，它是与大多数金属相反的非晶态原子排列的合金。BMG具有理想的性能，例如能够像塑料一样进行加工，由于制造效率的显著提高，使其成为许多应用(如下一代智能手机外壳)的诱人候选者。然而，对BMG破碎性的实验测量显示出很大的差异，限制了它们的使用。为了克服这些局限性，有必要建立BMG弹塑性的预测理论和计算模型。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转化为弹塑性。这些计算方法将与理论家和实验学家合作，研究BMG的断裂特性。其最终目的是提供一种实用的工程工具，用于预测弹塑性材料何时断裂，以及如何最佳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，并将涉及在新英格兰的外展活动，包括当地图书馆的讲座系列。Chorin(1968)的投影法是模拟不可压缩流体流动的Navier-Stokes方程的成熟方法。这一建议是基于不可压缩极限中的流体和准静态极限中的弹塑性固体(当惯性可以忽略时)之间令人惊讶的数学对应。在这个方案中，利用这种对应关系将从Chorin投影法导出的几种现代数值方法转化为准静态弹塑性，从而产生了一套实用而强大的新模拟工具，用于解决不同类别的物理问题。与现有技术相比，所得到的数值方法很可能特别适合于涉及大塑性变形的问题。大块金属玻璃(BMG)是弹塑性材料的一个例子，它是具有许多良好性能的合金，如优异的强度和耐磨性。本文开发的数值方法将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的韧性。PI计划扩大数值方法方面的研究生课程，以满足这一领域的迫切需求。开源软件将作为该项目的一部分发布，PI将对学生进行最佳实践培训，使软件可供广大受众使用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.