From Discrete Dislocation Dynamics to Crystal Plasticity - A Spatio-Temporal Coarse-Graining Approach

这项工作的目标是开发和展示一种预测的、下一代的计算工具的基础，用于对受机械应力和变形的现代机器的金属部件进行微结构敏感设计。金属材料在宏观尺度上的一般不可逆变形称为塑性变形，是由于其底层规则晶体结构中大量缺陷在外加载荷作用下的运动而产生的。这种缺陷运动导致多个缺陷形成特殊的空间组织模式，称为微观结构，已知这种缺陷微观结构对材料的强度和延展性等宏观性能有重要影响。与宏观观察到的塑性变形相比，单个缺陷运动发生在更小的长度尺度和更快的时间尺度上，因此需要一个平均的数学过程来从微观动力学中定义缺陷集体行为的宏观模型。该奖项支持通过多尺度数学和计算建模从微观基础开发和验证这种宏观模型。该项目解决了一个对社会有重大影响的关键科学问题。大量的现代制造、轻量化、高强度结构材料的设计以及用于能源和航空航天应用的燃气轮机材料都依赖于对微观结构演变及其对材料性能影响的理解。目前广泛使用的工业设计规范并未采用微结构敏感材料模型。这项研究将开发一种真正的材料微观结构敏感模型，用于塑性强度和位错微观结构的演变，这有可能彻底改变许多行业金属材料的设计方式，为美国经济带来显著的成本和能源节约。这项研究紧密地整合了应用数学、材料力学和材料科学领域，这将导致这些领域之间富有成效的跨学科思想相互作用，所有这些都指向非常实际的结果，包括培养学生，这将增强美国在科学和技术方面的劳动力。将努力在NSF支持的匹兹堡超级计算中心存档已开发的计算机代码，使感兴趣的从业者能够系统地、免费地、广泛地访问这些代码。这项工作的挑战是开发一种新的计算工具来精确模拟晶体材料的塑性和位错微观结构的演变。解决的任务将是计算塑性强度和相关的微观结构的材料在中观和宏观尺度上直接从晶体缺陷的潜在运动。这个应用程序是一个典型的复杂系统，具有巨大的实际意义。具体来说，将利用一个精确但非封闭的基于偏微分方程的理论来表示时空平均位错动力学的演化，该理论包含定义良好的基于微观位错动力学输入的占位符。这些输入，通常被规定为现象学本构假设，在这项工作中，将被精心设计的耦合所取代，在中宏响应的“慢”时间尺度上，与时间平均响应的“快”，局部（宏观尺度上）离散位错动力学模拟。整体策略是基于新颖而健全的连续介质力学原理，如晶体缺陷携带的拓扑电荷守恒声明与宏观弹性耦合，以及现代数学工具，如平均非线性常微分方程的多时间尺度响应的杨测量理论。有趣的是，这种解决明显实际问题的方法涉及将有限维动力系统输入到总体无限维动力系统中。

The objective of this work is to develop and demonstrate the fundamentals of a predictive, next-generation, computational tool for microstructure-sensitive design of metallic components of modern machines that are subjected to mechanical stress and deformation. The generally irreversible deformation of metallic materials at the macroscopic scale called plastic deformation arises due to the motion of vast numbers of defects in their underlying regular crystalline structure under applied loading. Such defect motions lead to special spatial patterns of organization of multiple defects called microstructure, and it is known that this defect microstructure has significant effect on the material's macroscopic properties like strength and ductility. The individual defect motions occur at much smaller length scales and much faster time scales than those of macroscopically observed plastic deformation, thus requiring a mathematical process of averaging to define the macroscopic model of collective behavior of defects from the microscopic dynamics. This award supports the development and validation of such a macroscopic model from microscopic fundamentals through multiscale mathematical and computational modeling. The project addresses a key scientific question with significant impact on society. A great deal of modern manufacturing, design of light-weight, high-strength structural materials and materials for gas turbines for energy and aerospace applications all depend upon understanding of microstructure evolution and its effect on material properties. Widely used industrial design codes do not at present make use of microstructure sensitive material models. This research will develop a truly material microstructure-sensitive model for plastic strength and dislocation microstructure evolution that has the potential to revolutionize the way metallic materials are designed in many industries, bringing about significant cost and energy savings for the US economy. The research tightly integrates the fields of Applied Mathematics, Mechanics of Materials, and Materials Science, and this will lead to fruitful interdisciplinary interplay of ideas between these fields, all directed to very practical outcomes, including the training of students which will enhance the US workforce in science and technology. Efforts will be made to archive developed computer codes at the NSF supported Pittsburgh Supercomputing Center to enable systematic, free, widespread access to interested practitioners.The challenge of this work is to develop a novel computational tool for accurate multi-scale simulations of plasticity and dislocation microstructure evolution in crystalline materials. The tasks addressed will be the computation of plastic strength and associated microstructure of a material at the meso and macroscale directly from the underlying motion of crystal defects. This application is a paradigmatic complex system, with immense practical relevance. Specifically, an exact, but non-closed, partial differential equation based theory representing the evolution of space-time averaged dislocation dynamics will be utilized, that contains well-defined place-holders for microscopic dislocation dynamics based input. These inputs, typically prescribed as phenomenological constitutive assumptions, will be replaced in this work by a carefully designed coupling, on the "slow" time-scale of meso-macro response, with time-averaged response of "fast", local (on the macroscopic scale) Discrete Dislocation Dynamics simulations. The overall strategy is based on novel and sound continuum mechanical principles like a conservation statement for topological charge carried by crystal defects coupled to macroscopic elasticity as well as modern mathematical tools like Young Measure theory for averaging multi-time scale response of nonlinear Ordinary Differential Equations. Interestingly, this approach for solving a patently practical problem involves finite dimensional dynamical system input into an overarching infinite dimensional dynamical system.