Understanding Continuum Models of Elasto-Plastic Deformations via Multiscale Analyses

金属的塑性变形过程涉及广泛的技术应用，包括制造过程、能量吸收系统和永久夹具。因此，人们在开发能够准确描述金属宏观行为的模型方面付出了巨大的努力。然而，对这些连续体模型背后的运动学假设及其与微观尺度过程的关系的完整理解仍然缺乏。本研究的目的是修正晶体塑性背景下的关键模型假设，建立一个更普遍适用和更准确的理论。本研究将使用的技术和方法有可能对非弹性变形做出更广泛的贡献，非弹性变形通常基于类似的运动学假设。这包括经历生长的生物组织模型、损伤模型和粘弹性理论。这项研究的结果将包括在本科和研究生教育中，并将作为跨学科努力的一个例子，将力学与材料科学和应用数学结合起来。晶体材料的弹塑性变形的运动学在原子和介观水平上得到了很好的理解，其中位错完全解决了。然而，宏观尺度下的有效运动学描述仍然是当前争论的话题。本工作的研究目标是通过升级程序严格推导出描述塑性变形的连续体运动学关系。这一目标将通过中尺度弹塑性变形的仔细数学表征和变异演算中均质化技术的使用来实现。建立离散到连续体的联系所需要的假设将清楚地决定连续体关系的适用范围。新的研究生水平课程的开发、学生的指导以及与同行社区的有针对性的互动将有助于本研究产生更广泛的影响。

Plastic deformation processes in metals are relevant to a wide range of technological applications including manufacturing processes, energy absorption systems, and permanent fixtures. As a result, there has been tremendous efforts in developing models that can accurately describe the macroscopic behavior of metals. However a complete understanding of the kinematic hypothesis underlying these continuum models and their relationship to processes at the microscale is still lacking. The objective of this research is to revise key model assumptions in the context of crystal plasticity and to establish a more general applicable and more accurate theory. The techniques and approaches that will be used for this investigation have the potential to contribute more broadly to inelastic deformations, which is generally based on similar kinematic assumptions. This includes models of biological tissues that experience growth, damage models and viscoelasticity theories. The results of this investigation will be included in the undergraduate and graduate education and will serve as an example of interdisciplinary effort that combines mechanics with material science and applied mathematics. The kinematics of elasto-plastic deformations in crystalline materials is very well understood at the atomistic and mesoscopic level where dislocations are fully resolved. However, the effective kinematic description at the macroscopic scale is still a topic of current debate. The research objective of this work is to rigorously derive the continuum kinematic relations describing plastic deformation from its discrete counterparts via an upscaling procedure. This goal will be achieved via a careful mathematical characterization of elasto-plastic deformations at the mesoscale and by use of homogenization techniques in the calculus of variations. The assumptions required to establish the discrete to continuum link will clearly determine the limits of applicability of the continuum relations. The development of new graduate level courses, the mentoring of students, and the targeted interaction with the peer community will contribute to the broader impact of this research.