Understanding Continuum Models of Elasto-Plastic Deformations via Multiscale Analyses

金属的塑性变形过程与广泛的技术应用有关，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做出了巨大的努力。然而，对这些连续体模型背后的运动学假设及其与微观过程的关系仍然缺乏完整的理解。这项研究的目的是在晶体塑性的背景下修正关键的模型假设，并建立更普遍、更适用和更准确的理论。将用于这项研究的技术和方法有可能对非弹性变形作出更广泛的贡献，而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。这项调查的结果将被纳入本科生和研究生教育，并将成为将力学与材料科学和应用数学相结合的跨学科努力的典范。晶体材料中弹塑性变形的运动学在原子和介观水平上被很好地理解，在那里位错被完全分解。然而，在宏观尺度上有效的运动学描述仍然是当前争论的主题。这项工作的研究目标是严格地推导出描述塑性变形的连续介质运动学关系，通过放大过程从离散的对应关系出发。这一目标将通过在中尺度上对弹塑性变形进行仔细的数学描述和使用变分中的均匀化技术来实现。建立从离散到连续联系所需的假设将清楚地确定连续关系的适用范围。新研究生水平课程的开发、对学生的指导以及与同伴社区的有针对性的互动将有助于这项研究的更广泛影响。

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.