Collaborative Research: A Sweeping Process Framework to Control the Dynamics of Elastoplastic Systems

在极端条件下准确有效地预测材料的机械行为对于解决安全，能源和健康方面的重大挑战的新型材料的设计变得越来越重要。这些例子从微芯片上微米大小的焊点到飞机的关键结构部件。材料在循环荷载作用下产生的局部塑性（即不可逆）变形是导致材料性能丧失和失效的最典型途径。近年来，连接弹簧格被广泛用于模拟现代材料在循环荷载作用下的塑性变形。然而，在目前可用的理论范围内，只能控制晶格弹簧模型的弹性（即可逆）变形。该奖项支持一种数学理论的发展，该理论能够预测和影响允许弹性和塑性变形的晶格弹簧模型的渐近行为（称为弹塑性）。新的数学框架将提供一种革命性的工具，以加速计算塑性变形集中的区域（已知会导致裂纹初始化），并将使设计具有优越使用寿命的材料在计算上可行。所设计的材料（例如，超级抗疲劳合金）最终可用于影响航空航天、汽车、微电子和生物医学等行业。因此，这项研究的结果将有利于美国社会和国家安全。多学科合作将有助于扩大代表性不足的群体在研究中的参与，并对数学和工程教育产生积极影响。具有运动多面体约束的微分方程（通常称为扫描过程）将用于模拟循环加载下弹塑性弹簧的晶格。通过发展清扫过程的稳定性和分岔理论，该项目将确定晶格弹簧模型的力学参数，以确保给定循环载荷的唯一周期响应（有限时间稳定或渐近）或共存周期响应（隔离或不隔离）。所发现的动力行为将用于有效地计算塑性变形的渐近分布。该工具的性能将通过将其应用于此类异质材料的设计来证明，这些材料的塑性变形分布（在对循环加载的响应中）尽可能保持均匀。在本设计中，将使用体积补偿晶格-粒子方法将材料微观结构的数字表示映射到晶格弹簧模型。该设计将使用3d打印样品复合材料进行实验验证。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Accurate and efficient prediction of the mechanical behavior of materials under extreme conditions is becoming increasingly crucial for the design of novel materials that address the grand challenges in security, energy and health. The examples range from micron-sized solder joints in micro-chips to crucial structural parts of airplanes. Localized plastic (i.e. irreversible) deformations that the material develops under cyclic loading represent the most typical route to the loss of performance and material's failure. Recently, lattices of connected springs became widely used to model plastic deformations of modern materials under cyclic loading. However, only elastic (i.e. reversible) deformations of lattice spring models can be controlled within the currently available theory. This award supports the development of a mathematical theory with the capability to predict and influence the asymptotic behavior of lattice spring models that are allowed to deform both elastically and plastically (termed elastoplastically). The new mathematical framework will provide a revolutionary tool to accelerate computation of the regions where the plastic deformations concentrate (known to cause crack initialization) and will make it computationally feasible to design materials with superior service lifetime. The designed materials (e.g., super fatigue resistant alloys) can be eventually manufactured to impact such industries as aerospace, automobile, microelectronics and biomedical. Therefore, the results from this research will benefit the U.S. society and national security. The multi-disciplinary collaboration will help broaden participation of underrepresented groups in research and positively impact mathematical and engineering education.Differential equations with moving polyhedral constraints (commonly known as sweeping processes) will be used to model the lattices of elastoplastic springs under cyclic loading. By developing a theory of stability and bifurcations for sweeping processes, this project will identify the mechanical parameters of lattice spring models that ensure a unique periodic response (finite-time stable or asymptotic) or co-existing periodic responses (isolated or not) to a cyclic loading given. The dynamical behavior found will be used to efficiently compute the asymptotic distribution of plastic deformations. The performance of this tool will be demonstrated by applying it to the design of such heterogeneous materials for which the distribution of plastic deformations (in the response to cyclic loading) stays as uniform as possible. In this design, the Volume-Compensated Lattice-Particle method will be utilized to map the digital representation of the material microstructure to a lattice spring model. The design will be experimentally validated using 3D-printed sample composite materials.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.