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

准确有效地预测材料在极端条件下的力学行为对于设计新型材料以应对安全、能源和健康方面的重大挑战变得越来越重要。这些例子从微芯片中的微米级焊点到飞机的关键结构部件。材料在循环载荷下产生的局部塑性（即不可逆）变形是导致性能损失和材料失效的最典型途径。最近，连接弹簧网格被广泛用于模拟现代材料在循环载荷下的塑性变形。然而，只有弹性（即可逆）变形的晶格弹簧模型可以控制在目前可用的理论。该奖项支持数学理论的发展，该理论能够预测和影响允许弹性和塑性变形（称为弹塑性）的晶格弹簧模型的渐近行为。新的数学框架将提供一个革命性的工具，以加速塑性变形集中的区域（已知会导致裂纹初始化）的计算，并将使其计算可行的设计材料与上级的使用寿命。设计的材料（例如，超级抗疲劳合金）最终可以被制造以影响诸如航空航天、汽车、微电子和生物医学的工业。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研究中代表性不足的群体的参与，并对数学和工程教育产生积极影响。带有移动多面体约束的微分方程（通常称为扫描过程）将用于模拟循环载荷下的弹塑性弹簧网格。通过发展一个稳定性和分叉理论的扫描过程，这个项目将确定格形弹簧模型的机械参数，确保一个独特的周期性响应（有限时间稳定或渐近）或共存的周期性响应（孤立或不）的循环加载。所发现的动力学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塑性变形分布（响应于循环载荷）尽可能均匀的非均质材料的设计来证明。在该设计中，将利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。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.