Collaborative Research: A New Framework for Prediction of Buckling and Other Critical Transitions in Nonlinear Structural Mechanics

除了对大型工程结构稳定性和坍塌的传统关注外，屈曲现象在许多其他自然和工程系统中也起着关键作用，例如，植物对刺激的响应，宏观工程结构的变形，周期性多孔材料的图案，以及从2D纳米结构设计3D形状。更一般地说，预测势能井的逃逸是一种普遍的练习，并且控制着许多在面对干扰时无法保持稳定的物理系统的行为。所有这些情况本质上都是由系统状态的大规模组织所主导的，因为系统会找到一种方法来退出一种状态（即行为）并进入另一种状态。这种转变通常是突然的，而这种逃避行为的后果可能是灾难性的。关注低阶实验屈曲系统是从抽象的理论概念过渡到实际和面向设计的指导方针的关键中间步骤。该奖项支持一项应用新范式的努力，通过一种新的理论计算方法，评估在统计意义上可以通过实验测试逃逸问题的方法，从而为预测、设计和某些情况下的预防提供框架，为更复杂的设计和控制这类系统奠定基础。这个项目汇集了两位背景互补的研究人员。虽然该框架将在非线性结构动力学领域内发展，但数学还有许多其他潜在的应用：化学反应，纳米结构，地震工程，船舶动力学，仅举几例，甚至可能在工程之外的用途：某些生物和生态系统。这个研究项目的范围和范围将为本科生和研究生提供一个强大的训练环境。研究小组还将制定一个积极的推广计划，通过在高中使用3D打印机技术的快速现象和自主探究来激发年轻人对动力学的兴趣。该奖项将支持理论和应用力学方面的努力，以开发一种创新的统一方法来研究实验非线性结构屈曲，使用高维圆柱状相空间结构作为理解相空间中潜在井（平衡状态）之间运动的动力学的基本基础。这些管状特征在整体意义上组织了相空间中轨迹的演化。这一观点将被应用于理解一些不同复杂程度的轴向载荷细长机械结构的行为。对于具有时变作用力的单自由度系统，先前已经考虑并确定了从潜在井中逃生的标准和路线，并与实验有一定的一致性。然而，当存在两个或多个自由度时，情况就变得更加复杂。然而，即使在更高维度的情况下，最近的工作表明，一个理论计算框架的开始，用于确定这些轨迹的边界，这些轨迹将很快逃逸（或等价地，在多井系统中，井之间的过渡）。对于确定性系统，这些方法本质上是几何的，当噪声和随机效应被纳入时，这些方法自然地融合到概率框架中。该研究为更深入地理解一类重要的结构力学问题提供了一致的方法，具有潜在的应用于传统工程结构的屈曲预测，以及设计可弯曲、折叠和扭曲的自适应结构的新机会，即可控地将结构变形成所需的形状以实现某些目标。

In addition to the conventional concern of stability and collapse in large-scale engineering structures, buckling phenomena play a key role in many other natural and engineered systems, e.g., plants responding to stimuli, deformations of macro-scale engineering structures, patterning of periodic porous materials, and design of 3D shapes from 2D nanostructures. More generally, predicting the escape from a potential energy well is a universal exercise, and governs behavior in many physical systems that are unable to maintain stability in the face of disturbances. All these situations are essentially dominated by the large-scale organization of system states, as the system finds a way to exit one state (i.e., behavior) and get to another. Often this transition is sudden, and the consequences of this escaping behavior may be catastrophic. Focusing attention on low-order experimental buckling systems is the key intermediate step in transitioning from an abstract theoretical concept to practical and design-oriented guidelines. This award supports an effort to apply a new paradigm, by way of a fresh theoretical-computational approach, to assess the ways in which the problem of escape can be tested experimentally, in a statistical sense, and hence provide a framework for prediction, design, and in some cases prevention, laying the groundwork for more sophisticated design and control of such systems. This project brings together two researchers from complementary backgrounds. Although the framework will be developed within the realm of nonlinear structural dynamics, there are many other potential applications of the mathematics: chemical reactions, nanostructures, earthquake engineering, ship dynamics, to name a few, and even possible utility beyond engineering: certain biological and ecological systems. The broad range and reach of this research project will provide a strong training environment for undergraduate and graduate students. The research team will also develop an active outreach program to excite young minds about dynamics through snap-through phenomena and self-guided inquiry using 3D printer technology at high schools.This award will support an effort in theoretical and applied mechanics to develop an innovative unified approach to experimental nonlinear structural buckling, using high dimensional cylinder-like phase space structures as the fundamental basis for understanding the dynamics governing motion between potential wells (equilibrium states) in phase space. These tube features organize the evolution of trajectories in phase space in a global sense. This point-of-view will be applied to understand the behavior of a number of axially loaded slender mechanical structures, of varying degrees of complexity. Criteria and routes of escape from a potential well have previously been considered and determined for one degree of freedom systems with time-varying forcing, with reasonable agreement with experiments. However, when there are two or more degrees of freedom, the situation becomes more complicated. Yet, even for the higher dimensional case, recent work suggests the beginnings of a theoretical-computational framework for determining the boundary of those trajectories which will soon escape (or equivalently, transition between wells in multi-well systems). These methods are geometric in nature for deterministic systems, merging naturally into a probabilistic framework when noise and stochastic effects are incorporated. The research develops a consistent approach to a deeper understanding of an important class of structural mechanics problems, with potential applications to buckling prediction for traditional engineered structures, and emerging opportunities to design adaptive structures that can bend, fold, and twist, i.e., controllably morphing structures into a desired shape to achieve some objective.