DynSyst_Special_Topics:Collaborative Research: Fundamental and Applied Dynamics of Granular Crystals: Disorder, Localization and Energy Harvesting

本计画的研究目标是研究粒状晶体中可能出现的非线性波形，包括行波、缺陷模和离散呼吸子。它还打算提供一个路线图，以确定这些不同的结构何时会出现，系统地建立在复杂性，从单体晶格，以调光器，微调和异构的，从一个到多个缺陷链，甚至随机晶格。粒状晶体由相互作用的粒子链组成，当它们碰撞时会发生弹性变形。它们的性质（材料类型，尺寸，形状）是非常可调的，它们的动态响应可以被修改为弱或强非线性。该项目将涉及从动力系统，非线性常微分方程和偏微分方程，渐近分析和理论方面的数值计算的工具和方法的富有成效的交叉授粉。这些将与物理实验交织在一起，这些实验将试图激发和识别相关的非线性波，并研究它们的特性（振幅，宽度，能量，寿命，相互作用，高维类似物等）。数学理论、数值模拟和物理实验之间的持续反馈回路将成为研究的驱动力。预计该项目将产生重大的社会影响，因为可能发现可能影响日常生活的机械系统。特别是，本地化模式将被研究在能量收集系统，延迟线和冲击保护或振动吸收材料的潜在用途。 此外，相关的努力将涉及学生研究人员与积极的教育组成部分，包括正式（与课程和研究会议相关）和非正式（组间交流和访问）的教育机会。它将汇集来自不同背景（数学和航空/航天工程）的学生，为他们提供丰富的跨学科经验，为学习和科学交流/合作提供良好的框架。该计划还将促进两个机构之间的合作，以及与其他大学（英国牛津大学，普林斯顿大学和雅典大学）的合作。我们预计，一些会议报告，期刊文章和专利申请将出现从这项研究。

The research objective of the present proposal is to study nonlinear waveforms that may arise in granular crystals, including traveling waves, defect modes, and discrete breathers. It also intends to provide a roadmap for identifying when these different structures will arise, systematically building up in complexity, from monomer lattices, to dimmer, trimmer and heterogeneous ones, and from one- to multiple-defect chains and even random lattices. Granular crystals consist of chains of interacting particles that deform elastically when they collide. Their properties (material types, sizes, shapes) are extremely tunable and their dynamic response may be modified to be weakly or strongly nonlinear. The project will involve a fruitful cross-pollination of tools and methods from dynamical systems, nonlinear ordinary and partial differential equations, asymptotic analysis and numerical computation on the theoretical side. These will be interwoven with physical experiments that will attempt to excite and identify the relevant nonlinear waves and to investigate their properties (amplitude, width, energy, lifetime, interactions, higher-dimensional analogs etc.). A continuous feedback loop between mathematical theory, numerical simulation and physical experimentation will be the driving force of the study.It is anticipated that this project will have a significant societal impact due to the potential discovery of mechanical systems that may affect aspects of everyday life. In particular, localized modes will be studied for potential use in energy harvesting systems, delay lines and shock protective or vibration absorption materials. Additionally, the relevant effort will involve student researchers with a vigorous educational component including both formal (associated with coursework and research conferences) and informal (inter-group exchanges and visits) educational opportunities. It will bring together students from fundamentally different backgrounds (mathematics and aeronautical/aerospace engineering) providing them with a rich interdisciplinary experience that will offer an excellent framework for learning and scientific exchange/collaboration. The program will also foster collaborations between the two institutions, but also with other Universities (Oxford University in the UK, Princeton University and the University of Athens). We expect that several conference presentations, journal articles and patent applications will emerge from this study.