Theory and Applications of Coupled Mathematical Models for Environmentally-friendly Multiscale Materials Systems

目前许多先进技术严重依赖于对环境有害的成分。其中一种成分是铅，其含量在商业上可用的尖端材料（如压电材料）中可能超过重量的60%。这些材料及其复合材料在许多需要机械和电场之间强耦合的应用中表现出优异的性能。然而，在不影响材料性能的情况下，消除对环境有害成分的动力越来越强，环保法规也越来越严格。这项任务涉及许多挑战，其中一些需要应用数学家更加密切的关注。基于我们在发展耦合多尺度数学模型方面的成功工作，拟议的研究计划将其扩展到新的领域，这对发展最先进的数学理论和创新应用都至关重要。该计划旨在进一步推进环境友好型多尺度材料系统分析的耦合数学模型的开发和应用。主要目标是系统地解释这一发展中问题的耦合和多尺度性质。应用方面，重点将放在几个关键类别的多尺度环境友好材料系统，如压电复合材料。在第一种近似中，优雅简单的数学概念和公式可以应用于一些这样的系统，例如，通过将它们视为由两个主要成分组成的系统-被认为是背景材料的矩阵和基于晶体颗粒的内含物。然而，这一领域的许多数学挑战仍处于几乎未被探索的领域。为了实现其长期目标，该计划将首先允许对此类系统的性质进行系统研究，考虑非线性和非局部耦合效应，以及夹杂物的多晶结构。其次，将对开发的模型进行微观结构和性能增强纳米添加剂的数学和数值分析。第三，它将系统地研究一类重要的几何定制系统。最后，鉴于其普遍存在的性质，预计在这个项目中开发的模型和工具将有助于解决其他具有挑战性的数学问题及其应用。事实上，结果可能不仅仅局限于给出的例子，而且可以用于研究科学和工程中的其他重要系统。该计划的核心包括数学模型的基础和应用研究，这将导致最先进的科学进步，并将对重要的全球挑战产生积极影响。培养高素质人才和扩大国际合作将与项目的发展有效结合。

Many current advanced technologies rely heavily on environmentally harmful components. One of such components is lead whose contents may exceed 60 weight percent in commercially available cutting-edge materials such as piezoelectrics. These materials and their composites demonstrate excellent performance in many applications where strong coupling between mechanical and electric fields is required. However, there is a strong impetus and increasingly strict environmental regulations to eliminate the environmentally harmful components without compromising the materials performance. This task is connected with a multitude of challenges, some of which require much more close attention of applied mathematicians. Building on our successful work in the development of coupled multiscale mathematical models, the proposed research program expands it to new areas, crucial for both the development of state-of-the-art mathematical theories and innovative applications. The program is aimed at further advancement in the development and applications of coupled mathematical models for the analysis of environmentally-friendly multiscale materials systems. The main goal is to account systematically for coupling and multiscale nature of the problem in this development. Application-wise, major focus will be given to several key classes of multiscale environmental-friendly materials systems such as piezoelectric composites. In the first approximation, elegantly simple mathematical concepts and formulations can be applied to some such systems, e.g., by viewing them as those consisting of two main components - a matrix, considered as a background material, and crystalline-particle-based inclusions. However, many mathematical challenges in this field still are on only scarcely explored horizons. In addressing its long-term goals, the program will, firstly, allow a systematic study of properties of such systems, accounting for nonlinear and nonlocal coupled effects, and for the polycrystalline structure of inclusions. Secondly, it will carry out mathematical and numerical analysis of the developed models with microstructures, as well as with performance-enhancing nano-additions. Thirdly, it will systematically study an important class of geometrically-tailored systems. Finally, given their ubiquitous nature, it is expected that the models and tools developed in this program will assist in addressing other challenging problems of mathematics and its applications. Indeed, the results may not be restricted to just the exemplifications given and can be useful in studying other important systems in science and engineering. The program's core includes fundamental and applied research in mathematical modelling that will lead to state-of-the-art scientific advances and will entail positive impacts on important global challenges. Training of highly qualified personnel and expanding international collaboration will be effectively integrated with the development of the program.