Limiting Theories in Material Science: Mathematical derivation and Analysis

材料科学的极限理论：数学推导与分析

• 批准号: 313878761
• 负责人: Professor Dr. Peter Bella
• 金额: --
• 依托单位: Lehrstuhl I: Analysis
• 依托单位国家: 德国
• 项目类别: Independent Junior Research Groups
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/313878761?language=en
• 关键词: Limiting; Theories; Material; Science; Mathematical

Recent technological advances have allowed for engineering of materials at ever decreasing scales toward broad applications, ultrathin films being one of the examples. The presence of different length scales often makes the numerical study of models in material science prohibitively expensive. Instead of treating them numerically one first studies them analytically to obtain some understanding of their solutions and then uses the acquired insight to pave the way for development of more effective numerical methods. Our goal is to rigorously analyze a few such problems. In the first part of the project we study the wrinkling patterns in compressed thin elastic sheets. In some situations the wrinkling could be non-uniform and the actual pattern shows branching. To understand wrinkling microstructure we consider a variational viewpoint, and identify and analytically study the next-order expansion of the energy (the subdominant energy) in the limit of vanishing sheet thickness. Within this framework we will study several physical situations, a model describing graphene nanoribbons being one of them. In the second part we study elliptic systems with random and rapidly oscillating coefficients, with the application to the model describing heterogeneous linearly-elastic materials in mind. Though the microscopic behavior could be quite complicated, due to stochastic cancellations the macroscopic behavior should be much simpler and deterministic, a process called homogenization. We will use PDE methods to study quantitative aspects of the stochastic homogenization for elliptic systems. The last area of research concerns behavior of compressible viscous fluids in domains with rough boundaries. Rather than study the problem in a rough domain, one poses the problem in a smooth domain where the roughness of the original boundary is reduced to an effective boundary law. Using the concept of relative energy inequality for dissipative solutions to the Navier-Stokes system our aim is to rigorously derive these effective boundary conditions and analyze the error one makes by taking this approach.

最近的技术进步使得材料工程在不断缩小的规模下走向广泛的应用，超薄膜就是其中一个例子。不同长度尺度的存在往往使材料科学模型的数值研究代价高昂。人们不是用数值方法来处理它们，而是首先用分析方法来研究它们，以获得对它们的解的一些理解，然后利用获得的洞察力为开发更有效的数值方法铺平道路。我们的目标是严谨地分析几个这样的问题。在项目的第一部分，我们研究了压缩弹性薄板的起皱模式。在某些情况下，褶皱可能是不均匀的，实际的图案显示出分支。为了理解起皱微结构，我们从变分的角度出发，识别并分析研究了在薄板厚度消失极限下能量的次阶扩展。在这个框架内，我们将研究几种物理情况，描述石墨烯纳米带的模型就是其中之一。在第二部分中，我们研究了具有随机和快速振荡系数的椭圆系统，并将其应用于描述非均质线弹性材料的模型。尽管微观行为可能相当复杂，但由于随机消去，宏观行为应该更简单和确定，这一过程称为均质化。我们将使用偏微分方程方法来研究椭圆系统随机均匀化的定量问题。最后一个研究领域涉及可压缩粘性流体在粗糙边界域中的行为。而不是在粗糙域研究问题，而是在光滑域提出问题，其中原始边界的粗糙度被简化为有效的边界律。利用Navier-Stokes系统耗散解的相对能量不等式的概念，我们的目的是严格推导这些有效的边界条件，并分析采用这种方法所产生的误差。