Multi-scale analysis of field behavior inside heterogeneous media for local and nonlocal continuum theories

利普顿DMS-0807265研究人员开发了新的数学和计算工具，用于定量描述复合材料结构中的多尺度现象。这项工作的重点是与故障引发相关的属性。失效通常是由于局部场在显微水平上的极端偏移而引起的。进行了三个项目，试图在局部变形、应变和应力场作为外加边界载荷和初始条件的函数之间建立新的联系。第一个项目研究了当子结构仅在统计意义上已知时，分层结构的长度尺度之间的荷载传递。我们的目标是提供新的变分工具，用于梳理出将局域场响应与所施加的长波长负载联系起来的关系。其目的是利用这些关系来发现在复合介质中所施加的载荷的明确标准，而这些载荷不是导致复合介质内部破坏的必要条件。第二个项目试图了解在存在残余应力的情况下，非均匀介质中的场波动如何依赖于边界数据。该项目的目标是：i)量化边界数据、微观几何形状和残余应力对统计定义介质内高应力区渗透的作用；ii)了解残余应力对复合域高频边界数据衰减的影响。了解这些现象有助于设计用于航空和基础设施的坚固复合材料结构。第三个项目使用弹性相互作用的非局部模型来研究非均匀介质的动力学。其目标是开发一种多尺度周期动力学公式，用于模拟纤维增强层合板内部的变形场。这一表述提供了捕捉小尺度和大尺度之间的动态相互作用所必需的数学框架和相关的数值方法。预计基本的数学工作和随后的数值研究可以解决导致纤维增强层合板分层和失效的新的动态现象。为了定量描述复合材料结构中的多尺度现象，研究者开发了新的数学和计算工具。复合材料正迅速成为结构应用的首选材料。这是因为它们重量轻，硬度高。为了加速它们的部署，需要根据它们的微观结构几何形状来更好地了解它们的强度特性。该提案集中在与故障引发相关的性质上。破坏往往是由于局部场在显微几何水平上的极端偏移而引起的。该方案包括三个方案，它们将局部变形、应变和应力场的偏移作为施加的边界载荷和初始条件的函数建立新的联系。选择这些项目是为了解决实际感兴趣的问题，如果解决这些问题，可以加快轻质复合材料在航空和基础设施中的应用。

LiptonDMS-0807265 The investigator develops new mathematical and computationaltools for the quantitative description of multi-scale phenomenaseen in composite structures. The effort focuses on propertiesrelated to failure initiation. Failure is often precipitated byextreme excursions of the local fields at the level of themicrogometry. Three projects are carried out that seek to makenew connections between the excursions of local deformation,strain, and stress fields as functions of applied boundary loadsand initial conditions. The first project examines load transferbetween length scales for hierarchical structures when thesubstructure is known only in a statistical sense. The goal isto provide new variational tools for teasing out relationshipsthat connect the local field response to the applied longwavelength loads. The objective is to use these relationships todiscover explicit criteria on the applied loads that arenecessary for failure initiation inside composite media. Thesecond project seeks to understand how field fluctuations insideheterogeneous media depend on the boundary data in the presenceof residual stress. The goals of this project are to: i)quantify the roles of boundary data, microgeometry and residualstress on the penetration of high stress zones insidestatistically defined media and ii) understand the effect ofresidual stress on the decay of high frequency boundary datainside the composite domain. Understanding these phenomenafacilitates the design of tough composite structures for use inaviation and infrastructure. The third project investigates thedynamics of heterogeneous media using nonlocal models for elasticinteractions. The goal is to develop a multi-scale peridynamicformulation for modeling deformation fields inside fiberreinforced laminates. This formulation provides the mathematicalframework and associated numerical methods necessary to capturethe dynamic interactions between small and large length scales. It is anticipated that the basic mathematical work and subsequentnumerical investigations can resolve new dynamic phenomena thatcontribute to the delamination and failure of fiber reinforcedlaminates. The investigator develops new mathematical and computationaltools for the quantitative description of multi-scale phenomenaseen in composite structures. Composite materials are rapidlybecoming the materials of choice for structural applications. This is due to their light weight and superior stiffnessproperties. In order to accelerate their deployment an improvedunderstanding of their strength properties based upon theirmicrostructural geometry is required. This proposal focuses onproperties related to failure initiation. Failure is oftenprecipitated by extreme excursions of the local fields at thelevel of the microgometry. The proposal consists of threeprojects that make new connections between the excursions oflocal deformation, strain, and stress fields as functions ofapplied boundary loads and initial conditions. The projects arechosen to address issues of practical interest that if solved canaccelerate the use of light weight composite materials forapplications in aviation and infrastructure.