Mathematical Sciences: Nonlocal Models in the Mechanics of Materials

数学科学：材料力学中的非局部模型

• 批准号: 9531925
• 负责人: Roger Fosdick
• 金额: $ 12.86万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531925&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlocal; Models; Mechanics

9531925 Fosdick This research program is concerned with fundamental issues of material behavior that are produced from constitutive structures that sense the nonlocal, long range particle-particle interactions within a body. It represents an attempt to generalize and to place into perspective the somewhat limited spatial gradient theories of local interaction. The investigation is theoretical and is based on the idea of energy minimization. While the generic problems contained in this work are nonconvex, and must address this level of difficulty, to a large extent the interests and objectives lie in understanding the "penalty" due to the global nonlocal structure. The project involves an investigation of the regularity and existence of minimizers. It includes a proposal to describe their detailed form, and a plan to understand how the number of transition boundaries that a body can coincidentally support relates to the degree of nonlocal influence and to the size of the body. The identification of length scales and the application of scaling transformations are suggested as avenues for understanding the way in which gradient theories may be considered as approximations. The overall project is expected to lead to a model for understanding the organization of material phase structures in bodies at finite scales. The detailed investigations are expected to form a bridge in the "regularity gap" that now exists between the purely local, strongly irregular, theories commonly associated with the classical ideas of Gibbs, and the relatively smooth gradient theories historically attributed to van der Waals, Korteweg, Ginzburg and Landau, and Cahn and Hilliard. %%% Understanding the behavior of materials which can change their internal structure when loaded, heated, or placed in chemical or electrical environments, is important to the development of advanced applications in materials science---applications which impact many diverse areas of society, from energy conservati on to micromachines for medical devices. The idea that nondestructive, reversible phase changes can be used to obtain a desired sensible material behavior is of great potential technological importance, and understanding how to affect and control such processes is an essential step in the transfer of this science into technology for the benefit of society. Electrorheological fluids sense a small scale structure and stiffen and become solid-like in the presence of an electric field, and the design of micro-switches in the area of thin film technologies also depends upon a similar small scale phenomenon. There is a need to improve our understanding and predictive capabilities of such materials through the development of relevant mathematical models. The existence of microscopic domains in ferromagnetic and ferroelastic materials, shear bands in ductile solids, finely twinned microstructural laminates in martensite (shape memory alloys, etc...), and recently discovered high transition temperature superconductors all underscore the importance of internal, small scale structures. Experimental observations have provided a wealth of evidence that relates the gross response of a material body to its underlying microstructure. This research program investigates the role that long range particle-particle interactions play in the development of microstructure. It is based on the global optimization of energy and the idea that the amount of energy that is stored in each element of a body is influenced by not only the state of the element, but also by the state of the other elements of the body, in a way similar to what molecular theory might suggest. The theory is phenomenological and, as such, is contained within the framework of nonlinear continuum mechanics. The aim is to determine the significance of nonlocal theory in describing the behavior of materials which support coexistent phase structures---materials which depend upon the existence of these microstructures for their novel responses. ***

小行星9531925 该研究项目关注的是材料行为的基本问题，这些材料行为是由本构结构产生的，这些结构可以感知体内的非局部，长程粒子-粒子相互作用。它代表了一种尝试，概括和透视的局部相互作用的空间梯度理论有些有限。调查是理论上的，是基于能量最小化的想法。 虽然在这项工作中所包含的一般问题是非凸的，必须解决这个困难的水平，在很大程度上的利益和目标在于理解的“惩罚”，由于全球非局部结构。该项目涉及的规律性和存在的极小的调查。 它包括一个建议，以描述其详细的形式，并计划了解如何过渡边界的数量，一个机构可以巧合地支持涉及到非本地的影响程度和身体的大小。长度尺度的识别和尺度变换的应用被认为是理解梯度理论可以被认为是近似的途径。整个项目预计将导致一个模型，用于了解在有限尺度机构的物质相结构的组织。详细的调查预计将形成一个桥梁之间的“正则性差距”，现在存在的纯本地，强烈的不规则，理论通常与经典的想法吉布斯，和相对光滑的梯度理论历史上归因于货车德瓦尔斯，Korteweg，金兹伯格和朗道，卡恩和Hilliard。 了解材料的行为，可以改变其内部结构时，加载，加热，或放置在化学或电气环境中，是重要的材料科学的先进应用程序的发展-应用程序，影响社会的许多不同领域，从节能到医疗设备的微机械。 非破坏性的，可逆的相变可以用来获得所需的敏感材料行为的想法是巨大的潜在技术重要性，了解如何影响和控制这样的过程是将这门科学转化为造福社会的技术的重要一步。 电流变流体感测小尺度结构并在电场存在下变小并变成固体状，并且薄膜技术领域中的微动开关的设计也取决于类似的小尺度现象。有必要通过开发相关的数学模型来提高我们对此类材料的理解和预测能力。铁磁和铁弹性材料中微观畴的存在，韧性固体中的剪切带，马氏体（形状记忆合金等）中精细孪晶微观结构层片，和最近发现的高转变温度超导体都强调了内部小尺度结构的重要性。 实验观察提供了大量的证据，将物质体的总体响应与其底层的微观结构联系起来。本研究计划调查的作用，长程粒子间的相互作用在微观结构的发展发挥。它基于能量的全局优化和这样的想法，即储存在物体每个元素中的能量的量不仅受到元素状态的影响，而且还受到物体其他元素状态的影响，这与分子理论可能提出的方式类似。该理论是唯象的，因此包含在非线性连续介质力学的框架内。目的是确定非局部理论在描述支持共存相结构的材料的行为中的意义--这些材料依赖于这些微结构的存在以获得其新颖的响应。 ***