Mathematical Sciences: Nonlocal Models in the Mechanics of Materials

这个研究项目关注的是材料行为的基本问题，这些问题是由本构结构产生的，这些本构结构可以感知物体内部的非局部、远距离粒子-粒子相互作用。它代表了对局部相互作用的有限空间梯度理论进行概括和透视的尝试。这项研究是理论性的，基于能量最小化的思想。虽然本工作中包含的一般问题是非凸的，并且必须解决这个难度水平，但在很大程度上，兴趣和目标在于理解由于全局非局部结构而导致的“惩罚”。该项目涉及对最小值的规律性和存在性的调查。它包括描述其详细形式的建议，以及了解一个机构可以碰巧支持的过渡边界的数量如何与非局部影响的程度和机构的大小相关的计划。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能被视为近似的途径。整个项目预计将导致一个模型，以理解在有限尺度的身体材料相结构的组织。这些详细的研究有望在“规律性差距”之间形成一座桥梁，这种差距目前存在于纯粹局部的、强烈不规则的、通常与吉布斯的经典思想有关的理论，以及历史上归因于范德华斯、科尔特维格、金兹堡和兰道以及卡恩和希利亚德的相对平滑的梯度理论之间。了解材料在加载、加热或放置在化学或电气环境中时可以改变其内部结构的行为，对于材料科学中先进应用的发展非常重要，这些应用影响着社会的许多不同领域，从能源保护到医疗设备的微型机器。无损的、可逆的相变可以用来获得理想的可感知的材料行为，这一想法在技术上具有巨大的潜在重要性，了解如何影响和控制这些过程是将这一科学转化为造福社会的技术的重要一步。电流变流体感知小尺度结构并在电场存在下变硬并变成固体状，薄膜技术领域的微开关设计也依赖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这类材料的理解和预测能力。铁磁和铁弹性材料中存在微观畴，韧性固体中的剪切带，马氏体中的细孪晶微结构层压板（形状记忆合金等），以及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。实验观察提供了丰富的证据，将物质体的总响应与其底层微观结构联系起来。本研究计划探讨粒子间的远距离相互作用在微观结构发展中的作用。它的基础是能量的全局优化，以及人体每个元素中储存的能量不仅受到元素状态的影响，还受到身体其他元素状态的影响，这与分子理论可能提出的观点类似。该理论是现象学的，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行为中的意义-材料依赖于这些微观结构的存在来获得新的响应。＊＊＊

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. ***