Mathematical Sciences: Multidimensional Problems in Dynamic Plasticity

摘要：动态塑性中的多维问题这个跨学科的提案解决了关于颗粒流动动力学的三个广泛的科学问题：(1)不稳定，包括不稳定后状态下的模式形成；(2)波动；(3)多尺度颗粒流动的计算。目前和计划进行的实验包括：(a)使用具有测量声速和用x射线连续监测变形能力的双轴装置进行本构试验，(b)进一步研究孔隙率波，(c)旨在分离振荡颗粒材料不稳定原因的实验，以及(d)探测颗粒流动波动的各个方面的实验，包括应力链和1/f噪声。基于非联想塑性的主导偏微分方程在中等应变下变得不适定这一事实，目前的分析工作寻求(a)在数学考虑和为二维和三维现象的数值模拟建立健全理论框架的需要的驱动下，概括以前的一维工作；(b)将不适定性与各种实验现象（如孔隙波和剪切带）联系起来。金属塑性的相关问题也在进行分析研究。数值工作包括(a)连续体和(b)分子动力学（MD）计算。前者的关键工作是完成模拟剪切带形成和传播的代码，特别是在双轴试验中；这个代码包括前跟踪和网格细化在剪切带。MD计算的直接目标是收集颗粒流动波动的定量信息，特别是这种波动随长度尺度的变化。从长远来看，计划开发一种混合代码，在解光滑的区域求解连续统方程，在快速变化的区域调用MD。应用工程的许多领域都将受益于本项目所解决的基本问题的进展，包括(1)颗粒处理和运输，(2)土力学，(3)材料成型，以及(4)岩土工程。下面详细阐述第(1)部分。据估计，化学工业增加值的40%，即610亿美元，与颗粒技术有关。兰德公司（Rand Corporation）的一项研究发现，由于无法准确预测粉末的性能，生产固体的制造工厂的平均产能利用率为63%，而生产液体的工厂的平均产能利用率为84%。从经济角度来看，这种差异是惊人的。（关于未来的竞争力，美国应该注意到德国和日本在粒子技术研究方面处于世界领先地位。）对颗粒物料流动的基本理解将有助于(a)找到控制工业问题的方法，(b)开发新的、更有效的工业工艺。为了说明(a)：颗粒流动的困难之一是，在表面上相同的情况下，可能会产生完全不同的行为，特别是当涉及到放大时（例如，从实验室规模的实验到工业筒仓）。大多数现有的理论仅仅试图用平均量来描述流体，而忽略了这些平均量的偏差。该项目对波动和长度尺度的关注提供了能够预测和设计真实材料处理系统的全部行为的可能性。为了说明(b)：在这个项目中，用摇晃过的颗粒材料进行的实验产生了两个计划申请专利的想法，一个是获得多尺寸颗粒的均匀混合，另一个是将大的表面积暴露给周围的气体（就像在流化床中一样，但不需要流体流动）。这些应用正在与工业界的研究人员共同探索。

ABSTRACT: MULTIDIMENSIONAL PROBLEMS IN DYNAMIC PLASTICITY This interdisciplinary proposal addresses three broad scientific issues regarding the dynamics of granular flow: (1) instability, including pattern formation in the post-instability regime, (2) fluctuations, and (3) computation of granular flows with multiple scales. Current and planned experiments include: (a) constitutive tests using a biaxial apparatus with the capability of measuring the speed of sound and of continuously monitoring the deformation with x-rays, (b) further study of porosity waves, (c) experiments directed toward isolating the causes of the instabilities of shaken granular material, and (d) experiments probing various aspects of fluctuations in granular flow, including stress chains and 1/f noise. Based on the fact that the governing PDE of nonassociative plasticity become ill-posed at moderate strains, current analytical work seeks (a) to generalize previous one-dimensional work, driven by both mathematical considerations and the need to establish a sound theoretical framework for numerical simulations of two-and-three-dimensional phenomena and (b) to relate ill-posedness to various experimental phenomena such as porosity waves and shear banding. Related problems in metal plasticity are also being studied analytically. Numerical work includes both (a) continuum and (b) molecular-dynamics (MD) computations. The key effort in the former is to complete a code for simulating shear-band formation and propagation, especially in the biaxial test; this code includes front tracking and mesh refinement at the shear band. MD computations have the immediate goal of gathering quantitative information about fluctuations in granular flow, particularly the variation of such fluctuations with length scale. In the long range, it is planned to develop a hybrid code that solves continuum equations in regions where the solution is smooth and invokes MD in regions of rapid change. Many areas of applied engineer ing stand to benefit from progress on the fundamental questions addressed in this project, including (1) particle handling and transport, (2) soil mechanics, (3) materials forming, and (4) geotechnical engineering. The following elaborates on area (1). An estimated 40%, or $61 billion, of the value added by the chemical industry is linked to particle technology. A study by the Rand Corporation found that, because of inability to accurately predict powder behavior, solids-producing manufacturing plants performed on average at 63% of design capacity, compared to 84% for liquids-producing plants. In economic terms, this difference is staggering. (Regarding future competitiveness, the U.S. should note that Germany and Japan lead the world in particle-technology research.) Fundamental understanding of the flow of granular materials would help (a) in finding ways to control industrial problems and (b) in developing new, more efficient industrial processes. To illustrate (a): one of the difficulties of granular flow is that quite different behavior may result from apparently identical circumstances, especially when scale-up is involved (e.g., from laboratory-scale experiments to an industrial silo). Most existing theories attempt to describe the flow only in term of average quantities, ignoring deviations from these averages. The focus in this project on fluctuations and length scales offers the possibility of being able to predict, and design for, the full range of behavior of real materials-handling systems. To illustrate (b): the experiments in this project with shaken granular material have led to ideas for two planned applications for patents, one in obtaining uniform mixing of multi-sized particles and the other in exposing a large surface area to the surrounding gas (as in a fluidized bed but not requiring fluid flow). These applications are being explored in concert with researchers in industry.