Mathematical Sciences: Multidimensional Problems in Dynamic Plasticity

摘要：动态塑性中的多维问题这一跨学科的提案涉及关于颗粒流动力学的三个广泛的科学问题：(1)不稳定性，包括不稳定后区域的图案形成；(2)涨落；(3)多尺度颗粒流的计算。目前和计划中的实验包括：(A)使用能够测量声速和用X射线连续监测变形的双轴仪器的本构测试；(B)进一步研究孔隙率波；(C)旨在隔离摇动颗粒材料不稳定原因的实验；以及(D)探索颗粒流动波动的各个方面的实验，包括应力链和1/f噪声。基于非缔合塑性的主导偏微分方程在中等应变时变得不适定的事实，目前的分析工作试图(A)推广以前的一维工作，既有数学上的考虑，也有为二维和三维现象的数值模拟建立健全的理论框架的需要，以及(B)将不适定与各种实验现象联系起来，如孔隙率波和剪切带。对金属塑性的相关问题也进行了分析研究。数值计算包括(A)连续体和(B)分子动力学(MD)计算。前者的关键工作是完成剪切带形成和传播的模拟程序，特别是在双轴试验中；该程序包括剪切带上的锋面跟踪和网格加密。MD计算的直接目标是收集有关颗粒流中波动的定量信息，特别是这种波动随长度尺度的变化。在长期内，计划开发一种混合代码，在解顺利的区域求解连续方程，并在快速变化的区域调用MD。许多应用工程领域将受益于该项目中涉及的基本问题的进展，包括(1)颗粒处理和运输，(2)土力学，(3)材料成型，和(4)岩土工程。以下就第(1)方面进行详细阐述。据估计，化学工业增加值的40%，即610亿美元与颗粒技术有关。兰德公司的一项研究发现，由于无法准确预测粉末行为，固体生产工厂的平均表现为设计产能的63%，相比之下，液体生产工厂的平均表现为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。)对颗粒状材料流动的基本了解将有助于(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。举例说明(A)：颗粒流的困难之一是，明显相同的情况下可能会产生非常不同的行为，特别是当涉及放大时(例如，从实验室规模的实验到工业筒仓)。大多数现有的理论试图仅用平均量来描述流动，而忽略了与这些平均数的偏差。本项目的重点是波动和长度标尺，为能够预测和设计真实物料搬运系统的全范围行为提供了可能性。为了说明(B)：这个项目中关于摇动颗粒材料的实验已经产生了两个计划中的专利申请的想法，一个是获得多种尺寸颗粒的均匀混合，另一个是将大表面积暴露在周围的气体中(就像在流态化床中，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.