Mathematical Sciences: Multidimensional Problems in Dynamic Plasticity

这个跨学科的持续项目的中心目标是发展对干燥颗粒材料缓慢流动现象的基本理解。本研究的第一部分，即两部分中的第一部分，重点关注具有两个附加创新特征的双轴本构试验：(1)连续测量粒状材料在变形过程中声速作为试样中累积变形的函数和声波传播方向的能力；(2)使用先进的x射线成像设备连续测量密度变化和粒子速度的能力。第一部分的目标包括(I)实验，构建和操作仪器和（ii）计算，模拟仪器中样品的变形，包括叠加声波，直至和超越剪切带的形成。数学分析在从实验数据中提取本构信息和制定适当的数值方案方面起着重要的辅助作用。研究的第二部分由几个子项目组成，这些子项目对前一个资助期发现的现象进行了更深入的探讨。其中包括：(i)研究作为筒仓振动可能原因的颤振不稳定性，（ii）进一步实验研究和分析理解卸料筒仓中的孔隙波，（iii）更全面地表征和解释卸料筒仓中随时间变化的应力功率谱。（iv）探索“尺度不变”初值问题作为弹塑性中可能的范式问题（类似于气体动力学中的黎曼问题）。在应用方面，该项目与筒仓内颗粒物料的流动密切相关。然而，有许多颗粒材料的地球物理出现，从这个项目中收集到的关于声音传播的丰富本构信息将与这些有关。此外，剪切带的模拟具有远远超出本项目的意义-数学上是因为剪切带涉及改变类型的PDE，科学上是因为剪切带是导致许多材料失效的普遍机制。

The central goal of this interdisciplinary, continuing project is to develop fundamental understanding of phenomena involving the slow flow of dry granular materials. Part I of the research, the first of two parts, focuses on a biaxial constitutive test with two additional, innovative, features: (i) the capacity to continuously measure the speed of sound in deforming granular material as a function of the accummulated deformation in the sample and the direction of propagation of the waves and (ii) the capacity of continuously measure density changes and particle velocities with sophisticated X-ray imaging equipment. The goals of Part I include (i) experimentally, to construct and operate the apparatus and (ii) computationally, to simulate deformation of a sample in the apparatus, including superimposed acoustic waves, up to and beyond the formation of shear bands. Mathematical analysis has a major supporting role to play in extracting constitutive information from the experimental data and in developing an appropriate numerical scheme. Part II of the research consists of several subprojects which probe more deeply phenomena discovered in the previous grant period. These include (i) to study the flutter instability as a possible cause of vibrations in silos, (ii) to further investigate experimentally and to understand analytically porosity waves in a discharging silo, (iii) to characterize more fully and to explain the power spectra of time-dependent stresses in a discharging silo, and (iv) to explore "scale-invariant" initial value problems as a possible paradigm problem in elastoplasticity (analogous to the Riemann problem in gas dynamics). Regarding applications, this project has been most closely associated with the flow of granular materials inside silos. However, there are many geophysical occurrences of granular materials, and the rich supply of constitutive information on sound propagatioyn to be gathered from this project will be relevant for these. Moreover, the simulation of shear banding has implications far beyond this project - mathematically because shear banding involves PDE which change type and scientifically because shear banding is a widespread mechanism leading to failure in many materials.