Mathematical Sciences: Multidimensional Problems in Dynamic Plasticity

这个跨学科的中心目标，继续 项目是发展对现象的基本理解 包括干燥颗粒材料的缓慢流动。 第I部 研究，两个部分的第一部分，重点是双轴 具有两个额外的创新特征的本构测试：（i） 连续测量声速的能力， 使颗粒材料变形， 样品中的变形和传播的方向， 波和（ii）连续测量密度的能力 变化和粒子速度与复杂的X射线成像 设备. 第一部分的目标包括（i）实验， 构造和操作该装置，以及（ii）在计算上， 在所述装置中模拟样品的变形，包括 叠加的声波，直到和超越的形成 剪切带 数学分析具有主要的辅助作用 在提取构成信息中发挥作用， 实验数据，并在开发一个适当的数值 阴谋 研究的第二部分由几个子项目组成 它更深入地探索了以前发现的现象， 补助期。 这些措施包括（i）研究颤振不稳定性 作为筒仓振动的可能原因，（ii）进一步 实验研究和分析理解 孔隙率波在排放筒仓，（iii）以表征更多 并解释了时变应力的功率谱 在卸料筒仓中，以及（iv）探索“标度不变” 初始值问题作为一个可能的范式问题， 弹塑性（类似于气体中的黎曼问题 动力学）。 在应用方面，该项目与 与筒仓内颗粒状物料的流动有关。 然而，在地球物理学中， 材料，并提供丰富的构成信息， 从这个项目中收集到的良好宣传将是 与这些有关。 此外，剪切带的模拟 其影响远远超出了这个项目-数学上， 剪切带涉及偏微分方程， 因为剪切带是导致 在许多材料中失败。

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.