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。研究的第一部分，是两部分中的第一部分，重点是双轴本构试验，其中增加了两个创新的，其特点是：(I)能够根据样品中的累积变形和波的传播方向连续测量颗粒材料变形中的声速，以及(Ii)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)在实验上建造和操作该装置，以及(Ii)在计算上模拟该装置中样品的变形，包括叠加的声波，直至和超出剪切带的形成。数学分析在从实验数据中提取本构信息和开发合适的数值格式方面起着重要的支持作用。研究的第二部分由几个子项目组成，这些子项目更深入地探讨了在前一批赠款期间发现的现象。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了解卸料筒仓中的孔隙率波，(Iii)更全面地描述和解释卸料筒仓中随时间变化的应力的功率谱，以及(Iv)将“标度不变”初值问题作为弹塑性力学中可能的范例问题(类似于气体动力学中的Riemann问题)进行探索。在应用方面，该项目与筒仓内颗粒状材料的流动关系最为密切。然而，存在许多颗粒状物质的地球物理产状，本项目将提供丰富的声传播本构信息，这将与这些有关。此外，剪切带的模拟的意义远远超出了这个项目--从数学上讲，因为剪切带涉及改变类型的偏微分方程，并且科学地因为剪切带是导致许多材料破坏的普遍机制。

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.