Mathematical Sciences: Multidimensional Problems in Dynamic Plasticity

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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.