Harvard University, SEAS
Room 312, Pierce Hall
29 Oxford St.
Cambridge, MA
02139
My name is Ken Kamrin and I am an Applied Mathematics Lecturer and Postdoctoral Researcher at Harvard University. My past and current interests lie primarily in the development of mathematical models to explain the deformation and flow properties of various materials in various phases of matter. I received my PhD in Applied Mathematics from MIT in May 2008 (advised by Professor Martin Z. Bazant in the MIT Dry Fluids Group). My dissertation looked at dense material flows, with a focus on granular materials like sand, gravel, etc. Two distinct theoretical models for granular flow were developed, each capable of accurately predicting flows in multiple geometries. These models are very different and yet are both based on intuitive concepts with one treating flow as a stochastic process and the other a continuum deformation process. For more information please visit the Thesis Research page, or see below for a summary. |
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Brief Dissertation Summary
Model 1: The Stochastic Flow Rule (SFR)
This work models granular flow as a sequence of localized collective displacements of grains. These meso-scopic grain collections are known as "spots". The Spot Model was first described by Bazant in 2000 in which flow out of a silo apparatus is viewed as the superposition of many random spot displacements. However, the Spot Model is completely devoid of any mechanics therefore constricting any general applicability. The SFR is technically a plasticity "flow rule" in that it connects spot motion directly to the material stresses thus generalizing and extending the Spot Model well beyond silo flow to any geometry with a computable stress field. To approximate the stress profile in a slow flowing granular assembly, we utilize the Slip-Line Theory of solid mechanics. The SFR then describes quantitatively how to convert the slip-line field and stresses into the necessary parameters to fully define a spot's random trajectory through the material and generate a steady flow profile.
Model 2: Nonlinear Granular Elasto-Plasticity
The work of many French physicists on granular rheology hit a high point in the summer of 2006 with the results of P. Jop et al. which showed that a Bingham fluid flow model was in fact able to describe a 3D granular flow. On another front, Y. Jiang and M. Liu detailed a functioning granular elasticity model in the winter of that year which determines stresses in a static granular assembly. My work combines both of these models into one universal granular constitutive law, capable of predicting both flowing regions and stagnant zones simultaneously in any arbitrary flow geometry. The model is implemented as a user material subroutine in the Finite Element Method software package ABAQUS. The combined effect of elastic stresses and plastic yielding resolves features such as shear bands which many previously believed to be outside the realm of a continuum model. This work adheres to the rules of finite deformation elasto-plasticity theory, a framework which has developed through the ages to become a highly rigorous and complex mathematical science.