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Dissertation
Complex averages of particle quantities and equations of balance
Washington State University
Doctor of Philosophy (PhD), Washington State University
05/2010
DOI:
https://doi.org/10.7273/000005663
Abstract
We study a new complex continuum quantity Q[tau][eta](t; x, z) and its applications for efficient particle system simulation. Function Q[eta](t; x, z) is constructed using both velocities and positions of particles. It carries more information than the standard quantities density, linear momentum, and kinetic energy, and, therefore, it can give better results in various applications. The standard quantities can be derived from Q[tau][eta](t; x, z). The proposed quantity and its derivatives are bounded independently of the number of particles, and can be used for numerical modeling. Several 1D particle systems are studied using Q[tau][eta](t; x, z), and an approximate closure is presented based on the examples. Research of Q[tau][eta](t; x, z) can be continued on a 2D example presented in the last chapter. The model in the example describes a so called bistable material. Bistable material is represented by a two-dimensional triangular lattice made of bistable rods. Each rod has two equilibrium lengths, and thus its energy has two equal minima. A rod undergoes a phase transition when its elongation exceeds a critical value. The lattice is subject to a homogeneous strain and is periodic with a sufficiently large period. The effective strain of a periodic element is defined. After phase transitions, the lattice rods are in two different states and lattice strain is inhomogeneous, the Cauchy-Born rule is not applicable. We show that the lattice has a number of deformed still states that carry no stresses. These states densely cover a neutral region in the space of entries of effective strains. In this region, the minimal energy of the periodic lattice is asymptotically close to zero. The compatibility of the partially transited lattice is studied. We derive compatibility conditions iiifor lattices and demonstrate a family of compatible lattices (strips) that densely covers the flat bottom region. Under an additional assumption of the small difference of two equilibrium lengths, we demonstrate that the still structures continuously vary with the effective strain and prove a linear dependence of the average strain on the concentration of transited rods.
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Details
- Title
- Complex averages of particle quantities and equations of balance
- Creators
- Andrei Kouznetsov
- Contributors
- Alexander Panchenko (Chair)Robert Dillon (Committee Member) - Washington State University, Department of Mathematics and StatisticsValipuram S Manoranjan (Committee Member) - Washington State University, Department of Mathematics and Statistics
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 72
- Identifiers
- 99901054738501842
- Language
- English
- Resource Type
- Dissertation