Brown University
Joint Solid Mechanics/Materials Science Seminar Series
Doshisha University, Japan (presently on leave at MIT) | |
Abstract: Understanding mechanisms of dislocation patterning like those into cells, PSB ladders and veins is still one of the most challenging as well as attractive problems in plasticity, which substantially requires us to treat many-body-type problem including collective and cooperative effects produced by extremely large numbers of interacting dislocations. Particularly, process of cellular pattern formation would be the most difficult example to be dealt with because it contains many complicated factors, i.e., multiple slip, dynamic recovery, short range interaction like Lomer-Cottrell junction formation, and long range stress field. This lecture intends to provide an updated approach toward the collective behavior of dislocations leading to patterning based on the method of field theory. The effective theory for a macroscopic order parameter field from a microscopic theory is firstly presented focusing on the cell formation process, where the importance of the coarse-graining will be emphasized. Secondly, basic idea and concept of the gauge field theory of dislocations and defects are briefly shown, which provides us with the foundation for writing down the dislocation theoretical first principle Hamiltonian as a point of departure of the present approach. Lastly, details of thus derived effective theory having a form of Ginzburg-Landau type equation and a model for the cell formation process are extensively discussed referring to some numerical simulation results on patterning.
Stochastic Systems Seminar
(Please note new room for this academic year!) |
Brown Analysis Seminar
Scientific Computing Seminar
PDE Seminar
Abstract: An increasing quantity of phenomena is being classified under the terms {\it Phase Transition}. Conservation laws provide models for several of these phenomena. The present talk is concerned with recent results related to systems of hyperbolic conservation laws that develop phase transitions.
More precisely, we will consider
* liquid-vapor phase transitions;
* a combustion model;
* phase transitions in traffic flow.
In spite of the deep physical differences, the three cases above all fit in the same theoretical framework: phase transitions in hyperbolic conservation laws.
First, a classical model of a two-phase fluid will be briefly considered. It is well known that suitable admissibility conditions are necessary to single out a unique solution to Riemann Problems. Usually, these conditions are chosen {\it a priori} thanks to physical considerations. On the contrary, we will select {\it a posteriori} those conditions that lead to well posed problems.
Secondly, a combustion model is considered. Here, the combustion front is sonic with respect to the unburnt gas. A unique solution to the Riemann Problem is selected without the introduction of any admissibility condition. This Riemann solver leads to an existence result for the corresponding Cauchy Problem.
Finally, we consider phase transitions in traffic flow, as recently high-lighted in the specialized literature. We propose a new model that provides a description for the transitions from {\it free} to {\it congested} flow. This model consists of a scalar conservation law coupled with a 2 x 2 system of conservation laws. The whole is proved to be well posed. Furthermore, it is consistent with various qualitative behaviors of real traffic flow.
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