Citation
Craciun, Bogdan (2002) Phase Boundary Propagation in Heterogeneous Media. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:10082010142653040
Abstract
There has been much recent progress in the study of free boundary problems motivated by phase transformations in materials science. Much of this literature considers fronts propagating in homogeneous media. However, usual materials are heterogeneous due to the presence of defects, grains and precipitates. This thesis addresses the propagation of phase boundaries in heterogeneous media. A particular motivation is a material undergoing martensitic phase transformation. Given a martensitic material with many nontransforming inclusions, there are well established microscopic laws that give the complex evolution of a particular twin or phase boundary as it encounters the many inclusions. The issue of interest is the overall evolution of this interface and the effect of defects and impurities on this evolution. In particular, if the defects are small, it is desirable to find the effective macroscopic law that governs the overall motion, without having to follow all the microscopic details but implicitly taking them into account. Using a theory of phase transformations based on linear elasticity, we show that the normal velocity of the martensitic phase or twin boundary may be written as a sum of several terms: first a homogeneous (but nonlocal) term that one would obtain for the propagation of the boundary in a homogeneous medium, second a heterogeneous term describing the effects of the inclusions but completely independent of the phase or twin boundary and third an interfacial energy term proportional to the mean curvature of the boundary. As a guide to understanding this problem, we begin with two simplified settings which are also of independent interest. First, we consider the homogenization for the case when the normal velocity depends only on position (the heterogeneous term only). This is equivalent to the homogenization of a HamiltonJacobi equation. We establish several variational principles which give useful formulas to characterize the effective Hamiltonian. We illustrate the usefulness of these results through examples and we also provide a qualitative study of the effective normal velocity. Second, we address the case when the interfacial energy is not negligible, so we keep the heterogeneous and curvature terms. This leads to a problem of homogenization of a degenerate parabolic initial value problem. We prove a homogenization theorem and obtain a characterization for the effective normal velocity, which however proves not to be too useful a tool for actual calculations. We therefore study some interesting examples and limiting cases and provide explicit formula in these situations. We also provide some numerical examples. We finally address the problem in full generality in the setting of antiplane shear. We explicitly evaluate the term induced by the presence of the inclusions and we propose a numerical method that allows us to trace the evolution of the phase boundary. We use this numerical method to evaluate the effect of the inclusions and show that their effect is quite localized. We use it to explain some experimental observations in NiTi.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Applied Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Applied And Computational Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  29 October 2001 
Record Number:  CaltechTHESIS:10082010142653040 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:10082010142653040 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6122 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  11 Oct 2010 03:22 
Last Modified:  10 Feb 2017 22:18 
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