NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
AND APPLICATIONS TO MATERIALS
Lectures will be in room EE/CS 3-180, on the ground floor
of the EE/CS building, located two blocks
east of the Radisson Metrodome Hotel. Registration and breaks
will be in the adjacent room EE/CS 3-176.
A phase-field model of the rapid solidification of a binary alloy
Abstract:
During the rapid solidification of a binary alloy, solute may be incorporated
into the solid phase at a concentration significantly different than
that predicted by equilibrium thermodynamics.
This process, known as solute trapping, leads to a progressive reduction in
the concentration change across the interface as the solidification rate
increases. Theoretical treatments of rapid solidification
using traditional sharp-interface descriptions
require the introduction of separately-derived non-equilibrium
models for the behavior of the interfacial temperature and
solute concentrations. In contrast, phase-field models
employ a diffuse-interface description, and eliminate
the need to specify interfacial conditions separately.
While at low solidification rates equilibrium behavior is recovered,
at high solidification rates non-equilibrium effects naturally emerge from
these models. We characterize the dependence of the characteristic speed for
solute trapping on the equilibrium partition
coefficient k_E that shows good agreement with experiments
by Smith and Aziz [Acta metall. mater.42 (1994) 3515].
We also show that in the phase-field model there is a dissipation
of energy in the interface region resulting in a solute drag,
which we quantify by determining the relationship between the
interface temperature and velocity.
Abstract: We consider a free boundary PDE problem
with highly oscillating
free boundary and highly oscillating boundary condition. We
derive the homogenized approximation. Some of the
results are applied for a control problem in chemical vapor
deposition on a semiconductor substrate.
2:55 -- 3:30 pm: coffee break
3:30 -- 4:20 pm: Mete Soner,
Carnegie-Mellon/Princeton University
Functions of Bounded Higher Variations
Abstract:
Functions of bounded variation (BV) play a central role in the
analysis of variational problems modelling phase transformation when the
order parameter is scalar valued. For vector-valued problems, such as the
Ginzburg-Landau model for superconductivity, BV is no longer the
appropriate space. Functions of higher bounded variations, BnV, is
introduced to overcome this difficulty. An n-valued function is defined
to be in BnV if its weak Jacobian is a Radon measure. These functions
have properties very similiar to those of BV. In particular, the
celebrated rectifiability theorem of De Giorgi and the coarea formula of
Fleming and Rishel extends to BnV. This is joint work with R. Jerrard of
University of Illinos.
High-frequency behavior of the nonlinear Schroedinger
equation with random inhomogeneities
Abstract: I will present some recent work, jointly with Shi Jin,
on the effect of random inhomogeneities on the
focusing of the nonlinear Schroedinger equation, in the high-frequency
limit. I will discuss both analytical and numerical results.
Abstract:
Martensitic transformations are shape-deforming phase transitions
which can be induced in certain materials as a result of changes in
the imposed strains, stresses or temperatures. In this talk we will
examine, in three concrete examples, the ways in which temperature
together with the elastic and dissipated energies determine the
dynamics and equilibria associated with the phase change.
Exact relations and fast numerical schemes for composites
Abstract:
Typically, the elastic and electrical properties of
composite materials are strongly microstructure
dependent. So it comes as a nice surprise to come across
exact formulae for (or linking) effective moduli that are
universally valid no matter what the microstructure. Such
exact formulae provide useful benchmarks for testing
numerical and actual experimental data, and for evaluating
the merit of various approximation schemes. Classic examples
include Hill's formulae for the effective bulk modulus of a
two-phase mixture when the phases have equal shear moduli,
Levin's formulae linking the effective thermal expansion
coefficient and effective bulk modulus of two-phase
mixtures, and Dykhne's result for the effective conductivity
of an isotropic two-dimensional polycrystalline material.
Here we present the first systematic theory of exact relations
embracing the known exact relations and establishing new ones.
The search for exact relations is reduced to a search for
tensor subspaces satisfying a certain algebraic condition.
One of many new exact relations
is for the effective shear modulus of a class of three-dimensional
polycrystalline materials.
The series expansions which prove useful for establishing exact relations
also turn out to have especially rapid convergence properties,
and provide the foundation for a fast numerical scheme for
computing the response of composites.
This is joint work with David Eyre, Yury Grabovsky and
Dan Sage.
Deformable thin films: from macroscale to microscale and from nanoscale to microscale
Abstract:
There is widespread interest, and major experimental programs
underway, for the design and construction of electromechanical "machines"
at nanometer to millimeter scale. This work is proceeding largely
without guidance from mathematical theory. The building blocks for such
machines
currently are thin films, nanotubes and nanorods, objects with at least
one nanoscale or microscale dimension but also one or more relatively large
dimensions.
Using a macroscale-to-microscale approach, we describe a direct
derivation of a theory of single-crystal thin films, starting from
three-dimensional
nonlinear elasticity including a term for interfacial energy. The derivation
relies on Gamma-convergence arguments, and yields a frame-indifferent
Cosserat membrane theory. We highlight some predictions that are
specifically linked to size and have no bulk analogs. We construct simple
energy minimizing deformations --- tents and tunnels --- which could be the
basis of simple microactuators or micropumps.
Alternatively, beginning at the atomic level, we face the presence of at
least one large dimension, which frustrates purely atomic-scale approaches, but
which provides a glimmering opportunity for analysis. We present a scheme for
the direct passage from atomic to continuum scale, applicable to cases
in which one or two dimensions remain at atomic scale. The scheme
is based on (1) limited distortion of unit cells; and (2) many atoms in
certain directions. It gives rise to weak convergence problems. The
continuum theories that emerge are completely nonstandard.
Joint work with Kaushik Bhattacharya and Gero Friesecke.
Stability of Microstructure in Martensitic Materials
Abstract:
We present a theory for the stability of simply laminated microstructure
in martensitic crystals. The analysis of stability becomes more difficult
for crystals with phase transformations which undergo a greater loss of
symmetry
(and hence have more variants) since the crystal then has more freedom to
deform without the cost of additional energy.
We have recently given an analysis of all of the tetragonal to monoclinic
transformations (which have four variants) and for a cubic to orthorhombic
transformation (which has six variants). We have shown for these
transformations that the simply laminated microstructure is stable,
except for special lattice
constants at which the simply laminated microstructure is not stable.
The analysis of the tetragonal to monoclinic transformations is joint work with
Pavel Belik,
and the analysis of the cubic to orthorhombic transformation
is joint work with Kaushik Bhattacharya and Bo Li.
Abstract: The microscopic mechanisms of epitaxial growth have been
known for at least 50 years. The mesoscopic consequences of these
mechanisms are however not yet well understood. It is natural to explore
this issue by deriving a partial differential equation for the
surface height, thus linking the behavior on different length scales.
I will discuss the current understanding of the 1+1 dimensional
case, including recent work by Politi and Villain and current work by
Aaron Yip, Weinan E, and Tim Schulze.