Mathematics
of Materials and Macromolecules: Multiple Scales, Disorder,
and Singularities, September 2004  June 2005
Abstracts and Talk
Materials
IMA Symposium: Prospects for Mathematics and Mechanics
upon the 80th Birthday of
Jerry Ericksen
November
56, 2004
program
web page
Kaushik Bhattacharya
(Department of Mechanics and Materials Science, California Institute of Technology)
bhatta@caltech.edu http://www.mechmat.caltech.edu
Ferroelectric ceramics
This talk will outline recent progress in understanding ferroelectric perovskite
ceramics with an emphasis on their electromechanical behavior.
Howard Brenner
(Department of Chemical Engineering, Massachusetts Institute of Technology)
hbrenner@MIT.EDU
NavierStokesFourier revisited
This talk builds upon the pioneering work of Dan Joseph and coworkers in clarifying
the notion of what is meant by an "incompressible" fluid when density gradients
are present. In particular, we introduce the notion of volume as an extensive
transportable physical property of a fluid continuum in both liquids and gases.
Of special interest is the kinematical notion of the "diffuse" transport of
volume, above and beyond the conventional (albeit implicit) view of convective
volume transport, the latter being simply and inseparably linked to mass transport
through the agency of the fluid's density; that is, in the presence of density
gradients, volume can be transported through space without a concomitant movement
of mass. Beyond the purely kinematical aspects of volume transport reflected
in the work of Joseph et al., the diffuse transport of volume is accompanied
by both momentum and energy transport in amounts above and beyond the amounts
heretofore considered in standard continuummechanical theories of diffuse momentum
and energy transport. This leads to constitutive revisions of both Newton's
law of viscosity governing the diffuse transport of momentum and Fourier's law
of heat conduction governing the diffuse transport of energy (the latter when
a clear distinction is drawn between the respective fluxes of heat and internal
energy). Experimental evidence based upon the phenomena of thermophoresis and
thermal transpiration in singlecomponent gases undergoing heat transfer, together
with replacement of the noslip massvelocity condition by a comparable noslip
volumevelocity condition, is used to quantitatively support the proposed constitutive
revisions to Newton's and Fourier's laws.
Jim Casey
(Department of Mechanical Engineering, University of California  Berkeley)
jcasey@me.berkeley.edu
A theory of pseudorigid bodies
Bodies that are somehow capable of keeping their deformation fields homogeneous
have been studied extensively in the literature. Such "pseudorigid" bodies,
or "Cosserat points," have been used successfully in a variety of applications.
The main question addressed in this lecture is: How, in principle, can a 3dimensional
continuum, subjected to arbitrary applied loads, keep its deformation field
homogeneous? The homogeneity condition is regarded as a "global constraint,"
and a system of indeterminate reactive stresses is introduced. The remaining
part of the stress tensor is specified by a constitutive equation. The reactive
stresses play the same role as in rigid body dynamics. It is also shown how
a pseudorigid body can be represented by a point moving in a 12dimensional
Euclidean space, the metric of which is determined by the radius of gyration
of the body. In the presence of holonomic constraints, the configuration manifold
is Riemannian, and a set of Lagrange's equations emerge as the covariant components
of the governing balance equation.
Patricia E.
Cladis (Advanced Liquid Crystal Technologies, Inc.)
cladis@alct.com
Hedgehogantihedghog
annihilation to a static soliton
Slides: pdf
Nematic liquid crystals are liquids with orientational order in
a preferred direction denoted by n, a unit pseudovector i.e. with neither head
nor tail. We call one of its point defects a hedgehog (H) because, in that case,
n radiates from a point so is reminiscent of a
hedgehog's quills when in a defensive
posture. The other point defect is then, by default, an antihedgehog (.H), because
it annihilates with a hedgehog to leave behind a static soliton [1].
Hedgehogs and antihedghogs are a direct consequence of the intrinsic
nonlinear elastic properties of nematics in a cylindrical geometry to break
spontaneously the symmetry imposed by the boundary condition. This was first
pointed out by me and Maurice Kléman in 1972 [2] then subsequently dubbed "escape
into the third dimension" [3]. While the statics of nematic point defects is
relatively well understood (or so we thought), their surprising dynamics is
of general interest in fields extending beyond liquid crystals.
As Jerry asked (based on preliminary observations Mayola Walters
and I found at Bell Labs in the late '70's using my first computer controlled
imaging system}[4]: Why is H.H annihilation dynamics so nonlinear? Indeed, why
do point defects move at all when their range of interactions is limited by
the cylinder radius, R, so that they should be "asymptotically free" when separated
by many times R?
Most recently, using observations from my latest imaging system
at ALCT, Helmut Brand and I found a new, unprecedented result: during annihilation,
the hedgehog always moved faster than the antihedgehog. This gave us an idea
that, after many referee battles, we finally published [5]. This talk is a tribute
to Jerry's long standing affection and profound appreciation for all hedgehogs
big and small.
References
[1] C. E. Williams, P. Pieranski and P. E. Cladis, Phys.
Rev. Lett. 29,
90 (1972).
[2] P. E. Cladis and M. Kléman, J.
de Phys. (Paris),
33, 591 (1972).
[3] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon
Press, Oxford (1993).
[4] J.L. Ericksen in Nonlinear Effects in Fluids and Solids, M.M. Carroll
and M.A. Hayes (eds), Plenum, New York (1996).
[5] P.E. Cladis and H. R. Brand, Hedgehogantihedgehog pair annihilation to
a static soliton, Physica
A326, 322 (2003).
Masao Doi
(Department of Applied Physics, University of Tokyo) doi@rheo.t.utokyo.ac.jp
A variational principle in dissipative systems
In constructing kinetic equations which describe the motion of
complex fluids (fluid mixtures, liquid crystals, polymer solutions and gels),
thermodynamics gives us an important guide. The way how one uses the principle
of thermodynamics, however, seems to have various versions. In this talk, I
would like to discuss a way which I found useful. It is based on Rayleigh's
extension of the Lagrangean mechanics to dissipative systems. In this talk,
I explain the method and discuss its applications to several systems (two fluid
model, electrolyte solutions, gels etc).
Richard D. James
(Department of Aerospace Engineering and Mechanics, University of Minnesota)
james@aem.umn.edu http://www.aem.umn.edu/people/faculty/bio/james.shtml
Hysteresis and geometry: a way to search for new materials with "unlikely"
physical properties
These thoughts begin with the observation by physicists, probing new phenomena
through the use of first principles' studies, that the simultaneous occurrence
of ferromagnetism and ferroelectricity is unlikely. While these studies do not
consider the possibility of a phase transformation, there is a lot of indirect
evidence that, if the lattice parameters are allowed to change a little, then
one might have coexistence of "incompatible properties" like ferromagnetism
and ferroelectricity. Thus, one could try the following: seek a reversible first
order phase transformation, necessarily also involving a distortion, from, say,
ferroelectric to ferromagnetic phases. If it were highly reversible, there would
be the interesting additional possibility of controlling the volume fraction
of phases with fields or stress. Thus, one could imagine having a strong magnet;
apply stress to it and it becomes a strong ferroelectric. The key point is reversibility.
Even big first order phase changes can be highly reversible (liquid water
to ice, some shape memory materials), and we argue that it is the nature of
the shape change that is critical. We suggest, based on a close examination
of measured hysteresis loops in various martensitic systems, that reversibility
is governed by the presence of certain special relations among lattice parameters.
While these relations are naively geometric, their fundamental status is not
clear, but they likely relate to a concept of metastability for an energy functional
that includes both interfacial and bulk energy. Fundamentally, we lack the ability
to formulate the appropriate concept of metastability because we do not really
understand how to model interfacial energy, as we explain.
Acknowledgment: John Ball, Karin Rabe, Jerry Zhang.
David
Kinderlehrer (Department of Mathematical Sciences, Carnegie
Mellon University ) davidk@andrew.cmu.edu
http://www.math.cmu.edu/people/fac/kinderlehrer.html
Issues for interfaces in polycrystals
Nearly all technologically useful materials are polycrystalline.
Their ability to meet system level specifications of performance and reliability
is influenced by the types of grain boundaries present and their connectivity.
We explore the role of mesoscale theory and experiment designed to establish
predictive models of material behavior. Traditionally we have studied geometrybased
statistics, like relative area statistics or distributions of numbers of sides
of grains. With the advent of automated data acquisition we now have the possibility
of obtaining large quantities of both geometric and crystallographic information.
In particular we shall discuss two new results which lead to the astonishing
conclusion that a polycrystal may leave its "footprint" in a microsopic scan:
simple analysis of the scan reveals the identity of the material.
This is part of the CMU MRSEC project.
Fanghua Lin
(Courant Institute of Mathematical Sciences, New York University) linf@courant.nyu.edu
A mathematical journey on defects motions
In this lecture,I recollect a few personel interactions with Jerry
Ericksen on a few mathematical issues concerning liquid crystals. In particular,I
shall describe briefly some thoughts and works related to the problem of the
defect motion law for flows of liquid crystal.
Gianni
RoyerCarfagni (CivilEnvironmental Engineering and Architecture,
Universita di Parma (CNR)) royer@aem.umn.edu
The role of stress on chemical transformations in an elastic
bar with nonconvex chemomechanical free energy
The equilibrium of an elastic bar capable of undergoing chemomechanical
transformations and in contact with a chemically aggressive environment is considered.
In the proposed model, stable equilibrium states are identified with minimizers
of a specific free energy functional, which depends upon the axial strain of
the bar and the extent of reaction with an external agent, which is dispersed
in a surrounding vapor or liquid solution with assigned chemical potential.
In general, the corresponding minimization problem is nonconvex and, therefore,
it predicts the coexistence of equilibrium phases. This work is related to the
now classical problem of Ericksen for an elastic bar stretched in either a "hard"
or a "soft" testing machine. However, here, the presence of an additional internal
variable, which represents the extent of reaction, allows for phase transformations
which are stressinduced and/or driven by changes in the chemical composition
of the surrounding environment. We discuss a characterization of "hard" or "soft"
environmental chemical boundary conditions. The model is germane to the description
of several phenomena, such as the swelling of ionic gels under chemomechanical
actions, or the formation of expanding crusts in stone monuments due to acid
rain or an otherwise polluted atmosphere.
Anja
Schlömerkemper (Institut fuer Analysis, Dynamik und Modellierung,
Universität Stuttgart) schloeme@mathematik.unistuttgart.de
http://www.mathematik.unistuttgart.de/~schloeme
About magnetic force formulae
The formula for the magnetic force that is exerted by a magnetic
field on a single magnetic dipole is well accepted. On the other hand, the formulation
of magnetic force formulae for macroscopic magnetic systems has been under debate
for a long time. A final answer to this question is of interest in the context
of deformable magnetic bodies as for instance of ferromagnetic shape memory
alloys.
In the first part of the talk, a brief overview of Brown's [1]
approach is given and related work that was initiated by Brown's approach is
mentioned.
Secondly, we focus on the formula for the magnetic force between
rigid magnetic bodies which was derived from a lattice of magnetic dipoles in
a continuum limit [2]. The main ideas of this approach and of the mathematical
proof are presented. In addition to the classical magnetic force formulae, one
obtains a surface force density which depends on the underlying lattice structure
and includes short range contributions of the magnetic interaction at interfaces.
In the final part of the talk we address the question of whether
this magnetic force formula describes nature well and compare it with Brown's
formula.
[1] Brown, W.F., Magnetoelastic Interactions, SpringerVerlag,
Berlin, 1966
[2] Schlömerkemper, A., Mathematical derivation of the continuum
limit of the magnetic force between two parts of a rigid crystalline material,
accepted for publ. in Arch. Rational Mech. Anal.
Lev Truskinovsky
(Laboratory of Solid Mechanics, École Polytechnique) trusk@lms.polytechnique.fr
Thermodynamics of rate independent plasticity
We show that the singular dissipative potential of the phenomenological
rate independent plasticity can be obtained by homogenization of a micromodel
with quadratic dissipation. The essential ingredient making this reduction possible
is a rugged energy landscape at the microscale, generating under external loading
a regular cascade of subcritical bifurcations. Such landscape may appear as
a result of a sufficiently strong pinning or jamming of defects, leading to
elastic micrometastability. The rate independent plastic deformation emerges
in this description as a continuous succession of infinitesimal viscous events;
the limiting procedure presumes the elimination of small time and length scales.
We present an explicit example of a simple viscoelastic massspring system whose
macroscopic dissipative behavior is plastic rate independent.
Anna Vainchtein
(Department of Mathematics, University of Pittsburgh) annav@euler.math.pitt.edu
http://www.math.pitt.edu/~annav/
Kinetics of lattice phase transitions
Understanding the origin of energy dissipation and the associated
kinetics of phase boundaries remains an important open problem in modeling lattice
phase transitions in martensites. Following the pioneering work of Ericksen
[1975], it has become common to model these materials by an updownup stressstrain
relation in the framework of continuum elasticity theory. The corresponding
dynamic problem changes type and is illposed; however, it may be regularized
by prescribing an additional kinetic relation between the driving force and
the velocity of a phase boundary. This relation is usually either postulated
or derived from a phenomenological model accounting for dispersive and dissipative
effects.
In this talk we will describe how one can avoid introducing additional
phenomenological parameters and instead obtain a kinetic relation by replacing
the continuum model with its natural discrete analog. We consider a lattice
model of martensitic phase transition which takes into account longrange interactions
of an arbitrary range. Although the model is Hamiltonian at the microscale,
it generates a nontrivial macroscopic kinetic relation. The apparent dissipation
is due to the induced radiation of lattice waves carrying energy away from the
front.
This is joint work with Lev Truskinovsky
(Ecole Polytechnique, France).
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