Composites play a vital role in industry, from carbon-fibre materials, to polycrystalline
alloys with the crystal microstructure tailored to achieve desired design parameters,
to rocket fuels of metallic particles in an oxidizing matrix, to porous materials
for filtering and storage, to electro and magneto rheological fluids, to photonic
and phononic band gap structures, and to novel nanostructured materials. Composites
are also a source of fascinating mathematics in the quest to understand how
features of the microstructure determine the overall macroscopic properties
of a material. This tutorial/workshop will consist of three parts. On the first
day there will be a series of tutorials, as listed below, on problems of direct
interest to industry. On the second day 5 or 6 industrial scientists will make
presentations on their work which involves multiscale modelling. These will
mix research reports with the posing of problems. On the third day we will break
out into groups where we will discuss the problems posed and techniques that
could be applied to solve them. The day will finish with each discussion group
giving a report.
Biography: M. Gregory Forest
is Grant Dahlstrom Distinguished Professor in the Department of Mathematics
at University of North Carolina at Chapel Hill, where he also serves as Co-Director
of the Institute for Advanced Materials, NanoScience and Technology, and the
founding leader of the Program in Applied Mathematics. He holds a PhD in Applied
Mathematics from the University of Arizona, and has served on the faculty of
Ohio State University and extensively consulted to industrial and government
research laboratories. His current research efforts are in complex fluids and
soft matter applied to high performance materials and biological systems. He
is on the editorial board of SIAM Journal on Applied Mathematics and Continuum
Mechanics and Thermodynamics.
Lecture Title: Modeling the pipeline
of high performance, nano-composite materials and effective properties
Part I Slides: htmlpdfpsppt
Part II Slides: htmlpdfpsppt
Abstract. We focus these lectures on the class
of nano-composites comprised of nematic polymers, either rod-like or platelet-like
macromolecules, together with a matrix or solvent. These materials are designed
for high performance, multifunctional properties, including mechanical, thermal,
electric, piezoelectric, aging, and permeability. The ultimate goal is to prescribe
performance features of materials under conditions they are likely to be exposed,
and then to reverse engineer the pipeline by picking the composition and processing
conditions which generate properties with those performance characteristics.
These lectures will address two critical phases of this nano-composite materials
pipeline. First, we model flow processing of nematic polymer films, providing
information about anisotropy, dynamics, and heterogeneity of the molecular orientational
distributions and associated stored elastic stresses. Second, we determine various
effective property tensors of these materials based on the processing-induced
orientational distribution data. Underlying these technological applications
is a remarkable sensitivity of nematic polymer liquids to shear-dominated flow,
which must be understood from rigorous multiscale, multiphysics theory, modeling
and simulation in order to approach the ultimate goal stated above.
This research is based on multiple collaborations and supported by various federal
sponsors, to be highlighted during the lectures.
Robert P. Lipton
(Department of Mathematics, Louisiana State University)
Biography: Robert Lipton
is Professor of Mathematics and founding member of the Mathematical Materials
Science Group at Louisiana State University. Currently a visiting scholar at
the Division of Engineering and Applied Sciences at Harvard University, he obtained
his Ph.D. from the Courant Institute of Mathematical Sciences in 1986 and, after
a postdoc at the Mathematical Sciences Institute at Cornell University, became
Charles B. Morrey Assistant Professor at the University of California at Berkeley
in 1988. He served on the Mathematical Sciences Faculty at WPI from 1990-2001.
He collaborates and consulted with with scientists at Wright Patterson Air Force
Base. His current research is in the area of failure initiation in composite
Lecture Title: Composite properties and
Abstract. We begin with an overview of composite
materials and their effective properties. Most often only a statistical description
of the microstructure is available and one must assess the effective behavior
in terms of this limited information. To this end approximation schemes such
as effective medium schemes and differential schemes are discussed. Variational
methods for obtaining tight bounds on effective properties for statistically
defined microgeometries are reviewed. Formulas for the effective properties
of extremal microgeometries are presented. Such microgeometries include layered
materials and sphere and ellipsoid assemblages.
Next we focus on physical situations where the interface between component
materials play an important role in determining effective transport properties.
This is relevant to the study of nanostructured materials in which the interface
or interphase between materials can have a profound effect on overall transport
properties. Variational methods and bounds are presented that illuminate the
effect of particle size and shape distribution inside random composites with
coupled heat and mass transport on the interface.
We conclude by introducing methods for quantifying load transfer between length
scales. This is motivated by the fact that many composite structures are hierarchical
in nature and are made up of substructures distributed across several length
scales. Examples include aircraft wings made from fiber reinforced laminates
and naturally occurring structures like bone. From the perspective of failure
initiation it is crucial to quantify load transfer between length scales. The
presence of geometrically induced stress or strain singularities at either the
structural or substructural scale can have influence across length scales and
initiate nonlinear phenomena that result in overall structural failure. We examine
load transfer for statistically defined microstructures. New mathematical objects
beyond the well known effective elastic tensor are presented that facilitate
a quantitative description of the load transfer in hierarchical structures.
Several physical examples are provided illustrating how these quantities can
be used to quantify the stress and strain distribution inside multi-scale composite
(Department of Physics, Hong Kong University of Science and Technology)
Biography: Ping Sheng
is head of the Physics Department and director of the Institute of Nano Science
and Technology at the Hong Kong University of Science and Technology. He obtained
his PhD in physics from Princeton University in 1971, and later worked at the
Institute for Advanced Study, the RCA David Sarnoff Research Center, and the
Exxon Corporate Research Lab, where he headed the theory group from 1982-86.
Professor Sheng's research interests include many areas of composites and materials
science. He is a fellow of the American Physical Society, a member of the Asia
Pacific Academy of Materials, and was elected the 2001 Technology Leader of
the Year by the Sing Tao Group of Hong Kong.
Abstract. In this talk I wish to tell the story
of a 10-year effort in search of a better electrorheological (ER) fluid material,
leading to the discovery of the giant ER effect, and the crucial role that mathematics
and simulations has played in the whole process.
Electrorheology denotes the control of a material's flow properties (rheology)
through the application of an electric field. ER fluid was discovered sixty
years ago. In the early days the ER fluids, generally consisting of solid particles
suspended in an electrically insulating oil, exhibited only a limited range
of viscosity change under an electric field, typically in the range of 1-3 kV/mm.
The study of ER fluid was revived in the 1980's, propelled by the envisioned
potential applications, as well as the successful fabrication of new ER solid
particles that, when suspended in a suitable fluid, can "solidify" under an
electric field, with the strength of the high-field solid state characterized
by a yield stress (breaking stress under shear). However, further progress was
hindered by the barrier of low yield stress (typically in the range of a few
Starting in 1994, we have adapted the mathematics of composites, in particular
the Bergman-Milton representation of effective dielectric constant, to the study
of ER mechanism(s) [1-4]. The questions we aim to answer are: (1) the role of
conductivity in the ER effect, (2) the role multipole interaction, (3) the ground
state microstructure of the high-field state and most importantly (4) the upper
bounds in the yield stress and shear modulus of the high field solid state.
Finding the answer to (4) led to the suggestion of the coating geometry for
the ER solid particles which can optimize the ER effect, but at the same time
also pointed out the limitation of the ER mechanism based on induced polarization.
The subsequent study of adding controlled amount of water to the ER fluid pointed
to the intriguing possibility of using molecular dipoles as the new "agent"
for enhancing the ER effect . Working along this direction, the experimentalist
W.J. Wen was able to synthesize urea-coated nanoparticles of barium titanyl
oxalate which exhibited yield stress in excess of 100 kPa, breaking the yield
stress upper bound and pointing to a new paradigm in ER effect in which the
molecular dipoles can be harnessed to advantage in controllable, reversible
liquid-solid transitions with a time constant on the order of 1 msec. We propose
the model of aligned surface dipole layers in the contact area of the coated
nanoparticles to explain the observed giant ER effect , with the electric-field
induced dissociation (the Poole-Frenkel effect) of the molecular dipoles accounting
for the observed ionic conductivity. Quantitative agreement between theory and
experiment was obtained. The talk concludes with an outline of the intriguing
questions yet to be answered, and the problems to be solved before ER fluids
can become a commercial reality.
 H.R. Ma, W.J. Wen, W.Y. Tam, and P. Sheng, Phys. Rev. Lett. 77, 2499
 W.Y. Tam, G.H. Yi, W.J. Wen, H.R. Ma, M.M. T. Loy, and P. Sheng,
Phys. Rev. Lett. 78, 2987 (1997).
 W.J. Wen, N. Wang, H.R. Ma, Z.F. Lin, W.Y. Tam, C.T. Chan, and P.
Sheng, Phys. Rev. Lett. 82, 4248 (1999).
 H.R. Ma, W.J. Wen, W.Y. Tam and P. Sheng, Adv. Phys. 52, 343 (2003).
 W.J. Wen, H.R. Ma, W.Y. Tam and P. Sheng, Phys. Rev. E55, R1294 (1997).
 W.J. Wen, X.X. Huang, S.H. Yang, K.Q. Lu and P. Sheng, Nature Materials
2, 727 (2003).