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IMA Tutorial/Workshop:

Composites: Where Mathematics Meets Industry

Composites: Where Mathematics Meets Industry

February 7-9, 2005

**Organizer: **

**Graeme Walter Milton**

Distinguished Professor and Chairman

Department of Mathematics

University of Utah

milton@math.utah.edu

http://www.math.utah.edu/~milton/

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.

** Modeling the pipeline of high performance,
nano-composite materials and effective properties **

**M. Gregory
Forest**

Professor of Mathematics

Applied Mathematics Program Leader Co-Director, Institute for Advanced Materials,
Nanoscience & Technology University of University of North Carolina at Chapel
Hill

forest@amath.unc.edu

http://www.amath.unc.edu/Faculty/forest/

**Composite properties and microstructure**

**Robert
P. Lipton**

Professor of Mathematics

Louisiana State University

lipton@math.lsu.edu

http://www.math.lsu.edu/~lipton/

** Nanoparticle suspensions with giant
electrorheological response **

**Ping Sheng
** Professor of Physics and Department Head

Director, Institute of Nano Science and Technology

Hong Kong University of Science and Technology

sheng@ust.hk

http://physics.ust.hk/department/staff_detail.php?action=1

** M. Gregory Forest**
(Mathematics, University of North Carolina at Chapel Hill) http://www.amath.unc.edu/Faculty/forest/

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: html
pdf
ps
ppt

Part II Slides: html
pdf
ps
ppt

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
materials.

Lecture Title: **Composite properties and
microstructure **

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 structures.

** Ping Sheng**
(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.

Lecture Title: **Nanoparticle suspensions
with giant electrorheological response
**Slides: html
pdf
ps
ppt

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
kPa).

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 [5]. 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 [6], 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.

[1] H.R. Ma, W.J. Wen, W.Y. Tam, and P. Sheng, Phys. Rev. Lett. 77, 2499
(1996).

[2] 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).

[3] 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).

[4] H.R. Ma, W.J. Wen, W.Y. Tam and P. Sheng, Adv. Phys. 52, 343 (2003).

[5] W.J. Wen, H.R. Ma, W.Y. Tam and P. Sheng, Phys. Rev. E55, R1294 (1997).

[6] W.J. Wen, X.X. Huang, S.H. Yang, K.Q. Lu and P. Sheng, Nature Materials
2, 727 (2003).

Monday | Tuesday | Wednesday | |||||||
---|---|---|---|---|---|---|---|

Monday February 07, 2005 | |||||||

8:30am-9:15am | Coffee and registration | EE/CS 3-176 | |||||

9:15am-9:30am | Welcome and introduction | Douglas Arnold (University of Minnesota, Twin Cities) | EE/CS 3-180 | ||||

9:30am-10:30am | Composite properties and microstructure | Robert Lipton (Louisiana State University) | EE/CS 3-180 | ||||

10:30am-11:00am | Coffee break | EE/CS 3-176 | |||||

11:00am-12:00pm | Composite properties and microstructure | Robert Lipton (Louisiana State University) | EE/CS 3-180 | ||||

12:00pm-1:30pm | Lunch break | ||||||

1:30pm-2:30pm | Nanoparticle suspensions with giant electrorheological response | Ping Sheng (Hong Kong University of Science and Technology) | EE/CS 3-180 | ||||

2:30pm-3:00pm | Coffee break | EE/CS 3-176 | |||||

3:00pm-4:00pm | Nanoparticle suspensions with giant electrorheological response | Ping Sheng (Hong Kong University of Science and Technology) | EE/CS 3-180 | ||||

4:15pm-4:30pm | Group photo | Lind Hall 400 | |||||

4:30pm-6:00pm | IMA Tea and more | Lind Hall 400 | |||||

Tuesday February 08, 2005 | |||||||

9:00am-9:30am | Coffee | EE/CS 3-176 | |||||

9:30am-10:30am | Modeling the pipeline of high performance, nano-composite materials and effective properties | M. Gregory Forest (University of North Carolina) | EE/CS 3-180 | ||||

10:30am-11:00am | Coffee break | EE/CS 3-176 | |||||

11:00am-12:00pm | Modeling the pipeline of high performance, nano-composite materials and effective properties | M. Gregory Forest (University of North Carolina) | EE/CS 3-180 | ||||

12:00pm-1:30pm | Lunch break | ||||||

1:30pm-4:00pm | Industrial problems presentationsEE/CS 3-180
| ||||||

Wednesday February 09, 2005 | |||||||

9:00am-9:30am | Coffee | EE/CS 3-176 | |||||

9:30am-12:00pm | Moderated break-out sessions | EE/CS 3-180 | |||||

12:00pm-1:30pm | Lunch break | ||||||

1:30pm-3:00pm | Wrap-up and follow-up recommendations | EE/CS 3-180 | |||||

3:00pm-3:15pm | Closing remarks | EE/CS 3-180 |

NAME | DEPARTMENT | AFFILIATION |
---|---|---|

Sharf Ahmed | Global Nonwoven | H.B. Fuller Company |

Douglas Arnold | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Donald Aronson | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Gerard Awanou | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Joseph Brennan | Department of Mathematics | North Dakota State University |

Robert Burgmeier | Materials Research and Development | Boston Scientific |

Carme Calderer | School of Mathematics | University of Minnesota, Twin Cities |

Qian-Yong Chen | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Brian DiDonna | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

David Dobson | Department of Mathematics | University of Utah |

Ryan Elliott | Department of Aerospace Engineering and Mechanics | University of Michigan |

M. Gregory Forest | Department of Mathematics | University of North Carolina |

Paul Fussell | Department of Mathematics and Computing Technology | The Boeing Company |

Babu Gaddam | Corporate Materials Research Laboratory | 3M |

Eugene Gartland | Department of Mathematical Sciences | Kent State University |

Robert Gulliver | School of Mathematics | University of Minnesota, Twin Cities |

Qun Huo | Department of Polymers and Coatings | North Dakota State University |

Richard James | Department of Aerospace Engineering and Mechanics | University of Minnesota, Twin Cities |

Robert Jennings | Corporate Research Materials Lab | 3M |

Xiaoshi Jin | Department of Research and Development | Moldflow Corporation |

Sookyung Joo | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Chiu-Yen Kao | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Richard Kollar | Institute of Mathematics and its Application | University of Minnesota, Twin Cities |

Matthias Kurzke | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Frédéric Legoll | University of Minnesota, Twin Cities | |

Debra Lewis | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Xiantao Li | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Robert Lipton | Department of Mathematics | Louisiana State University |

Chun Liu | Department of Mathematics | The Pennsylvania State University |

Hailiang Liu | Department of Mathematics | Iowa State University |

Mitchell Luskin | School of Mathematics | University of Minnesota, Twin Cities |

Suping Lyu | Materials and Biosciences Center | Medtronic |

Christopher Macosko | University of Minnesota, Twin Cities | |

Miao-Jung Ou | Department of Mathematics | University of Central Florida |

Jinhae Park | School of Mathematics | University of Minnesota, Twin Cities |

Lyudmila Pekurovsky | CRML | 3M |

Peter Philip | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Amy Rovelstad | Department of Modeling and Simulation | Corning Incorporated |

Piotr Rybka | Institute of Applied Mathematics | University of Warsaw |

Rolf Ryham | Department of Mathematics | The Pennsylvania State University |

Fadil Santosa | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Arnd Scheel | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |

Ping Sheng | Department of Physics | Hong Kong University of Science and Technology |

Valery Smyshlyaev | Department of Mathematical Sciences | University of Bath |

James Sorensen | Metal Matrix Composites Lab | 3M |

Vladimir Sverak | School of Mathematics | University of Minnesota, Twin Cities |

Peter Takac | Fachbereich Mathematik | Universität Rostock |

Darrel Untereker | Corporate S&T | Medtronic |

Qi Wang | Department of Mathematics | Florida State University |

Baisheng Yan | Department of Mathematics | Michigan State University |

Nung Kwan (Aaron) Yip | Department of Mathematics | Purdue University |

Emmanuel Yomba | Faculty of Science | Ngaoundere University |

Hui Zhang | School of Mathematical Sciences | Beijing Normal (Teachers) University |

Xiaoyu Zheng | Department of Mathematics | University of North Carolina |

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