Read a recap of the conference in SIAM News.
The late 1940's saw the beginning of new trends in the development of a continuum theory of complex materials, and the foundation of a theory that bridges continuum mechanics and atomistic theory. This led to the emergence of modern applied mathematics in which new concepts in geometry, analysis, partial differential equations, numerical and computational mathematics and stochastic mathematics are developed synergistically with continuum mechanics, molecular theory, and multiscale analysis. These early innovative research directions were, in many ways, precursors of a mathematical approach to the study of materials science, and of current programs such as the Materials Genome Initiative (MGI).
This workshop will celebrate the 90th birthday of Jerry Ericksen and will gather several academic generations of researchers, to showcase the work of young researchers influenced by these developments and to explore future trajectories. It will also allow the younger research community to network with the senior one, with the long term goal of helping to overcome the challenges of accessing interdisciplinary research at the interface of mathematics, the sciences, and engineering.
Scope
Nonlinear elasticity emerged in the mathematical community as a result of the works by Ericksen and Rivlin, in the 1950s and 1960s. Ericksen's research gave a deep understanding of how principles of invariance served to identify special solutions. A landmark paper from Ericksen's body of work of this period delivers an explicit classification of all deformations possible in every incompressible nonlinear elastic material, regardless the specific form of the free energy function. The combination of nonlinear elasticity models with earlier work by Morrey on calculus of variations initiated the fertile mathematical link between the direct method of the calculus of variations and the stability of materials, developed to a high level in the 1970's by researchers such as Ball and Antman. These methods have recently been used to study multiscale problems, thin structures and a variety of reduced models, with Ennio De Giorgi’s method of Gamma convergence assuming a central role in these developments. Owing to this body of research, we now have a rigorous understanding of the formerly disparate theories of mechanics, and, more and more, this insight underlies the design and discovery of both new materials and new device concepts. In parallel with this understanding of energy minimization, research on timedependent problems led to a deeper understanding of nonlinear hyperbolic partial differential equations, notably represented by the contributions of Dafermos. Frequent crossfertilization with closely related lines of research, represented notably by the work of Cahn, Frank and Mullins, led to body of work now associated with Mathematics of Materials Science.
In the 1960s and 1970s work on liquid crystals by Frank Leslie and Jerry Ericksen gave rise to the now renowned dynamical theory of liquid crystals. This theory featured highly nonstandard but physically relevant ingredients: non symmetric stresses, unusual kinetics, and an unconventional energy equation. From the point of view of statics, the problem of minimizing the OseenFrank energy of liquid crystals and the related Dirichlet problem prompted another line of research on calculus of variations related to the study of singularities and their associated defects. This work had a deep impact on degree theory and harmonic analysis, and the study of singularities in other condensed matter systems.
In the late 1980s, Ericksen formulated a theory for liquid crystal polymers involving a variable degree of orientation. On the one hand, this work prompted new studies of flow instabilities in liquid crystals and their impact in polymer processing, and secondly, the regularizing role of the order parameter in the model initiated a new way to study defect dynamics. This in turn sparked a new interest in the mathematical community towards the analysis of the Landaude Gennes model, based on an order tensor. More recently, the latter approach has successfully addressed longstanding difficulties inherent to the director model, such as the loss of orientability of the vector field that occurs in singularities of fractional degree. This work, combined with recently developed electrokinetic models of liquid crystals, is now giving important predictions on liquid crystal colloids, and and new understanding of their stunning nonlinear properties. These hold great promise for new kinds of microfluidic devices in mechanics and biology.
The simple but highly influential 1970s paper by Ericksen on Equilibrium of Bars prompted a new research line on phase transformations in crystals. Ericksen's formulation of the CauchyBorn rule and the the relation between lattice instabilities and the loss of ellipticity of free energy functions launched a vibrant area of research on the mechanics of crystals. This led to a new theory of phase transformations that continues to be one of the most vital in science and mathematics. This line of research is represented by the works of Ball, Bhattacharya, Fonseca, Kinderlerher, Kohn and James. A major advance was the reconciliation of the use of infinite symmetry groups of crystals with the finite symmetry groups used in nonlinear elasticity, achieved by the definition of the EricksenPitteri neighborhood. The mathematical tools involved have further developed for the study of shape memory alloys, ferroelectric and magnetic materials.
Another major advance, also spearheaded by Ericksen, was the generalization of the Landau theory of second order phase transformations to first order ones. This work evolved toward a body of mathematical concepts that turned out to be useful not only for the understanding of the behavior of materials, but also to guide the synthesis of radically new materials. Some of these materials offer a promise in the development of new concepts for clean energy recovery.
These new areas of inquiry, in turn, continue to have a tremendous impact in the development of new areas in numerical analysis, especially those bridging atomic to continuum scales, computer vision and image segmentation.
The close interaction between mathematical research and mechanics has inspired both scientists and mathematicians, with its influence now reaching deeply into both pure mathematics and materials science. The deep interdisciplinary character of this research offers a plethora of opportunities, but this crossing of disciplinary boundaries may present a challenge to new researchers. This conference will bring together pioneering researchers in the aforementioned fields and the new generations that they have mentored. It will celebrate prior accomplishments and will seek to identify prospects for future research on nonlinear materials, and to ensure that growing numbers of researchers, equally representing gender and background, keep emerging in the years to come.
Confirmed Speakers
Professor John Ball, Oxford University, Oxford, UK
Professor Kaushik Bhattacharya, California Institute of Technology, Pasadena, California
Professor Irene Fonseca, Carnegie Mellon University, Pittsburgh, PA
Professor Yi-Chao Chen, University of Houston, Houston, Texas
Professor Robert Hardt, Rice Univesity, Houston, Texas
Professor Richard James, University of Minnesota, Minneapolis, Minnesota
Professor David Kinderlehrer, Carnegie Mellon University, Pittsburgh, PA
Professor Bo Li, University of California San Diego, San Diego, California
Professor Chi-Sing Man, University of Kentucky, Lexington, Kentucky
Professor Lev Truskinovsky, Ecole Polytechnique, Paris, France
Professor Epifanio Virga, Università di Pavia, Italy
Professor Giovanni Zanzotto, Università di Padova, Italy
Professor Arghir Zarnescu, University of Sussex, UK and Romanian Academy
Acknowledgements
The workshop is co-organized by the Department of Aerospace Engineering and Mechanics of the University of Minnesota, the College of Science and Engineering (University of Minnesota), the Institute for Mathematics and its Applications (IMA, University of Minnesota), the Center for Nonlinear Analysis (Carnegie Mellon University), the SIAM Activity Group (SIAG) on Mathematical Aspects of Materials Sciences, the School of Mathematics of the University of Minnesota, and the Society for Natural Philosophy.
Lodging
The venue will be the Valley River Inn, a destination hotel and conference center in Eugene, Oregon. Participants will need to make their own reservations. A block of rooms has been reserved at each of these hotels. Special prices will be available until September 1, 2015. Please mention the "Math conference in the honor of Jerry Ericksen" when reserving. Available lodging options:
Valley River Inn
1000 Valley River Way
Eugene, OR 97401 |
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Phone number:
(541) 743-1000 |
Residence Inn, Marriott
25 Club Road
Eugene, OR 97401 |
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Phone number:
1-541-342-7171 |
Inn at the 5th
205 E. 6th Ave.
Eugene, OR 97401 |
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Phone number:
(541) 743-4099
(541) 743-0869 |
Phoenix Inn Suites
850 Franklin Blvd
Eugene, OR 97403 |
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Phone number:
(541) 344-0001 |
Travel Support
Travel support may be available for young researchers attending the conference. Please submit your application by September 23, 2015 for consideration.
Fees
A registration fee of $150 will be charged at the conference site. It will be used to cover coffee break and other conference administration expenses. In addition, an approximate amount of $40 will be charged for the banquet.
