Science at the Triple Point between Mathematics, Mechanics and Materials Science

In 2010 the US National Science Foundation (NSF) awarded a 'Partnerships for International Research and Education' (PIRE) grant for "Science at the Triple Point between Mathematics, Mechanics and Materials Science" from a highly competitive field of hundreds of applications. The grant was awarded to form a network of international working groups around major research areas in Mathematics, Mechanics and Materials Science. The University of Oxford is one of four named European Partners working with US Participating Institutions to develop research collaborations that will benefit both PIRE-funded participants and European researchers alike. This proposal is directed to issues in applied mathematics and mechanics which arise from materials science. Many contemporary problems in new and advanced materials are related to the variety of length and time scales and heterogeneities inherent in their fabrication and function. Predictive theories for these complex systems require new advanced mathematics whose discovery will be enhanced by international collaboration. New mathematics can abstract methods developed in one area and apply them to other areas, facilitating far-reaching cross-fertilization and the discovery of unanticipated linkages. The complementary strengths and combined expertise of the team will push applied analysis to these new frontiers.

The proposal concerns four main subjects:

1. Pattern Formation from Energy Minimization.
Several members of the PIRE team have organized their careers around problems at the interface between materials science and the calculus of variations. We have identified four topics that seem ripe for near-term development. Each is an area where several team members have expertise; therefore the enhanced communication associated with this PIRE will greatly accelerate our progress.

(a) Elastic sheets, leaves, and flowers.
(b) Dimension reduction
(c) Stressed epitaxial films
(d) Dislocation microstructures in crystal plasticity.

A recurring theme is the search for ansatz-free lower bounds. Guessing the minimum-energy state is usually easy (nature gives us a hint). Understanding why the guess is right - why no other state can do better - is typically much more difficult.

2. Challenges in Atomistic to Continuum Modeling and Computing.
Localized defects such as dislocations, crack tips, or grain boundaries interact across large length scales though elastic fields. Accurate simulation of localized defects requires an atomistic model - which however is too computationally demanding to be used for the entire system. Hence the attraction of atomistic-to-continuum coupling, which permits one to use the computationally intensive atomistic model only near the defects. Far away, where the deformation is nearly uniform, a continuum elastic model provides adequate resolution.

3 Prediction of Hysteresis.
For a solid-to-solid phase transformation, thermal hysteresis refers to a transformation temperature on cooling that differs from that on heating. Hysteresis also occurs during stress-induced transformation, with the stress needed to induce the forward transformation being different from that causing the reverse transformation. Similar effects occur in ferromagnetism and ferroelectricity. Recently this topic has acquired fresh significance in connection with materials for energy conversion, since the efficiency of a conversion process often depends on the size of an associated hysteresis loop.

4 Pattern Dynamics and Evolution of Material Microstructure.
Cellular and granular networks are ubiquitous in nature. They exhibit behaviour on many different length and time scales and are often found to be metastable. The energetics and connectivity of the ensemble of the grain and the boundary network during evolution play a crucial role in determining the properties of a material across a wide range of scales.

The proposal concerns fundamental research on material behaviour. Medium to long-term benefits can be expected for the discovery and optimization of new materials with special properties, such as enhanced shape-memory effects and low hysteresis. Some progress has already been made in these directions. The presence in the network of theoretical groups, an electron microscopy laboratory and the large industrial partner Robert Bosch GmbH will facilitate such developments.

This is fundamental research with numerous potential medium to long term impacts on materials science and engineering. On the one hand current developments in nano-technology require an increased understanding of mechanics on small length-scales, and on the other hand the macroscopic behaviour of materials is largely a result of their constituent microstructures. More immediate possible implications are for the design of nano-devices based on solid phase transformations and to the discovery of new alloys with special properties related to possible microstructures (such as low hysteresis) which can result from tuning deformation parameters. The research on atomistic-to-continuum coupling is particularly relevant for applications in which defects appear that may stabilize or destabilize such microstructures.

The proposed programme will build on our established collaborations with international colleagues and also lead to new engagements. Established arrangements will allow for Oxford based researchers, and the wider community to benefit from the existing PIRE communication and engagement infrastructure.

The impact on the academic community will be achieved via the well established dissemination mechanisms of journal and conference publications, conference and seminar talks, through the direct interaction and discussions with colleagues throughout the world on academic visits and at conferences, through engagement with PIRE working groups and through workshops organised annually, the first of which is being held in Oxford. All of these means of dissemination are known to be effective for reaching the academic community.

Communication and Engagement
The PIRE website (http://www.math.cmu.edu/PIRE/) contains information on research activities of the PIRE network. Details of summer schools, workshops and other events open to the wider community are available, including recorded seminars and presentations. Other information, including publications produced by the PIRE network is detailed on the website. Access to the information on the website is not restricted and so Oxford based researchers and the wider community will benefit from this facility.

Each topic of research will have at least one working group hosted by one of the four US sites. Each working group will focus on a PIRE research topic and will involve network collaboration. Visitor and member exchange will be encouraged. The groups will meet regularly and play an important educational, research, and administrative role, providing a forum for discussing open conjectures and new results. Dissemination of these results will be, amongst other streams, through PIRE research seminars, which will be digitally recorded and the video files posted on the PIRE website for Oxford based academics to share. This will create an environment that facilitates international engagement.
Workshops will be open to the wider scientific community, and will rotate among partner institutions from the PIRE network and provide an excellent platform from which to inform young researchers about key developments, attract them to the areas, and provided networking opportunities. The first workshop, "Pattern Formation and Multiscale Phenomena in Materials", will be held in Oxford with subsequent workshops being held at the Hausdorff Institute in Bonn in year 2, SISSA in year 3, Caltech in year 4, and NYU in year 5. Lectures at the workshops will be recorded and made available via the PIRE website.