Mathematical theory of polycrystalline materials

Metal and alloys typically form as polycrystals, consisting of different single crystal grains, each grain corresponding to a different orientation of the underlying atomic lattice. Under thermal treatment or applied forces alloys develop microstructure within each grain due to phase transformations involving a change of shape of the lattice, and to dislocations (defects in the lattice structure), this microstructure having to fit together geometrically both within each grain and across grain boundaries. The geometry of the grains, the orientation of the crystal lattice in each grain, and the microstructure within each grain, are all crucial for determining the macroscopic behaviour of the material.
The aim of the proposal is to analyze the behaviour of polycrystals using rigorous mathematical theory, covering all the aspects described above, and to apply the results to different metals and alloys, in particular to the development of new high-performance steels in collaboration with Tata Steel.

This project will have multiple types of impact: industrial, academic, and training.

(i) Industrial impact: Collaboration with Tata Steel.

A substantial part of this proposal is in collaboration with Carola Celada-Casero, Piet Kok and Wil Spanjer from Tata Steel. The ultimate goal of Tata Steel is to improve steel grades (alloys) and steel-forming processes for the automobile industry, which will lead to safer, lighter, and more fuel-efficient cars. There are several steps to this goal, which will only be achieved through a combined effort of industry, scientists and engineers. Designing better steels requires a deep understanding of its material properties, which are in turn determined by the geometry of its microstructure. The challenge is to incorporate information about the microstructure of multi-phase steels into models at the macroscopic, engineering scale. We will do this in several ways: create geometrical models of the grain structure of steels for computational homogenization, study the interaction of grain geometry and microstructure, study the interaction of dislocations and grain boundaries, incorporate the effect of dislocations in strain-gradient plasticity models, and perform rigorous homogenization.


(ii) Academic impact.

The project will deliver academic impact through dissemination of the results in mathematics, materials science and engineering journals, and by bringing together different academic communities. A crucial aim of this proposal is to facilitate knowledge exchange between mathematicians, engineers and industry. To this aim we will organise two interdisciplinary conferences. The first conference will be at the start of the project to identify the most pressing needs of practitioners and the tools that must be developed to meet these needs. The second conference will be at the end of the project to report on our progress. In addition to these conferences, we will also disseminate the results of this project through our wide network of collaborators, and at several international conferences.

(iii) Building capacity.

This proposal will contribute to the people pipeline and will consist of several training activities for postdocs and PhD students. The training will be relevant to future careers in both academia and industry.