Abstract of the talk
Polyhedral cell complex (PCC) is a discrete representation of a 3D space tessellation providing an excellent model of real polycrystalline materials 1 such as alloys and ceramics. From the combinatorial point of view, PCCs are just a family of sets containing graphs and hypergraphs as their subsets or skeletons. Algebraically, they can be expressed as a set of incidence matrices 2 or discrete operators acting on the k-cells of different dimensions k = {0, 1, 2, 3}. In addition to the scalar and vector parameters defined on a PCC’s cells, the present talk focuses on the labelling or indexing of sets in a PCC, which means an assignment for each set a label or several labels characterising its own type. The whole set of these labels assigned on each of the PCC skeletons forms a configuration state, which can be expressed by a single state vector encoding all the cell types similar to the well-known DNA sequences. The set of all such state vectors in a PCC one can regard as a structure defined on its elements. A succession of related structures, expressed by their state vectors, can be referred to as a process defined on a PCC's elements. Three different types of structures can be distinguished: directly assigned, imposed by the k-cells of other dimensions and induced in due course of some kinetic process which depends on the pre-existing assigned structure.
Such a discrete theoretical framework allows to describe and analyse appearance and evolution of a particular space ordering inside real polycrystalline materials based on their X-ray/EBSD maps by the well-developed tools of algebraic topology, statistics and graph theory. The present work primarily focuses on the processes of structure evolution, providing both mathematical tools and software 3 for the simulation of structure evolution of assigned, imposed and induced defect types. In particular, we perform an entropic (informational) and component (graph) analysis of grain structure evolution during continuous dynamic recrystallisation (CDRX) processes in copper and aluminium alloys 4,5 and of another structure of micro-cracks obtained as a result of kinetic fracture process and induced by the spatial arrangement of graphene inclusions in ceramic composites 6.
The authors acknowledge the financial support from EPSRC UK via grants EP/V022687/1 (PRISB) and EP/N026136/1 (GEMS).
1 E. N. Borodin, A. P. Jivkov, Evolution of triple junctions’ network during severe plastic deformation of copper alloys – a discrete stochastic modelling. Philosophical Magazine, 100 (2019) 467-485.
2 K. Berbatov, P.D. Boom, A.L. Hazel, A.P. Jivkov, Applied Mathematical Modelling 110 (2022) 172-192.
3 Elijah Borodin, Discrete Processing Design code (2023) URL: github.com/PRISBteam/PCC_Processing_Design
4 S. Zhu, E. Borodin, A. P. Jivkov, Topological phase transitions of grain boundary networks during severe plastic deformations of copper alloys. Acta Materialia (Under review), 2023.
5 S. Zhu, E. Borodin, A. P. Jivkov, Triple junctions network as the key pattern for characterisation of grain structure evolution in metals, Materials & Design, 198 (2021) 109352.
6 E. Borodin, A.P. Jivkov, A.G. Sheinerman, M. Yu Gutkin. Optimisation of rGO-enriched nanoceramics by combinatorial analysis. Materials & Design, 212 (2021) 110191.
About the speaker
Dr. Elijah Borodin is a staff at the Department of Solids and Structures in the School of Engineering.
