PSI - Issue 43

Andrey P. Jivkov et al. / Procedia Structural Integrity 43 (2023) 15–22 Author name / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 1. Mapping of material’s substructures to a cell 3-complex.

these elements, as they can either localise or inhibit the corresponding processes, e.g., heat conduction, mass di ff usion, charge transfer, mechanical deformation, damage, and fracture Tilley (2013). The current methods for analysis of materials’ behaviour can be divided into two large categories – continuum and discrete. Continuum methods are based on the approximation of materials as collections of infinitely many in finitesimal points, which leads to balance laws for scalar quantities and momenta written in terms of their densities using classical continuum calculus Van Groesen and Molenaar (2007). Numerical solutions of boundary value prob lems based on such balance laws are presently performed with a variety of numerical methods, most notably the finite element and boundary element methods Provatidis (2019), and the finite di ff erence and finite volume methods Hesthaven (2018). However, describing internal structures with localised or abruptly changing properties that cause localised or abruptly changing processes is not admitted by the classical continuum-based formulations. Discrete methods approximate materials as finite collections of particles with prescribed interactions and are closely related to molecular dynamics. They have been developed for modelling discontinuous problems, such as jumps in properties and / or fields that are challenging for continuum calculus. Examples include discrete element methods Onate and Owen (2011), peridynamics Madenci and Oterkus (2014) and smoothed particle hydrodynamics Filho (2019). While such methods are closer to the requirement to describe abruptly changing properties and analyse discontinuous processes, microstructural elements appearing as 2D, 1D, and 0D need to be represented by layers of particles, lines / curves of particles, or single particles with di ff erent properties / interactions, respectively. The problems become computationally intractable unless the analysed domains are reduced to sizes containing only a few elements. A method for exploring the e ff ects of the arrangement and interactions between elements of di ff erent dimensions on the material performance must reflect their finite, discrete nature, so that it can be used to analyse materials’ structures at any required length scale. Such a method must be based on a rational physical theory where balance laws are formulated from materials’ perspective, i.e., a background-independent theory describing relations between elements Smolin (2019). A background-independent theory can be constructed using di ff erential forms. In continuum calculus, these are fully antisymmetric multilinear functionals over vector fields. They are playing major role in the modernisation of fundamental physical areas Fortney (2018). Familiar examples in dimension n = 3 are 0-forms which are scalar fields, 1-forms which are co-vector fields, 2-forms which are antisymmetric tensor fields, and 3-forms which are pseudoscalar fields. The basic metric-independent operations with forms are the exterior derivative, which by acting on a p -form produces a ( p + 1)-form, and the exterior product, which by acting on a p -form and a q -form produces a ( p + q )-form. The basic metric-dependent operations with forms are the inner product, which by acting on two p -forms produces a scalar field, and the Hodge-star, which by acting on a p -form produces a ( n − p )-form by Hodge duality. A background independent, relational theory for materials with internal structures requires discrete analogues of forms and operations for formulating the fundamental balance laws of physics and mechanics. A material can be modelled by mapping the elements of its internal structure at any required length scale to components of a polyhedral assembly. This is a geometric realisation of a combinatorial structure, referred to as a 3-complex in algebraic topology May (1999). It is a collection of 0-cells representing vertices, 1-cells representing line segments, 2-cells representing polygonal areas, and 3-cells representing polyhedral volumes, as shown in Fig. 1(a). The mapping sends 0D components to some 0 cells, 1D components to some 1-cells, 2D components to some 2-cells, and 3D components to some 3-cells, as shown in Fig. 1(b-e). In all cases there maybe components of one and the same topological dimension but of di ff erent nature, forming distinct substructures illustrated with di ff erent colours. The bulk material is mapped to all unoccupied cells. Two notions related to cell complexes are required for this work: p -cochains, denoted by c p for p = 0 , 1 , 2 , 3; and coboundary operators, denoted by δ p for p = 0 , 1 , 2. A given c p can be considered as a map from p -cells to