PSI - Issue 43

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ScienceDirect

Procedia Structural Integrity 43 (2023) 15–22 Structural Integrity Procedia 00 (2023) 000–000 Structural Integrity Procedia 00 (2023) 000–000

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© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. © 2023 The Authors. Published by Elsevier B.V. is is an open access article under the CC BY-NC-ND license (http: // creativec mmons.org / licenses / by-nc-nd / 4.0 / ) r-review under the responsibility of SMF10 organizers. Keywords: Discrete structures; Topology; Metric; Balance laws; Boundary conditions Abstract Presented is a new mathematical framework for analysis of physical processes in solids with complex internal structures. Unlike the classical description of solids as continua, the solids here are treated as assemblies of discrete, finite entities that represent di ff erent microstructural elements. Scalar quantities and momenta are defined as discrete di ff erential forms on such assemblies, and their balances are formulated using topological and metric operations with such forms. The new description is background independent, i.e., it does not rely on structures external to the solid. The resulting boundary value problems are given by matrix equations with constraints. Contrary to the familiar numerical methods, these equations do not approximate continuum problems but represent the physics on discrete assemblies exactly. The method provides a unique modelling capability: elements of materials’ internal structures with di ff erent dimensions may have di ff erent physical properties. For example, in a polycrystalline assembly, the substance di ff usivity inside a crystal (bulk, 3D) can be di ff erent from its di ff usivity along a grain boundary (surface, 2D) and from its di ff usivity along a triple junction (curve, 1D), or these microstructural elements can have di ff erent mechanical properties. © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under the responsibility of MSMF10 organizers. Keywords: Discrete structures; Topology; Metric; Balance laws; Boundary conditions 10 th International Conference on Materials Structure & Micromechanics of Fracture Microstructures, physical processes, and discrete di ff erential forms Andrey P. Jivkov a, ∗ , Kiprian Berbatov a , Pieter D. Boom b , Andrew L. Hazel c a Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Oxford Road, Manchester M13 9PL, UK b Mechanical Engineering Department & Interdisciplinary Research Center for Construction and Building Materials, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia c Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK Abstract Presented is a new mathematical framework for analysis of physical processes in solids with complex internal structures. Unlike the classical description of solids as continua, the solids here are treated as assemblies of discrete, finite entities that represent di ff erent microstructural elements. Scalar quantities and momenta are defined as discrete di ff erential forms on such assemblies, and their balances are formulated using topological and metric operations with such forms. The new description is background independent, i.e., it does not rely on structures external to the solid. The resulting boundary value problems are given by matrix equations with constraints. Contrary to the familiar numerical methods, these equations do not approximate continuum problems but represent the physics on discrete assemblies exactly. The method provides a unique modelling capability: elements of materials’ internal structures with di ff erent dimensions may have di ff erent physical properties. For example, in a polycrystalline assembly, the substance di ff usivity inside a crystal (bulk, 3D) can be di ff erent from its di ff usivity along a grain boundary (surface, 2D) and from its di ff usivity along a triple junction (curve, 1D), or these microstructural elements can have di ff erent mechanical properties. 10 th International Conference on Materials Structure & Micromechanics of Fracture Microstructures, physical processes, and discrete di ff erential forms Andrey P. Jivkov a, ∗ , Kiprian Berbatov a , Pieter D. Boom b , Andrew L. Hazel c a Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Oxford Road, Manchester M13 9PL, UK b Mechanical Engineering Department & Interdisciplinary Research Center for Construction and Building Materials, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia c Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

1. Introduction 1. Introduction

The materials’ internal structures are formed of elements that, depending on the length scale of observation, may appear as three-dimensional (3D), two-dimensional (2D), one-dimensional (1D), and zero-dimensional (0D). For example, at the polycrystalline scale, metals appear as assemblies of 3D grains, 2D grain boundaries, 1D triple junc tions, and 0D quadruple points. Similar classification of elements into groups with di ff erent apparent dimensions can be made at lower length scales Phillips (2001). Elements of di ff erent dimensions can have di ff erent physical (e.g., thermal conductivity, mass di ff usivity, electrical conductivity) and mechanical (e.g., sti ff ness, strength, ductil ity) properties. The e ff ective (macroscopic) properties of the material depend on the arrangements and interactions of The materials’ internal structures are formed of elements that, depending on the length scale of observation, may appear as three-dimensional (3D), two-dimensional (2D), one-dimensional (1D), and zero-dimensional (0D). For example, at the polycrystalline scale, metals appear as assemblies of 3D grains, 2D grain boundaries, 1D triple junc tions, and 0D quadruple points. Similar classification of elements into groups with di ff erent apparent dimensions can be made at lower length scales Phillips (2001). Elements of di ff erent dimensions can have di ff erent physical (e.g., thermal conductivity, mass di ff usivity, electrical conductivity) and mechanical (e.g., sti ff ness, strength, ductil ity) properties. The e ff ective (macroscopic) properties of the material depend on the arrangements and interactions of

∗ Corresponding author. Tel.: + 44-161-306-3765. E-mail address: andrey.jivkov@manchester.ac.uk ∗ Corresponding author. Tel.: + 44-161-306-3765. E-mail address: andrey.jivkov@manchester.ac.uk

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 10.1016/j.prostr.2022.12.228 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under the responsibility of MSMF10 organizers. 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under the responsibility of MSMF10 organizers.