A mathematical model for elasticity using calculus on discrete manifolds

Ioannis Dassios, Gary O'Keeffe, Andrey Jivkov

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    Abstract

    We propose a mathematical model to represent solid materials with discrete lattices and to analyse their behaviour by calculus on discrete manifolds. Focus is given on the mathematical derivation of the lattice elements by taking into account the stored energy associated with them. We provide a matrix formulation of the non-linear system describing elasticity with exact kinematics, known as finite strain elastic- ity in continuum mechanics. This formulation is ready for software implementation, and may also be used in atomic scale models as an alternative to existing empirical approach with pair and cohesive potentials. An illustrative example, analysing a local region of a node, is given to demonstrate the model performance.
    Original languageEnglish
    Pages (from-to)9057-9070
    Number of pages14
    JournalMathematical Methods in the Applied Sciences
    Volume41
    Issue number18
    Early online date27 Apr 2018
    DOIs
    Publication statusPublished - 1 Dec 2018

    Keywords

    • discrete manifold, lattice model, steel microstructure, elasticity, energy, non-linear system

    Research Beacons, Institutes and Platforms

    • Manchester Energy
    • Advanced materials

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