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Approximation Schemes for Materials with Discontinuities

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Non-standard Discretisation Methods in Solid Mechanics

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 98))

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Abstract

Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution investigates and devises methods from mathematical and numerical analysis to quantify them: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in spaces of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed.

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Acknowledgements

The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non- standard discretization methods, mechanical and mathematical analysis” within the project “Reliability of Efficient Approximation Schemes for Material Discontinuities Described by Functions of Bounded Variation”—Project Number 255461777 (TH 1935/1-2 and BA 2268/2-2).

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  1. Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104, Freiburg im Breisgau, Germany

    Sören Bartels, Marijo Milicevic & Nico Weber

  2. Weierstrass Instinuite for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Marita Thomas & Sven Tornquist

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  1. Sören Bartels
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Correspondence to Sven Tornquist .

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  1. Institute of Mechanics, University of Duisburg-Essen, Essen, Germany

    Jörg Schröder

  2. Institute of Continuum Mechanics, Leibniz Universität Hannover, Hannover, Germany

    Peter Wriggers

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Bartels, S., Milicevic, M., Thomas, M., Tornquist, S., Weber, N. (2022). Approximation Schemes for Materials with Discontinuities. In: Schröder, J., Wriggers, P. (eds) Non-standard Discretisation Methods in Solid Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-030-92672-4_17

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