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Modelling of Biomaterials as an Application of the Theory of Mixtures

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Modeling Biomaterials

Part of the book series: Nečas Center Series ((NECES))

Abstract

An overview of the mixture theory is provided while building upon similarities with the classical single continuum theory. The mixture theory can be formulated on different levels of description, in terms of different state variables. The second law of thermodynamics is used as a fundamental constraint for obtaining the constitutive relations, the closures. For this purpose, one can either use a definition of entropy (Gibbs’ relation) or a definition of temperature, which is used to identify the entropy production. We discuss the significance and role of coupling in a model formulation and illustrate it using examples stemming from biology.

The theory is applied to the formulation of a biphasic model of cartilage. The superiority of mixture theory over the single continuum framework is evident, but there is a trade-off in terms of more parameters that need to be estimated and the number of boundary conditions. In the latter, the difficulties are inherent to the theory and remain an open problem. They are not derivable and require further modelling, although there are situations where boundary conditions can be assessed. Upscaling methods might provide answers in certain situations as well as a new idea within GENERIC framework.

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Acknowledgements

Václav Klika is grateful for support from the Czech Grant Agency, project number 20-22092S.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Václav Klika

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  1. Václav Klika
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Correspondence to Václav Klika .

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  1. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Josef Málek

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Endre Süli

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Klika, V. (2021). Modelling of Biomaterials as an Application of the Theory of Mixtures. In: Málek, J., Süli, E. (eds) Modeling Biomaterials. Nečas Center Series. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-88084-2_4

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