Multiscale modeling of calcified polymer hydrogels

• Applicant: Professorin Dr.-Ing. Sandra Klinge
• Subject Area: Mechanics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 420794479

• Final Report Year: 2023

Final Report Abstract

Calcified hydrogels present strongly heterogeneous microstructures in varying forms. The present project investigates the influence of that heterogeneity on the response of the material to mechanical loads, as well as the impact it has on the diffusion behaviour. Where the former is the primary motivation for developing these materials, the latter plays an important role in applications of the material in both biomedical and industrial settings. In the first stage of the project, the materials were modelled by developing RVEs consisting of two distinct phases. The modelling of the calcium phosphate phase is based on a neo-Hookean strain energy function, whereas the Ogden model is assumed for the hydrogel matrix. The stiffnesses of the three investigated materials PHEA (limited calcification), PDMA (strongly calcified) and PAAm (strongly calcified) are remarkably different. Simulated tests in tension, compression and shear loading modes successfully show that the difference in effective stiffnesses stems from the degree of calcification, as well as from the arrangement of the inorganic microstructure. PDMA includes spherical structures, whereas a honeycomb-like structure is found in PAAm. The latter provides more stability in bond with the hydrogel matrix and yields higher stresses than the spherical structure. This finding is also substantiated by the simulation of indentation tests for both strongly calcified hydrogels. In the second stage of the project, the diffusion hindrance factors caused by the heterogeneities in PDMA and PAAm following calcification are investigated. Based on a numerical asymptotic homogenisation technique, a finite element framework for the unit-cell problem in media with randomly-distributed inclusions and voids is developed. Here, RVEs need to be large enough for a statistically-accurate homogenisation, yet finely resolved enough to correctly model their behaviour and solve the finite element problem accurately. This was achieved by implementing an automatic refinement scheme, based on the Kelly Indicator, using the open-source package deal.II. Multiscale homogenisation for PDMA showed a strongly non-linear dependence of diffusion on the solute radius. For PAAm, it was shown that the primary dependent variable is the free volume, and the relationship to the diffusivity is also non-linear. These results can guide material designers when tuning calcification parameters for different applications.

Publications

• Multiscale modeling of calcified hydrogel networks. PAMM, 21(1).
  Aygün, Serhat & Klinge, Sandra
  
• Thermomechanical and multiscale modelling of polymeric materials with u complex microstructures. 91st GAMM Annual Meeting, Kassel, Germany, March 15-19, 2021.
  Aygün, S. & Klinge
  
• Homogenisation: Effective Diffusion Coefficients in Hydrogels with tunable mechanical Properties. 7th Common PhD-Candidates’ Seminar, Berlin, Germany, October 10-11, 2022.
  Graham, M.
  
• Mechanical modelling of calcified hydrogels by using homogenization u schemes. 92st GAMM Annual Meeting, Aachen, Germany, August 15-19, 2022.
  Aygün, S. & Klinge, S.
  
• Multiscale modelling of hydrogels with complex microstructures. 9th u GACM Colloquium on Computational Mechanics, Essen, Germany, September 21-23, 2022.
  Aygün, S. & Klinge, S.
  
• Thermomechanical and multiscale modelling of polymeric materials with complex microstructures. Joy of Mechanics, conference on occasion of the birthday of Prof. Paul Steinmann, Gothenburg, Sweden. 24.26. August, 2022.
  Klinge, S.
  
• Effective Diffusion Properties of Calcified Hydrogels Obtained by the Asymptotic Homogenisation. 5th SEECCM2023 ECCOMAS Conference, Vrnjacka Banja, Serbia, July 5-7, 2023.
  Graham, M.
  
• Effective Diffusion Properties of Calcified Hydrogels Obtained by the Asymptotic Homogenisation. 93rd GAMM Annual Meeting, Dresden, Germany, May 30-June 2, 2023.
  Graham, M.
  
• Effective Diffusion Properties of Calcified Hydrogels. Si-Mposium: Der Simulierte Mensch. Berlin, Germany. 1. September, 2023.
  Graham, M.
  
• Multiscale material modelling in engineering. Freie Universit¨t Berlin, Germany. 12. a January, 2023. (Invited talk)
  Klinge, S.
  
• Multiscale modelling of calcified polymer hydrogels. DGM-Meeting of the research group on the Microstructures Modeling. Aachen, Germany. 18. April, 2023. (Invited Talk)
  Klinge, S.
  
• Multiscale modelling of calcified polymer hydrogels. ICPDF International Conference on Placticity, Damage and Fracture. Punta Cana, Dominic Republic. 2.-9. January, 2023.
  Klinge, S.
  
• Multiscale modelling of polymeric materials with complex microstructures. TU Darmstadt, Germany. 7. June, 2023. (Invited talk)
  Klinge, S.
  
• Two‐scale computational homogenization of calcified hydrogels. Mathematical Methods in the Applied Sciences, 47(17), 13182-13198.
  Aygün, Serhat & Klinge, Sandra
  
• Multiscale homogenisation of diffusion in enzymatically-calcified hydrogels. Journal of the Mechanical Behavior of Biomedical Materials, 149, 106244.
  Graham, Marc & Klinge, Sandra