Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints

Research output: Preprint/Documentation/ReportPreprint

Contributors

  • Robert J. Martin - , University of Duisburg-Essen (Author)
  • Ionel-Dumitrel Ghiba - , Alexandru Ioan Cuza University of Iaşi (Author)
  • Maximilian Köhler - , Ruhr University Bochum (Author)
  • Daniel Balzani - , Ruhr University Bochum (Author)
  • Oliver Sander - , Chair of Numerical Methods of Partial Differential Equations (Author)
  • Patrizio Neff - , University of Duisburg-Essen (Author)

Abstract

We derive the quasiconvex relaxation of the Biot-type energy density ∥sqrt(Dφ^TDφ)−I_2∥^2 for planar mappings φ:R2→R2 in two different scenarios. First, we consider the case Dφ∈GL+(2), in which the energy can be expressed as the squared Euclidean distance dist2(Dφ,SO(2)) to the special orthogonal group SO(2). We then allow for planar mappings with arbitrary Dφ∈R2×2; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks.

Details

Original languageEnglish
PublisherarXiv
Number of pages29
Publication statusPublished - 18 Jan 2025
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External IDs

ORCID /0000-0003-1093-6374/work/178384018