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Approximation of incompressible large deformation elastic problems: some unresolved issues

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Abstract

Several finite element methods for large deformation elastic problems in the nearly incompressible and purely incompressible regimes are considered. In particular, the method ability to accurately capture critical loads for the possible occurrence of bifurcation and limit points, is investigated. By means of a couple of 2D model problems involving a very simple neo-Hookean constitutive law, it is shown that within the framework of displacement/pressure mixed elements, even schemes that are inf-sup stable for linear elasticity may exhibit problems when used in the finite deformation regime. The roots of such troubles are identified, but a general strategy to cure them is still missing. Furthermore, a comparison with displacement-based elements, especially of high order, is presented.

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Acknowledgments

The authors were partially supported by the European Commission through the FP7 Factory of the Future project TERRIFIC (FoF-ICT-2011.7.4, Reference: 284981), by the European Research Council through the FP7 Ideas Starting Grants n. 259229 ISOBIO and n. 205004 GeoPDEs, as well as by the Italian MIUR through the FIRB “Futuro in Ricerca” Grant RBFR08CZ0S and through the PRIN Project n. 2010BFXRHS. These supports are gratefully acknowledged.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile e Architettura, Università di Pavia, Pavia, Italy

    Ferdinando Auricchio & Alessandro Reali

  2. Center for Advanced Numerical Simulations—IUSS, Pavia, Italy

    Ferdinando Auricchio & Carlo Lovadina

  3. Dipartimento di Matematica, Università di Milano, Milan, Italy

    Lourenço Beirão da Veiga

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Carlo Lovadina

  5. Department of Civil and Environmental Engineering, University of California, Berkeley, CA, 94720, USA

    Robert L. Taylor

  6. Institute of Continuum Mechanics, Leibnitz University of Hannover, Hannover, Germany

    Peter Wriggers

Authors
  1. Ferdinando Auricchio
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  2. Lourenço Beirão da Veiga
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  3. Carlo Lovadina
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  4. Alessandro Reali
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  5. Robert L. Taylor
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  6. Peter Wriggers
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Correspondence to Carlo Lovadina.

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Auricchio, F., Beirão da Veiga, L., Lovadina, C. et al. Approximation of incompressible large deformation elastic problems: some unresolved issues. Comput Mech 52, 1153–1167 (2013). https://doi.org/10.1007/s00466-013-0869-0

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  • DOI: https://doi.org/10.1007/s00466-013-0869-0

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