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A semi-implicit finite strain shell algorithm using in-plane strains based on least-squares

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Abstract

The use of a semi-implicit algorithm at the constitutive level allows a robust and concise implementation of low-order effective shell elements. We perform a semi-implicit integration in the stress update algorithm for finite strain plasticity: rotation terms (highly nonlinear trigonometric functions) are integrated explicitly and correspond to a change in the (in this case evolving) reference configuration and relative Green-Lagrange strains (quadratic) are used to account for change in the equilibrium configuration implicitly. We parametrize both reference and equilibrium configurations, in contrast with the so-called objective stress integration algorithms which use a common configuration. A finite strain quadrilateral element with least-squares assumed in-plane shear strains (in curvilinear coordinates) and classical transverse shear assumed strains is introduced. It is an alternative to enhanced-assumed-strain (EAS) formulations and, contrary to this, produces an element satisfying ab-initio the Patch test. No additional degrees-of-freedom are present, contrasting with EAS. Least-squares fit allows the derivation of invariant finite strain elements which are both in-plane and out-of-plane shear-locking free and amenable to standardization in commercial codes. Two thickness parameters per node are adopted to reproduce the Poisson effect in bending. Metric components are fully deduced and exact linearization of the shell element is performed. Both isotropic and anisotropic behavior is presented in elasto-plastic and hyperelastic examples.

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Notes

  1. Note that these are relative degrees-of-freedom since the rotation occurs between configurations \(\Omega _{b}\) and \(\Omega _{a}\).

  2. Note that in this Section, and for reasons of clarity, we only consider conservative problems.

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Acknowledgments

The first author thanks Professor Carlos Tiago (IST, Lisbon) for in-depth advice leading to this work. The authors gratefully acknowledge financing from the “Fundação para a Ciência e a Tecnologia” under the Project PTDC/EME-PME/108751 and the Program COMPETE FCOMP-01-0124-FEDER-010267.

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Authors and Affiliations

  1. Department of Physics, Colégio Luís António Verney, University of Évora, Rua Romão Ramalho, 59, 7002-554, Évora, Portugal

    P. Areias

  2. ICIST, Lisbon, Portugal

    P. Areias

  3. Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstraße 15, 99423, Weimar, Germany

    T. Rabczuk

  4. Mechanical Engineering Department, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n, 4200-465, Porto, Portugal

    J. César de Sá & R. Natal Jorge

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  1. P. Areias
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  2. T. Rabczuk
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  4. R. Natal Jorge
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Correspondence to P. Areias.

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Areias, P., Rabczuk, T., de Sá, J.C. et al. A semi-implicit finite strain shell algorithm using in-plane strains based on least-squares. Comput Mech 55, 673–696 (2015). https://doi.org/10.1007/s00466-015-1130-9

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