Computational Mechanics

, Volume 55, Issue 4, pp 673–696 | Cite as

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

  • P. AreiasEmail author
  • T. Rabczuk
  • J. César de Sá
  • R. Natal Jorge
Original Paper


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.


Constitutive laws Large deformation Shell Plasticity Anisotropic 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • P. Areias
    • 1
    • 2
    Email author
  • T. Rabczuk
    • 3
  • J. César de Sá
    • 4
  • R. Natal Jorge
    • 4
  1. 1.Department of Physics, Colégio Luís António VerneyUniversity of ÉvoraÉvoraPortugal
  2. 2.ICISTLisbonPortugal
  3. 3.Institute of Structural MechanicsBauhaus-University WeimarWeimarGermany
  4. 4.Mechanical Engineering Department, Faculty of EngineeringUniversity of PortoPortoPortugal

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