Computational Mechanics

, Volume 59, Issue 4, pp 625–646 | Cite as

A quadrilateral shell element with degree of freedom to represent thickness–stretch

Original Paper


This paper presents a quadrilateral shell element incorporating thickness–stretch, and demonstrates its performance in small and large deformation analyses for hyperelastic material and elastoplastic models. In terms of geometry, the proposed shell element is based on the formulation of the MITC4 shell element, with additional degrees of freedom to represent thickness–stretch. To consider the change in thickness, we introduce a displacement variation to the MITC4 shell element, in the thickness direction. After the thickness direction is expressed in terms of the director vectors that are defined at each midsurface node, additional nodes are placed along the thickness direction from the bottom surface to the top surface. The thickness–stretch is described by the movement of these additional nodes. The additional degrees of freedom are used to compute the transverse normal strain without assuming the plane stress condition. Hence, the three dimensional constitutive equation can be employed in the proposed formulation without any modification. By virtue of not imposing the plane stress condition, the surface traction is evaluated at the surface where the traction is applied, whereas it is assessed at the midsurface for conventional shell elements. Several numerical examples are presented to examine the fundamental performance of the proposed shell element. In particular, the proposed approach is capable of evaluating the change in thickness and the stress distribution when the effect of the surface traction is included. The behavior of the proposed shell element is compared with that of solid elements.


Shell element Thickness–stretch Sheet forming Large strains Elastoplasticity Volumetric locking 



This work was supported by JSPS KAKENHI Grant Number JP16J07401.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Graduate School of Environment and Information ScienceYokohama National UniversityYokohamaJapan

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