A formulation for the 4‐node quadrilateral element

authored by: Ulrich Hueck, Peter Wriggers
Abstract: A formulation for the plane 4‐node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second‐order Taylor series in the physical co‐ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non‐linear analysis. There, an application is given for the calculation of inelastic problems in physically non‐linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero‐energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.
External Organisation(s): Technische Universität Darmstadt
Type: Article
Journal: International Journal for Numerical Methods in Engineering
Volume: 38
Pages: 3007-3037
No. of pages: 31
ISSN: 0029-5981
Publication date: 30.09.1995
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, Engineering(all), Applied Mathematics
Electronic version(s): https://doi.org/10.1002/nme.1620381802 (Access: Unknown)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{50d9696ed409483c98971986c2a6afc0,
title = "A formulation for the 4‐node quadrilateral element",
abstract = "A formulation for the plane 4‐node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second‐order Taylor series in the physical co‐ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non‐linear analysis. There, an application is given for the calculation of inelastic problems in physically non‐linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero‐energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.",
keywords = "incompatible modes, quadrilateral, stabilization matrix, underintegration",
author = "Ulrich Hueck and Peter Wriggers",
year = "1995",
month = sep,
day = "30",
doi = "10.1002/nme.1620381802",
language = "English",
volume = "38",
pages = "3007--3037",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "18",
}