Details
| Original language | English |
|---|---|
| Pages (from-to) | 3007-3037 |
| Number of pages | 31 |
| Journal | International Journal for Numerical Methods in Engineering |
| Volume | 38 |
| Issue number | 18 |
| Publication status | Published - 30 Sept 1995 |
| Externally published | Yes |
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
ASJC Scopus subject areas
- Mathematics(all)
- Numerical Analysis
- Engineering(all)
- Mathematics(all)
- Applied Mathematics
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In: International Journal for Numerical Methods in Engineering, Vol. 38, No. 18, 30.09.1995, p. 3007-3037.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A formulation for the 4‐node quadrilateral element
AU - Hueck, Ulrich
AU - Wriggers, Peter
PY - 1995/9/30
Y1 - 1995/9/30
N2 - 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.
AB - 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.
KW - incompatible modes
KW - quadrilateral
KW - stabilization matrix
KW - underintegration
UR - http://www.scopus.com/inward/record.url?scp=0006771329&partnerID=8YFLogxK
U2 - 10.1002/nme.1620381802
DO - 10.1002/nme.1620381802
M3 - Article
AN - SCOPUS:0006771329
VL - 38
SP - 3007
EP - 3037
JO - International Journal for Numerical Methods in Engineering
JF - International Journal for Numerical Methods in Engineering
SN - 0029-5981
IS - 18
ER -