Axisymmetric quadrilateral elements for large deformation hyperelastic analysis
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Axisymmetric quadrilateral elements for large deformation hyperelastic analysis G. H. Liu • K. Y. Sze
Received: 29 June 2010 / Accepted: 6 July 2010 / Published online: 18 July 2010 Ó Springer Science+Business Media, B.V. 2010
Abstract In this paper, axisymmetric 8-node and 9-node quadrilateral elements for large deformation hyperelastic analysis are devised. To alleviate the volumetric locking which may be encountered in nearly incompressible materials, a volumetric enhanced assumed strain (EAS) mode is incorporated in the eight-node and nine-node uniformly reduced-integrated (URI) elements. To control the compatible spurious zero energy mode in the 9-node element, a stabilization matrix is attained by using a hybrid-strain formulation and, after some simplification, the matrix can be programmed in the element subroutine without resorting to numerical integration. Numerical examples show the relative efficacy of the proposed elements and other popular eight-node elements. In view of the constraint index count, the two elements are analogous to the Q8/3P and Q9/3P elements based on the u–p hybrid/mixed formulation. However, the former elements are more straight forward than the latter elements in both formulation and programming implementation. Keywords Hybrid/mixed Finite element method Large deformation Hyperelastic Volumetric locking Enhanced assumed strain Stabilization
G. H. Liu K. Y. Sze (&) Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, People’s Republic of China e-mail: [email protected]
1 Introduction Hyperelastic constitutive models are commonly used for rubbery materials and biological soft tissues which are often assumed to be nearly incompressible. In finite element analysis of nearly incompressible materials, when the number of volumetric constraints per element is excessive, the element assemblage would exhibit the volumetric or dilatational locking which is signified by a non-physical and excessive stiffness as if the assemblage was locked. Meanwhile, the predicted pressure or mean stress may exhibit oscillation (Naylor 1974; Pian and Lee 1976; Malkus and Hughes 1978; Sani et al. 1981; Spilker 1981; Sani et al. 1981; Yu 1991; Yu et al. 1993; Sze et al. 1995; Zienkiewicz and Taylor 2000; Yu and Netherton 2000). In four-node quadrilateral elements, the latter phenomenon is known as the checkerboard mode (Sani et al. 1981; Spilker 1981). Note worthily, an effective way of tackling nearly incompressible material analyses is the u–p formulation in which a variational functional with independently assumed displacement and pressure is employed (Pian and Lee 1976; Sze et al. 1995; Zienkiewicz and Taylor 2000). Over the past decades, effort has been put into the development of high performance finite element formulations for axisymmetric problems. While incompatible plane and solid QM6 elements (Taylor et al. 1976) have been popularly employed by commercial software, the related axisymmetric element fails the patch test. On the other hand, a number
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