Trilinear Hexahedra with Integral-Averaged Volumes for Nearly Incompressible Nonlinear Deformation

Abstract

Many materials such as biological tissues, polymers, and metals in plasticity can undergo large deformations with very little change in volume. Low-order finite elements are also preferred for certain applications, but are well known to behave poorly for such nearly incompressible materials. Of the several methods to relieve this volumetric locking, the B method remains popular as no extra variables or nodes need to be added, making the implementation relatively straightforward and efficient. In the large deformation regime, the incompressibility is often treated by using a reduced order or averaged value of the volumetric part of the deformation gradient, and hence this technique is often termed an F approach. However, there is little in the literature detailing the relationship between the choice of F and the resulting B and stiffness matrices. In this article, we develop a framework for relating the choice of F to the resulting B and stiffness matrices. We examine two volume-averaged choices for F , one in the reference and one in the current configuration. Volume-averaged F formulation has the advantage that no integration points are added. Therefore, there is a modest savings in memory and no integration point quantities needed to be interpolated between different sets of points. Numerical results show that the two formulations developed give similar results to existing methods.

A new 3D finite element for the finite deformation of nearly incompressible hyperelastic solids

“…The displacement of point A obtained through the finite element discretisation of the block with the 27 noded hexahedral elements is shown in Figure 1. It can be seen that the standard method is showing slightly stiffening behaviour whereas tri-linear J-Bar is predicting the locking free response consistent with the Foster and Nejad (2015). It is also found that with the 8 × 4 × 4 mesh of 27 noded element using standard method, volumetric locking is eliminated.…”

“…Tri-linear J-Bar formulation is validated by considering the finite stretching of neo-Hookean rubber block (2 m × 1 m × 1 m) which is fixed at one end and subjected to deformation independent uniformly distributed tensile load at the other end. Material parameters for the neo-Hookean material model are K = 29,980 kPa and C 1 = 60 kPa (Foster and Nejad, 2015). The displacement of point A obtained through the finite element discretisation of the block with the 27 noded hexahedral elements is shown in Figure 1.…”

“…Incompressibility constraint imposes severe restriction on the displacement field resulting in the over stiffening response of the finite element mesh and is termed as volumetric locking. Volumetric locking in nearly incompressible solids is a current field of research in solid mechanics (Elguedj et al, 2008a;Wriggers and Carstensen, 2009;Bernardin et al, 2015a;Foster and Nejad, 2015). Many techniques have been developed to avoid volumetric locking viz.…”

“…In B-Bar method, strain displacement matrix is split into dilatational and deviatoric parts (Hughes, 1980). Foster and Nejad (2015) developed a volumetric locking free element based on the volumetric average of Jacobian within the framework of B-Bar method. F-Bar method of de Souza Neto et al (1996) assumed Jacobian as a constant evaluated at the centre of the element whereas the mean dilatational approach of Nagtegaal et al (1974) evaluated the Jacobian as the volume average within the element.…”

A new 3D finite element for the finite deformation of nearly incompressible hyperelastic solids

“…In this example, the displacement of point A obtained by discretising the block with 8 × 4 × 4 elements using eight noded hexahedral element is shown in Figure 3. It can be seen that the standard formulation without smoothening of J is showing stiff behaviour whereas J-Bar-based element eliminates the volumetric locking and the results match with Foster and Nejad (2015) B-Bar-based element. Now, we investigate the effect of mesh refinement on the accuracy of tip displacement of point A. Domain is discretised with 2 2 n n n × × mesh of eight noded brick element where n = 2, 4, 8, 16 is the number of divisions along the length of the block.…”