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

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.