Brick Elements

Abstract

So far, we have dealt with the finite element modelling of problems which were simplified to one-dimensional or two-dimensional descriptions. There remains a large class of problems which need to be addressed directly as three dimensional states of stress. Solid or three dimensional elements are needed to carry out the finite element modelling of such cases. Three dimensional solid elements can be classified as tetrahedral, triangular prism or hexahedral (see Fig. 9.1). In most general cases of three dimensional stress analysis, no problems are encountered with the use of any of these elements as long as a sufficiently large number of elements are used and no constrained media limits are approached. However, when such limits appear, e.g. in the modelling of slender beams, thin plates or shells, or at near incompressibility (i.e. Poisson’s ratio v → 0.5), the tetrahedral and triangular prism elements cannot be easily modified to avoid these difficulties. It is possible, by a variety of techniques, to improve the 8-node and 27-node elements so that they are free of locking. In this chapter, we shall examine these aspects closely. Once again, we shall emphasize that the consistency requirements for a general distorted brick element are very difficult to describe analytically and so most of the studies in this chapter will be based on the rectangular form of these elements. The 20-node element suffers from the same difficulties that we have seen for the 8-node plate element (Chapter 8) due to the missing nodes — a complete introduction of consistency thus becomes impossible to achieve for this element. In fact, there is also evidence [9.1] that in some 3D applications, the 8noded element augmented with bubble functions is superior in performance to the 20-noded element on the basis of the computational cost for a required accuracy.