A variational justification of the assumed natural strain formulation of finite elements—I. Variational principles

Abstract

The assumed natural strain (ANS) formulation of finite elements has undergone rapid development over the past five years. The key formulation step is the replacement, in the potential energy principle, of selected displacement-related strains by independently assumed strain fields in element natural coordinates. These strains are not generally derivable from displacements. This procedure was conceived as one of several competing methods with which to solve the element locking problem. Its most noteworthy feature is that, unlike many forms of reduced integration, it produces no rank deficiency; furthermore, it is easily extendible to geometrically non-linear problems. Many original formulations were not based on a variational principle. The objective of Part I is to study the ANS formulation from a variational standpoint. This study is based on two hybrid extensions of the Reissner-type functional that uses strains and displacements as independent fields. One of the forms is a genuine variational principle that contains an independent boundary traction field, whereas the other one represents a restricted variational principle. Two procedures for element-level elimination of the strain field are discussed, and one of them is shown to be equivalent to the inclusion of incompatible displacement modes. In Part II [C. Militello and C. A. Felippa, Comput. Struct.34, 439–444 (1990)], the four-node C0 plate bending quadrilateral element is used to illustrate applications of this theory.