Classification of stress modes in assumed stress fields of hybrid finite elements

A classiÞcation method is presented to classify stress modes in assumed stress Þelds of hybrid Þnite element based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there only exist m ("n!r) natural deformation modes in a hybrid Þnite element which has n degrees of freedom and r rigid-body modes. For a hybrid element, stress modes in various assumed stress Þelds proposed by di¤erent researchers can be classiÞed into m stress mode groups corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the ßexibility matrix [H] is a diagonal matrix, the classiÞcation of stress modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress modes that are interchangeable in an assumed stress Þeld and do not cause any kinematic deformation modes in the element. A necessary and su¦cient condition for avoiding kinematic deformation modes in a hybrid element is also presented. By means of the m classiÞed stress mode groups and the necessary and su¦cient condition, assumed stress Þelds with the minimum number of stress modes can be constructed and the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress Þeld can be constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node solid element are discussed