An Assumed Stress Field Method with Orthogonal Basic Deformation Modes for Zero-Energy Mode Free Hybrid Element Construction

Abstract

A novel method is presented to orthogonalize the basic deformation modes for hybord stress elements. It is shown that the resulting orthogonal basic deformation modes are independent of elastic constants and they can be used to investigate the hybrid elements with different assumed stress fields. It is noted that a deformation mode appears to be a zero‐energy mode when its strain‐driven stress mode is orthogonal to all the assumed stress modes. This provides an easy way to identify the spurious modes of a given element. In addition, based upon the orthogonality relationship between the given initial stress modes and the orthogonal basic deformation modes, the necessary stress modes to formulate a hybrid element free of spurious modes can be straightforwardly selected from the stress matrix including the given initial stress modes. Furthermore, it is found that the orthogonal condition of the basic deformation modes is not only necessary but also sufficient to derive the assumed stress field that can produce a hybrid element free of zero‐energy modes. Several numerical examples are illustrated in details to demonstrate the efficacy of the proposed basic deformation mode method.