Influence of Membrane Stresses on Postbuckling of Rectangular Plates Using a Nonlinear Elastic 3-D Cosserat brick Element

Abstract

This paper has two main objectives. The first is to examine the influence of membrane stresses on postbuckled deformations of nonlinear elastic isotropic rectangular plates. The second is to further examine the accuracy of a new 3-D Cosserat eight noded brick element (Nadler and Rubin in Int J Solids Struct 40: 2003) which was developed within the context of the theory of a Cosserat point. The equations of the Cosserat element include both material and geometric nonlinearities. A number of example problems are considered which examine predictions of the Cosserat element for beams and plates and comparison has been made with results from the commercial codes ANSYS and ADINA. Also, the approximate nonlinear postbuckling solution described in Timoshenko and Gere (Theory of elastic stability, Mc Graw-Hill, New York) is shown to be more limited than originally expected. These results suggest that the Cosserat element is robust, can perform well under extreme conditions and is capable of modeling combinations of threedimensional bodies with attached thin structures.

Modified torsion coefficients for a 3-D brick Cosserat point element

“…More specifically, it can be shown that the CPE will reproduce this exact simple solution only when the value of B is specified by the Bubnov-Galerkin value (16). The examples considered in the next section demonstrate that the simple torsion-like solutions (8) are basic solutions that should be reproduced exactly in order to improve the convergence properties for the more general pure torsion problem. Therefore, it is now suggested that the torsion coefficients (13) in the CPE theory be modified to use the value (16) instead of the original value (17).…”

“…Thus, the displacement field associated with the simple torsion-like solution (8) [with ω 1 = ω 2 = 0] can be expressed in terms of the convected coordinates by…”

An improved 3-D brick Cosserat point element for irregular shaped elements

“…However, in [12] the coefficients of the part of the strain energy function which controls the response to inhomogeneous deformations were determined by limiting attention to a rectangular parallelepiped, and by matching exact linear solutions to a class of problems that included pure bending, pure torsion and high-order hourglassing modes of deformation. It was observed in [10,11] that the 3-D brick CPE can be used to characterize the response of thin plates and shells with only one element through the thickness and that it produces robust accurate results for a difficult three-dimensional test problem of surface buckling of a column. In [17] the CPE has been used for axisymmetric problems and in [10] it has been used to analyze postbuckling of plates.…”

“…It was observed in [10,11] that the 3-D brick CPE can be used to characterize the response of thin plates and shells with only one element through the thickness and that it produces robust accurate results for a difficult three-dimensional test problem of surface buckling of a column. In [17] the CPE has been used for axisymmetric problems and in [10] it has been used to analyze postbuckling of plates. Also, in [9] it was shown that some element formulations based on hyperelastic material constitutive equations in commercial codes predict inelastic response and that only the CPE was able to predict physical shear buckling of a plate in plane strain and also predict accurate response for situations that typically exhibit locking.…”