Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates

Abstract

One objective of the present study is to use arbitrary complete 3-dimensional constitutive equations without reduction or manipulation in nonlinear plate and shell analysis. The obvious consequence, namely the extension of a conventional 5-parameter shell formulation with Reissner-Mindlin kinematics to a 6-parameter formulation including the full set of stress and strain state does not solve the problem because a significant error in bending dominated cases occurs. To avoid this error the transverse normal strain is allowed to vary linearly across the thickness. This so-called 7-parameter theory recently proposed in the group of the authors resorts to the Enhanced Assumed Strain concept and preserves the basic features of a displacement formulation.

The 7-parameter formulation is extended to the simulation of the response of laminated structures with arbitrarily large displacements and rotations. Following the main idea of the concept, it is sufficient to formulate a complete 3-dimensional material law which considers the layered setup of the shell.

Finally, a layer-wise so-called multidirector model is developed which is well suited to grasp local interlaminar effects. In this formulation the displacement interpolation across the thickness is extended to a C⁰-continuous field described by a layer-wise Reissner-Mindlin kinematics. The purpose of the multidirector formulation is twofold: Firstly the higher order kinematics satisfies the same requirements discussed for the 7-parameter theory and allows also to use complete 3-dimensional material laws. Secondly it is appropriate to simulate laminates with extreme differences of the thicknesses or dissimilar material properties of each layer with sufficient accuracy. These are situations where formulations with C¹-continuous displacement fields across the thickness fail.