${\infty }^6$ Mixed plate theories based on the Generalized Unified Formulation.

Abstract

The generalized unified formulation introduced in Part I for the case of composite plates and Reissner’s Mixed variational theorem is, for the first time in the literature, applied to the case of layerwise theories. Each layer is independently modeled. The compatibility of the displacements and the equilibrium of the transverse stresses between two adjacent layers are enforced a priori. Infinite combinations of the orders used for displacements $u_x$, $u_y$, $u_z$ and out-of-plane stresses ${\sigma }_{\mathit{zx}}$, ${\sigma }_{\mathit{zy}}$, ${\sigma }_{\mathit{zz}}$ can be freely chosen. ${\infty }^6$ layerwise theories are therefore presented. The code based on this capability can have all the possible ${\infty }^6$ theories built-in, thus, making the code a powerful and versatile tool to analyze different geometries, boundary conditions and applied loads. All ${\infty }^6$ theories are generated by expanding 13 $1\times 1$ invariant matrices (the kernels of the Generalized Unified Formulation). How the kernels are expanded and the theories generated is explained. Details of the assembling in the thickness direction and the generation of the matrices are provided.

Introduction

Equivalent single layer theories give a sufficiently accurate description of the global laminate response. However, these theories are not adequate for determining the stress fields at ply level. Layerwise theories assume separate displacement field expansions within each layer. The accuracy is then greater but the price is in the increased computational cost. Many layerwise plate models have been proposed in the past by applying classical plate theory or higher order theories at each layer. Generalizations of these approaches were also given, and the displacements variables were expressed in terms of Lagrange polynomials. Among the papers devoted on the subject of layerwise theories, Refs. [1], [2], [3], [4], [5] give an idea of the different approaches. Normally, displacement-based layerwise models do not a priori take into account the continuity of the transverse stresses between two adjacent layers. The problem of satisfying the interlaminar continuity of the transverse stresses a priori led to the derivation of mixed layerwise theories [6], [7], [8]. Layerwise theories could also be easily extended to the case of composite beams as explained in Ref. [9]. The conceptual differences between the displacement fields in layerwise and equivalent single layer theories are depicted in Fig. 1. Layerwise models are computationally more expensive than the less accurate equivalent single layer models. Therefore, layerwise models can be used in regions of the structure in which an accurate description is required [10], whereas equivalent single layer models are employed in other parts of the structure. It is also possible to develop quasi-layerwise theories in which some quantities are described using the layerwise approach and some are described using the equivalent single layer approach. These two categories of theories will be described in Part III Ref. [11] and Part IV Ref. [12] of this work.

The generalized unified formulation (GUF) [13], [14] is a new formalism and a generalization of Carrera’s Unified Formulation (CUF) [15]. GUF was introduced in the case of displacement-based theories. GUF was extended, for the first time in the literature, to the case of mixed theories (see Part I, Ref. [16]). In particular, Reissner’s mixed variational theorem (RMVT) (see [17], [18]) was employed. The unknown variables were the displacements and transverse stresses.

In the present work GUF will be extended to the case of composite multilayered structures analyzed with layerwise models. Each layer variable (a displacement or a transverse stress) will be independently expanded along the thickness leading to a very wide variety of new theories. Since each variable can be expanded in infinite different forms (by simply changing the order of the polynomial used in the expansion along the thickness), the case of RMVT-based theories leads to the writing of ${\infty }^6$ layerwise mixed theories. All the possible theories generated using GUF can be implemented in a single code without the requirements of new implementations or theoretical developments.

This is a new powerful methodology to create layerwise theories, and the details will be given in this work. In particular, it will be shown how to generate the layer matrices from the fundamental nuclei or kernels (introduced in Part I, Ref. [16]) and how to assemble these matrices. The interlaminar continuity of the displacements and transverse stresses is taken into account. Numerous examples clarify how to use GUF to generate a desired layerwise theory.