Historical review of Zig-Zag theories for multilayered plates and shells Erasmo Carrera Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; carrera@polito.it This paper gives a historical review of the theories that have been developed for the analysis of multilayered structures. Attention has been restricted to the so-called Zig-Zag theories, which describe a piecewise continuous displacement field in the plate thickness direction and fulfill interlaminar continuity of transverse stresses at each layer interface. Basically, plate and shell geometries are addressed, even though beams are also considered in some cases. Models in which the number of displacement variables is kept independent of the number of constitu- tive layers are discussed to the greatest extent. Attention has been restricted to those plate and shell theories which are based on the so-called method of hypotheses or axiomatic approach in which assumptions are introduced for displacements and/or transverse stresses. Mostly, the work published in the English language is reviewed. However, an account of a few articles originally written in Russian is also given. The historical review conducted has led to the fol- lowing main conclusions. 1 Lekhnitskii 1935 was the first to propose a Zig-Zag theory, which was obtained by solving an elasticity problem involving a layered beam. 2 Two other different and independent Zig-Zag theories have been singled out. One was developed by Am- bartsumian 1958, who extended the well-known Reissner-Mindlin theory to layered, aniso- tropic plates and shells; the other approach was introduced by Reissner 1984, who proposed a variational theorem that permits both displacements and transverse stress assumptions. 3 On the basis of historical considerations, which are detailed in the paper, it is proposed to refer to these three theories by using the following three names: Lekhnitskii Multilayered Theory, LMT, Ambartsumian Multilayered Theory AMT, and Reissner Multilayered Theory RMT. As far as subsequent contributions to these three theories are concerned, it can be re- marked that: 4 LMT although very promising, has almost been ignored in the open literature. 5 Dozens of papers have instead been presented which consist of direct applications or par- ticular cases of the original AMT. The contents of the original works have very often been ignored, not recognized, or not mentioned in the large number of articles that were published in journals written in the English language. Such historical unfairness is detailed in Section 3.2. 6 RMT seems to be the most natural and powerful method to analyze multilayered struc- tures. Compared to other theories, the RMT approach has allowed from the beginning devel- opment of models which retain the fundamental effect related to transverse normal stresses and strains. This review article cites 138 references. DOI: 10.1115/1.1557614 1 INTRODUCTION Two-dimensional 2D modeling of multilayered plates and shells requires appropriate theories. The discontinuity of physical/mechanical properties in the thickness direction makes inadequate those theories which were originally de- veloped for one-layered structures, eg, the Cauchy-Poisson- Kirchhoff-Love thin plate/shell theory 1–4, or the Reissner-Mindlin theory Reissner 5 and Mindlin 6, as well as higher order models such as the one by Hildebrand, Reissner, and Thomas 7. These theories are, in fact, not able to reproduce piecewise continuous displacement and transverse stress fields in the thickness direction, which are experienced by multilayered structures 8. In 9, these two effects have been summarized by the acronym C z 0 - requirements; that is, displacements and transverse stresses must be C 0 -continuous functions in the z -thickness direction. A qualitative comparison of displacement and stress fields in a single-layered and a multi-layered structure is shown in Fig. 1. This picture clearly shows that theories designed for single-layered structures are not suitable to analyze multilay- Transmitted by Associate Editor S Adali Appl Mech Rev vol 56, no 3, May 2003 © 2003 American Society of Mechanical Engineers 287