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 constitutive 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 following 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 Ambartsumian ~1958!, who extended the well-known Reissner-Mindlin theory to layered, anisotropic 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 remarked 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 particular 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 structures. Compared to other theories, the RMT approach has allowed from the beginning development 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#

Historical review of Zig-Zag theories for multilayered plates and shells

Three-dimensional Problems of the Theory of Elasticity. By A. I. Lur’e.1964. (Interscience Publishers)

A large variety of plate theories are described and assessed in the present work to evaluate the bending and vibration of sandwich structures. A brief survey of available works is first given. Such a survey includes significant review papers and latest developments on sandwich structure modelings. The kinematics of classical, higher order, zigzag, layerwise, and mixed theories is described. An exhaustive numerical assessment of the whole theories is provided in the case of closed form solutions of simply supported panels made of orthotropic layers. Reference is made to the unified formulation that has recently been introduced by the first author for a plate/shell analysis. Attention has been given to displacements, stresses (both in-plane and out-of-plane components), and the free vibration response. Only simply supported orthotropic panels loaded by a transverse distribution of bisinusoidal pressure have been analyzed. Five benchmark problems are treated. The accuracy of the plate theories is established with respect to the length-to-thickness-ratio (LTR) geometrical parameters and to the face-to-core-stiffness-ratio (FCSR) mechanical parameters. Two main sources of error are outlined, which are related to LTR and FCSR, respectively. It has been concluded that higher order theories (HOTs) can be conveniently used to reduce the error due to LTR in thick plate cases. But HOTs are not effective in increasing the accuracy of the classical theory analysis whenever the error is caused by increasing FCSR values; layerwise analysis becomes mandatory in this case.

A Survey With Numerical Assessment of Classical and Refined Theories for the Analysis of Sandwich Plates

Recent research advances in the dynamic behavior of shells: 1989–2000, Part 2: Homogeneous shells

A unified formulation to assess theories of multilayered plates for various bending problems

Design of corrugated circular plates under a local load as a flexible system of shells of revolution

Solution of problems of the statics of shallow shells and plates with hinged and rigidly-fastened opposing edges

A heterogeneous mathematical model is formulated. It permits us to use simultaneously the equations of the theories of elasticity and Timoshenko-type shells to describe different fragments of a structure. This model can be written as a closed system of differential equations of different dimensions with boundary conditions on the domain boundary and conjugate conditions on the surfaces where fragments are mated. A variational problem is formulated. The existence and uniqueness of the solution are analyzed. Numerical results demonstrate the efficiency of the approach

Stress-strain analysis of elastic bodies on the basis of a heterogeneous mathematical model

Use of spline approximation to study displacement and stress fields in cylinders with different boundary conditions on the ends

The effect of change in the curvature parameters of the stress state of concave corrugated hollow cylinders is studied. The change is attributed to variations in the radius of a moving circle and in the distance to its center. The problem is solved in spatial statement using analytical methods of separation of variables, approximation of functions by discrete Fourier series, and the numerical discrete-orthogonalization method. Results are presented in the form of plots demonstrating distributions of displacement and stress fields and are analyzed.

Ya. M. Grigorenko

Papers

Design of corrugated circular plates under a local load as a flexible system of shells of revolution

Solution of problems of the statics of shallow shells and plates with hinged and rigidly-fastened opposing edges

Stress-strain analysis of elastic bodies on the basis of a heterogeneous mathematical model

Use of spline approximation to study displacement and stress fields in cylinders with different boundary conditions on the ends

Effect of Change in the Curvature Parameters on the Stress State of Concave Corrugated Hollow Cylinders