This work is an overview of available theories and finite elements that have been developed for multilayered, anisotropic, composite plate and shell structures. Although a comprehensive description of several techniques and approaches is given, most of this paper has been devoted to the so called axiomatic theories and related finite element implementations. Most of the theories and finite elements that have been proposed over the last thirty years are in fact based on these types of approaches. The paper has been divided into three parts. Part I, has been devoted to the description of possible approaches to plate and shell structures: 3D approaches, continuum based methods, axiomatic and asymptotic two-dimensional theories, classical and mixed formulations, equivalent single layer and layer wise variable descriptions are considered (the number of the unknown variables is considered to be independent of the number of the constitutive layers in the equivalent single layer case). Complicating effects that have been introduced by anisotropic behavior and layered constructions, such as high transverse deformability, zig-zag effects and interlaminar continuity, have been discussed and summarized by the acronimC
 -Requirements. Two-dimensional theories have been dealt with in Part II. Contributions based on axiomatic, asymtotic and continuum based approaches have been overviewed. Classical theories and their refinements are first considered. Both case of equivalent single-layer and layer-wise variables descriptions are discussed. The so-called zig-zag theories are then discussed. A complete and detailed overview has been conducted for this type of theory which relies on an approach that is entirely originated and devoted to layered constructions. Formulas and contributions related to the three possible zig-zag approaches, i.e. Lekhnitskii-Ren, Ambartsumian-Whitney-Rath-Das, Reissner-Murakami-Carrera ones have been presented and overviewed, taking into account the findings of a recent historical note provided by the author. Finite Element FE implementations are examined in Part III. The possible developments of finite elements for layered plates and shells are first outlined. FEs based on the theories considered in Part II are discussed along with those approaches which consist of a specific application of finite element techniques, such as hybrid methods and so-called global/local techniques. The extension of finite elements that were originally developed for isotropic one layered structures to multilayerd plates and shells are first discussed. Works based on classical and refined theories as well as on equivalent single layer and layer-wise descriptions have been overviewed. Development of available zig-zag finite elements has been considered for the three cases of zig-zag theories. Finite elements based on other approches are also discussed. Among these, FEs based on asymtotic theories, degenerate continuum approaches, stress resultant methods, asymtotic methods, hierarchy-p,_-s global/local techniques as well as mixed and hybrid formulations have been overviewed.

Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells

A general two-dimensional shear deformation theory of laminated composite plates is presented. The theory account for a desired degree of approximation of the displacements through the laminate thickness. As special cases, the classical, first-order (Reissner–Mindlin) and other shear deformation theories available in the literature can be deduced from the present theory.

A generalization of two-dimensional theories of laminated composite plates†

The layerwise laminate theory of Reddy (1987) is used to develop a layerwise, two-dimensional, displacement-based, finite element model of laminated composite plates that assumes a piecewise continuous distribution of the tranverse strains through the laminate thickness. The resulting layerwise finite element model is capable of computing interlaminar stresses and other localized effects with the same level of accuracy as a conventional 3D finite element model. Although the total number of degrees of freedom are comparable in both models, the layerwise model maintains a 2D-type data structure that provides several advantages over a conventional 3D finite element model, e.g. simplified input data, ease of mesh alteration, and faster element stiffness matrix formulation. Two sample problems are provided to illustrate the accuracy of the present model in computing interlaminar stresses for laminates in bending and extension.

Modelling of thick composites using a layerwise laminate theory

One of the research direction of Horst Lippmann during his whole scientific career was devoted to the possibilities to explain complex material behavior by generalized continua models. A representative of such models is the Cosserat continuum. The basic idea of this model is the independence of translations and rotations (and by analogy, the independence of forces and moments). With the help of this model some additional effects in solid and fluid mechanics can be explained in a more satisfying manner. They are established in experiments, but not presented by the classical equations. In this paper the Cosserat-type theories of plates and shells are debated as a special application of the Cosserat theory.

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On generalized Cosserat-type theories of plates and shells: a short review and bibliography

It is 100 years since the well-know Mohr-Coulomb strength theory was established in 1900. A considerable amount of theoretical and experimental research on strength theory of materials under complex stress state was done in the 20th Century. This review article presents a survey of the advances in strength theory (yield criteria, failure criterion, etc) of materials (including matellic materials, rock, soil, concrete, ice, iron, polymers, energetic material, etc) under complex stress, discusses the relationship among various criteria, and gives a method of choosing a reasonable failure criterion for applications in research and engineering. Three series of strength theories, the unified yield criterion, the unified strength theory, and others are summarized. This review article contains 1163 references regarding the strength theories. This review also includes a biref discussion of the computational implementation of the strength theories and multi-axial fatigue.

Advances in strength theories for materials under complex stress state in the 20th Century

Nonlinear analysis of multilayered shells

The principles of symmetry as applied to structural mechanics are reviewed, symmetric and anti-symmetric forces and structures exhibiting various kinds of symmetries are defined, and a systematic procedure for the application of symmetry principles to the determination of existing and vanishing force and displacement components is presented with the objectives of: (1) reminding structural engineers of the availability of these powerful and extremely useful tools and encouraging them to review these principles and apply them in their daily activities; (2) illustrating the uses and advantages of symmetry arguments in applied and structural mechanics; and (3) arousing interest for increased use of, and research activities in the application of group-theoretic methods in structural and continuum mechanics.

Symmetry in structural mechanics

On the dynamic stability of viscoelastic perfect columns

The large deflection and stability behaviour of spherical inflatables subjected to a concentrated load applied at the apex and undergoing various degrees of wrinkling is analyzed. It is shown that for low profile inflatables, the behaviour of such structures under axi-symmetric concentrated loads is non-linear, without instability or limit point characteristics. On the other hand for high-profile membranes with height/span ratios greater than 0.5 and undergoing deflections equal to or greater than the initial height or radius of curvature of the structure, “snap through” behaviour is established. Ultimate loads for the totally wrinkled membranes are also established. The wrinkled region is considered in an Eulerian description satisfying equilibrium equations and the Gauss-Codazzi relations. The equations are solved numerically and the results are presented in non-dimensional form in a number of figures.

Peter G. Glockner

Papers

Nonlinear analysis of multilayered shells

Symmetry in structural mechanics

On the dynamic stability of viscoelastic perfect columns

Finite deformation and stability behaviour of spherical inflatables under axi-symmetric concentrated loads

The stability of viscoelastic perfect columns: A dynamic approach