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Theories and finite elements for multilayered, anisotropic, composite plates and shells

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Summary

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 0 z -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.

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Abbreviations

a, b, h :

plate/shell geometrical parameters (length, width and thickness)

k :

sub/super-script used to denote parameters related to thek-layer

N :

order of the expansions used for transverse stresses and displacements

N l :

Number of constituent layers of multilayered plate/shell

x,y,z :

coordinates of Cartesian reference systems used for plates

α, β,z :

curvilinear coordinates of reference systems used for shells

2D:

two-Dimensional

3D:

three-Dimensional

AWRD:

Ambartsumian-Whitney-Rath-Das theory

CLT:

Classical lamination Theory

ESLM:

Equivalent Single Layer Models

FEs:

Finite Elements

FEM:

Finite Element Method

FSDT:

First Shear Deformation Theory

HOT:

Higher Order Theories

HTD:

High Transverse Deformability

IC:

Interlaminar Continuity

KR:

Koiter's Recommendation

LR:

Lekhnitskii-Ren theory

LFAT:

Love First Approximation Theory

LSAT:

Love Second Approximation Theory

LWM:

Layer-Wise Models

RMC:

Reissner-Murakami-Carrera theory

RMVT:

Reissner's Mixed Variational Theorem

TA:

Transverse Anisotropy

VRT:

Vlasov-Reddy Theory

WFHL:

Weak Form of Hooke's Law

ZZ:

Zig-Zag.

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    E. Carrera

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Carrera, E. Theories and finite elements for multilayered, anisotropic, composite plates and shells. ARCO 9, 87–140 (2002). https://doi.org/10.1007/BF02736649

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