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

This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions; some quadratic forms have a clear energy meaning; variational properties characterize the solutions and other results, invalid in traditional boundary element methods enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, and computer implementations. Areas and aspects which at present require further research are identified, and comparative assessments are attempted with respect to traditional boundary integral-elements. This article includes 176 references.

https://hal.archives-ouvertes.fr/hal-00111259/document

Symmetric Galerkin Boundary Element Methods

This paper deals with the formulation of finite plate elements for an accurate description of stress and strain fields in multilayered, thick plates subjected to static loadings in the linear, elastic cases. The so-called zig-zag form and interlaminar continuity are addressed in the considered formulations. Two variational statements, the principle of virtual displacements (PVD) and the Reissner mixed variational theorem (RMVT) are employed to derive finite element matrices. Transverse stress assumptions are made in the framework of RMVT and the resulting finite elements describe a priori interlaminar continuous transverse shear and normal stresses. Both modellings which preserve the number of variables independent of the number of layers (equivalent single-layer models, ESLM) and layer-wise models (LWM) in which the same variables are independent in each layer, have been treated. The order N of the expansions assumed for both displacement and transverse stress fields in the plate thickness direction z as well as the number of element nodes Nn have been taken as free parameters of the considered formulations. By varying N, Nn, variable treatment (LW or ESL) as well as variational statements (PVD and RMVT), a large number of newly finite elements have been presented. Finite elements that are based on PVD and RMVT have been called classical and advanced, respectively.

In order to write the matrices related to the considered plate elements in a concise form and to implement them in a computer code (see Part 2), extensive indicial notations have been set out. As a result, all the finite element matrices have been built from only five arrays that were called fundamental nuclei (four are related to RMVT applications and one to PVD cases). These arrays have 3×3 dimensions and are therefore constituted of only nine terms each. The different formulations are then obtained by expanding the indices that were introduced for the N-order expansion, for the number of nodes Nn and for the constitutive layers Nl. Compliances and/or stiffness are accumulated from layer to multilayered level according to the corresponding variable treatment (ESLM or LWM). The numerical evaluations and assessment for the presented plate elements have been provided in the companion paper (Part 2), where it has been concluded that it is convenient to refer to RMVT as a variational tool to formulate multilayered plate elements that are able to give a quasi-three-dimensional description of stress/strain fields in multilayered thick structures. Copyright © 2002 John Wiley & Sons, Ltd.

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Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: Derivation of finite element matrices

The analysis of adhesively bonded joints started in 1938 with the closed-form model of Volkersen. The equilibrium equation of a single lap joint led to a simple governing differential equation with a simple algebraic equation. However, if there is yielding of the adhesive and/or the adherends and substantial peeling is present, a more complex model is necessary. The more complete is an analysis, the more complicated it becomes and the more difficult it is to obtain a simple and effective solution. The finite element (FE) method, the boundary element (BE) method and the finite difference (FD) method are the three major numerical methods for solving differential equations in science and engineering. These methods have also been applied to adhesive joints, especially the FE method. This book deals with the most recent numerical modelling of adhesive joints. Advances in damage mechanics and extended finite element method are described in the context of the FE method with examples of application. The classical continuum mechanics and fracture mechanics approach are also introduced. The BE method and the FD method are also discussed with indication of the cases they are most adapted to. There is not at the moment a numerical technique that can solve any problem and the analyst needs to be aware of the limitations involved in each case.

Advances in Numerical Modelling of Adhesive Joints

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

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Recent Advances and Emerging Applications of the Boundary Element Method

https://msvlab.hre.ntou.edu.tw/grades/bem(2008)/A%20FAST%203D%20DUAL%20BOUNDARY%20ELEMENT%20METHOD%20BASED%20ON%20HIERARCHICAL%20MATRICES.pdf

A fast 3D dual boundary element method based on hierarchical matrices

Multidomain boundary integral formulation for piezoelectric materials fracture mechanics

A direct approach is employed to obtain a general boundary integral formulation for the analysis of composite laminates subjected to uniform axial strain. The integral equations governing the problem are directly deduced from the reciprocity theorem, employing the generalized orthotropic elasticity fundamental solutions expressly inferred. The solution is achieved by the boundary element method, which gives, once the traction-free boundary conditions and the interfacial continuity conditions are enforced, a linear system of algebraic equations. The formulation does not present restrictions with regard to the laminate stacking sequence and it does not require any aprioristic assumption. The interlaminar stress field near the free edge of generally stacked composite laminates subjected to uniaxial extension is investigated through this boundary integral equation formulation. The numerical applications show good agreement with those already available in literature, and they demonstrate the accuracy and the efficiency of the proposed method. approach where a complete three-dimensional analysis is performed was presented by Pipes and Pagano,1 who employed the finite dif- ference technique to obtain the solution of the governing elastic- ity equations. Many solutions obtained by using the finite element method are available.2"1() These differ from each other in the formu- lation, in the kind of employed elements, and in the discretization schemes. In the literature analytical solutions of approximate the- ories are also present. The techniques employed to achieve these solutions include the perturbation method,11 series solution,12'13 Lekhintskii's complex stress potentials coupled with an eigenfunc- tion expansion14'15 or a polynomial expansion,16'll the extension of Reissner's variational principle,18'19 and the force balance method coupled with the minimum complementary energy principle.20'21 The stress distributions obtained with these approaches show good agreement between them for points away from the free edge. Con- siderable disagreement exists instead for points near the free edge location among the various numerical and analytical solutions. This is to be expected as a result of a priori assumptions or because the boundary conditions of the continuum problem have been trans- formed into conditions on the generalized data. Actually the numer- ical solution techniques are not able to predict the singular trend in the stress field at the free edge, and hence, to achieve satisfactory solutions a lengthy extrapolation procedure is required. On the other hand, the analytical approximate solutions often rest on assumptions about the problem unknowns that enforce the free edge stress field structure. In the present paper, the stress field in multilayered composite laminates under uniform axial extension is analyzed on the basis of the integral equation theory.22'23 The laminate is considered as composed by prismatic elements having different elastic proper- ties. The generic element or ply is treated as homogeneous, and in terms of constitutive equations it is described by a generalized orthotropic law. The integral equations governing the exact elastic- ity solution of the problem are directly obtained by applying the reciprocity theorem with the fundamental solutions of the general- ized orthotropic elasticity.24"27 The

Stress Fields in General Composite Laminates

A variational formulation for symmetric positive definite BEM is derived by using a hybrid displacement functional. The functional is expressed in terms of domain and boundary variables, assumed as independent from one another. The resulting equations involve only boundary displacement variables and the approach gives symmetric and positive definite matrices which can be combined with finite elements. The accuracy and efficiency of the method is tested through comparison of some typical results.

A hybrid displacement variational formulation of BEM for elastostatics

In this paper a regular variational boundary element formulation for dynamic analysis of two-dimensional magneto-electro-elastic domains is presented. The method is based on a hybrid variational principle expressed in terms of generalized magneto-electro-elastic variables. The domain variables are approximated by using a superposition of weighted regular fundamental solutions of the static magneto-electro-elastic problem, whereas the boundary variables are expressed in terms of nodal values. The variational principle coupled with the proposed discretization scheme leads to the calculation of frequency-independent and symmetric generalized stiffness and mass matrices. The generalized stiffness matrix is computed in terms of boundary integrals of regular kernels only. On the other hand, to achieve meaningful computational advantages, the domain integral defining the generalized mass matrix is reduced to the boundary through the use of the dual reciprocity method, although this implies the loss of symmetry. A purely boundary model is then obtained for the computation of the structural operators. The model can be directly used into standard assembly procedures for the analysis of non-homogeneous and layered structures. Additionally, the proposed approach presents some features that place it in the framework of the weak form meshless methods. Indeed, only a set of scattered points is actually needed for the variable interpolation, while a global background boundary mesh is only used for the integration of the influence coefficients. The results obtained show good agreement with those available in the literature proving the effectiveness of the proposed approach.

Giuseppe Davi

Papers

A fast 3D dual boundary element method based on hierarchical matrices

Multidomain boundary integral formulation for piezoelectric materials fracture mechanics

Stress Fields in General Composite Laminates

A hybrid displacement variational formulation of BEM for elastostatics

A regular variational boundary model for free vibrations of magneto-electro-elastic structures