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

Vital signs such as pulse rate and breathing rate are currently measured using contact probes. But, non-contact methods for measuring vital signs are desirable both in hospital settings (e.g. in NICU) and for ubiquitous in-situ health tracking (e.g. on mobile phone and computers with webcams). Recently, camera-based non-contact vital sign monitoring have been shown to be feasible. However, camera-based vital sign monitoring is challenging for people with darker skin tone, under low lighting conditions, and/or during movement of an individual in front of the camera. In this paper, we propose distancePPG, a new camera-based vital sign estimation algorithm which addresses these challenges. DistancePPG proposes a new method of combining skin-color change signals from different tracked regions of the face using a weighted average, where the weights depend on the blood perfusion and incident light intensity in the region, to improve the signal-to-noise ratio (SNR) of camera-based estimate. One of our key contributions is a new automatic method for determining the weights based only on the video recording of the subject. The gains in SNR of camera-based PPG estimated using distancePPG translate into reduction of the error in vital sign estimation, and thus expand the scope of camera-based vital sign monitoring to potentially challenging scenarios. Further, a dataset will be released, comprising of synchronized video recordings of face and pulse oximeter based ground truth recordings from the earlobe for people with different skin tones, under different lighting conditions and for various motion scenarios.

DistancePPG: Robust non-contact vital signs monitoring using a camera

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

Piezoelectric sensors are increasingly being used in active structural health monitoring, due to their durability, light weight and low power consumption. In the present work damage detection and characterization methodologies based on Lamb waves have been evaluated for aircraft panels. The applicability of various proposed delay-and-sum algorithms on isotropic and composite stiffened panels have been investigated, both numerically and experimentally. A numerical model for ultrasonic wave propagation in composite laminates is proposed and compared to signals recorded from experiments. A modified delay-and-sum algorithm is then proposed for detecting impact damage in composite plates with and without a stiffener which is shown to capture and localize damage with only four transducers.

https://iopscience.iop.org/article/10.1088/0964-1726/23/7/075007/pdf

Assessment of delay-and-sum algorithms for damage detection in aluminium and composite plates

A boundary integral formulation for the analysis of stress fields induced in composite laminates by initial strains, such as may be due to temperature changes and moisture absorption is presented. The study is formulated on the basis of the theory of generalized orthotropic thermo-elasticity and the governing integral equations are directly deduced through the generalized reciprocity theorem. A suitable expression of the problem fundamental solutions is given for use in computations. The resulting linear system of algebraic equations is obtained by the boundary element method and stress interlaminar distributions in the boundary-layer are calculated by using a boundary only discretization. The approach is general and it does not require a priori assumptions. Numerical results are presented to show the potential of the proposed approach.

Initial strain effects in multilayer composite laminates

This work presents a novel formulation for the linear buckling analysis of multilayered shells. The formulation employs high-order Equivalent-Single-Layer (ESL) shell theories based on the through-the-thickness expansion of the covariant components of the displacement field, whilst the corresponding buckling problem is derived using the Euler's method. The novelty of the formulation regards the solution of the governing equations, which is obtained via implicit-mesh discontinuous Galerkin (DG) schemes. The DG method is a high-order accurate numerical technique based on a discontinuous representation of the solution among the mesh elements and on the use of suitably defined boundary integrals to enforce the continuity of the solution at the inter-element interfaces as well as the boundary conditions. Owing to its discontinuous nature, the DG method may be naturally employed with non-conventional meshes and is combined in this work with the implicitly-defined mesh technique, whereby the mesh of the shell modelling domain is constructed by intersecting an easy-to-generate background grid and a level set function that implicitly represents the cutouts. Several numerical examples are considered. First, the buckling loads are computed for plates and cylindrical shells modelled by different ESL theories and characterized by various materials, geometry and boundary conditions. Then, the buckling load of a plate with a circular defined cutout is computed for different diameter-to-plate's width ratios. The obtained results are compared with those available in the literature or those obtained using finite-element analyses and demonstrate the accuracy and the robustness of the proposed approach. 

Buckling analysis of multilayered structures using high-order theories and the implicit-mesh discontinuous Galerkin method

Topic: Structures) In the field of smart material, magneto-electro-elastic media have gained importance due to the ability to convert energy among three distinct form: magnetic, electric and mechanical. Magneto-electro-elastic composites are build up by combining piezoelectric and piezomagnetic phases to obtain a smart composite that presents not only the electro-mechanical and magneto-mechanical coupling, characteristics of the constituent phases, but also a strong magneto-electric coupling (1). Several kinds of composite arrangements have been tested, including particulate, fibrous and layered, among which magneto-electric laminated composites have shown the highest magneto-electric voltage coefficient (2-4). Magneto-electro-elastic composites are exploited for the construction of magnetic field probes, electric packaging, hydrophones, medical ultrasonic imaging, wireless powering of MEMS, sensors and actuators (5-7). Because of their inherent multidisciplinary nature, a deep understanding of their intrinsic features and of the way they behave is mandatory. Many research activities on magneto-electro-elastic composites have been conducted, not only from the chemical or technological viewpoint (8-10) but also to develop analytical and numerical model (11- 16). In this paper an analytical model for magneto-electro-elastic bimorph beam is employed to study its response to mechanical and electro-magnetic time varying loads. The model is based on the Timoshenko's beam theory (17) to take into account the effect of shear strain in the dynamic behaviour of magneto-electro- elastic beam. By assuming that no density charge and no density current act on the analyzed domain and that the magneto-electric behaviour can be considered quasi-static with respect to the mechanical one, since electro-magnetic waves propagation velocity is several order of magnitude higher than the propagation velocity of elastic waves, the electric and magnetic fields are modelled by mean of scalar electric, ϕ(x, y, t), and magnetic, ψ(x, y, t), potential functions, respectively. On the other hand, the kinematical variables involved into the analysis are the beam mean-line transverse displacement v(x, t) and the cross sectional rotation ϑ(x, t). The equations of motion for the beam are derived under the hypothesis of mono-axial stress state and by neglecting the electric and magnetic fields components transverse to the thickness direction and are written in terms of the kinematical variables only. The free-vibration problem is firstly solved to compute the beam natural frequencies and mode shapes, then, to take into account the mechanical and electro- magnetic loads, the problem of forced vibration is solved by applying the procedure proposed by Mindlin and Goodman (18) to deal with time-dependent boundary conditions. In Fig.(1) the electric and mechanical responses of a cantilever bi-layer magneto-electro-elastic beam used as magnetic field probe are presented in terms of difference of electric potential Δϕ, along the beam thickness, and in terms of the free-end displacement, respectively. The lamination sequence is CoFe 2O4/BaTiO 3, that are piezomagnetic and piezoelectric, respectively. The beam length is 5 cm and its thickness is 0.5 mm. The driving magnetic induction field is

Forced Vibration Analysis of Magneto-Electro-Elastic beam

An alternative single domain boundary ele- ment formulation and its numerical implementation are presented for the analysis of two-dimensional cracked bodies. The problem is formulated employing the clas- sical displacement boundary integral representation and a novel integral equation based on the stress or Airy's function. This integral equation written on the crack pro- vides the relations needed to determine the problem so- lution in the framework of linear elastic fracture mechan- ics. Results are presented for typical problems in terms of stress intensity factors and they show the accuracy and efficiency of the approach. keyword: Fracture mechanics, Stress function, Stress intensity factor, Boundary element method. of BEM are given by the pointwise representation of the solution and the computational gain associated with the reduction in dimensionality due to boundary discretiza- tion (Banerjee and Butterfield (1981), Hong and Chen (1988), Aliabadi (2002)). These features give meaning-

https://msvlab.hre.ntou.edu.tw/cite/sdhl2006-citing.pdf

An Alternative BEM for Fracture Mechanics

In this paper a structural Finite Element analysis of a 50 ft pleasure vessel is presented. The study is performed under different loads conditions: modal analyses have been done in order to find the natural frequencies of the vessel, structural analyses to verify the strength of the vessel to design loads. The design loads for the vessel considered are computed according to RINA rules for the construction and classification of pleasure vessels [1]. Two different composites are used for the lamination: one is a monolithic sequence of short fibre and balanced glass lamina, used for the bottom of the vessel and for structural reinforcements, the other is a sandwich made of glass fibre composite skins and a PVC core, used for the main deck and sides of the vessel. All the analyses are performed by using Patran/Nastran™ finite element commercial software in order to identify critical areas where possible reinforcement or redesign needs to be considered.

Alberto Milazzo

Papers

Initial strain effects in multilayer composite laminates

Buckling analysis of multilayered structures using high-order theories and the implicit-mesh discontinuous Galerkin method

Forced Vibration Analysis of Magneto-Electro-Elastic beam

An Alternative BEM for Fracture Mechanics

Modal and Structural FEM Analysis of a 50 ft Pleasure Yacht