Scale effects in orthotropic composite assemblies as micropolar continua: A comparison between weak-and strong-form finite element solutions

The aim of the present work was to investigate the mechanical behavior of orthotropic composites, such as masonry assemblies, subjected to localized loads described as micropolar materials. Micropolar models are known to be effective in modeling the actual behavior of microstructured solids in the presence of localized loads or geometrical discontinuities. This is due to the introduction of an additional degree of freedom (the micro-rotation) in the kinematic model, if compared to the classical continuum and the related strain and stress measures. In particular, it was shown in the literature that brick/block masonry can be satisfactorily modeled as a micropolar continuum, and here it is assumed as a reference orthotropic composite material. The in-plane elastic response of panels made of orthotropic arrangements of bricks of different sizes is analyzed herein. Numerical simulations are provided by comparing weak and strong finite element formulations. The scale effect is investigated, as well as the significant role played by the relative rotation, which is a peculiar strain measure of micropolar continua related to the non-symmetry of strain and work-conjugated stress. In particular, the anisotropic effects accounting for the micropolar moduli, related to the variation of microstructure internal sizes, are highlighted

Three-Dimensional Boundary Element Analysis of DElaminated Composite Structures with Attached Piezoelectric Patch

Sommario Gli elevati valori di resistenza e rigidezza specifica unitamente alla possibilità di gestire i percorsi di carico rendono tali materiali particolarmente indicati nella progettazione delle strutture aeronautiche. Il crescente interesse nell'uso dei materiali compositi trova anche la sua ragion d'essere nell'opportunità di realizzare pannelli di grandi dimensioni con un minor numero di giunzioni rivettate, il che implica una semplificazione del componente da realizzare e da luogo ad una riduzione sia dei tempi che dei costi di manifattura ed ispezione. Indipendentemente dai vantaggi visti, grande attenzione va posta nell'uso di tali materiali in strutture molto caricate, specialmente in presenza di discontinuità geometriche, quali fori o cutout, poichè in prossimità di tali discontinuità il materiale composito tende a delaminare. Il comportamento dei materiali compositi risulta quindi molto complesso specialmente in presenza di danneggiamenti, per tale motivo risulta necessario stabilire delle strategie di modellazione numerica per caratterizzare la risposta strutturale in prossimità delle suddette discontinuità. Va infatti evidenziato che il comportamento a frattura per le delaminazioni, diversamente da quanto avviene per cricche in mezzi omogenei, è caratterizzato usualmente da una compresenza dei tre modi elementari di frattura e che, inoltre, i campi di tensione e deformazione al fronte di delaminazione sono caratterizzati da un andamento oscillatorio, oltre che singolare, che si esplica, matematicamente, in una rappresentazione complessa sia dei campi di tensione e deformazione che dei fattori d'intensificazione delle tensioni (SIFs). La rappresentazione in termini di funzioni complesse dei campi al fronte di cricca è in realtà un artefatto matematico introdotto dall'ipotesi di meccanica della frattura elastico lineare. Ciò causa anche delle difficoltà nel calcolare in modo univoco gli angoli di fase tra i modi di frattura, poichè tutte le possibili definizioni degli angoli di fase, basati cioè sui rapporti tra componenti di tensione, di campo di spostamenti o dei SIFs, producono risultati non-convergenti e che dipendono dall'entità della discontinuità elastica all'interfaccia. In questa tesi si è sviluppato un approccio agli elementi al contorno per l'analisi 3D di cricche all'interfaccia tra mezzi anisotropi ricorrendo alla implementazione multi-dominio delle equazioni BEM per modellare sia solidi non omogenei come i laminati sia le superfici di frattura all'interfaccia tra le lamine. Per caratterizzare il comportamento a frattura delle delaminazioni si è implementato un metodo basato sull'integrale di chiusura della cricca, così da sfruttare direttamente la rappresentazione al contorno dei campi tensione e spostamento in prossimità del fronte di cricca. La caratterizzazione a frattura di una cricca all'interfaccia tra materiali diversi è ottenuta intermini di rateo di rilascio energetico totale G ed in termini degli angoli di fase e che caratterizzano il mix tra i modi di frattura e taglio e di apertura. In particolare, ipotizzando che l'influenza della discontinuità elastica all'interfaccia possa essere ritenuta trascurabile quando introdotta solo da una diversa orientazione delle fibre delle lamine contenenti la delaminazione, gli angoli di fase sono definiti come la tangente inversa della radice dei rapporti tra le componenti di rateo di rilascio energetico GII e GIII rispetto a quella associata al modo di apertura GI. L'analisi di una double cantilever beam (DCB) per varie lunghezze di delaminazione e diversi materiali ortotropi è presentata per validare l'approccio BEM e le ipotesi proposte e mostrare la versatilità degli strumenti numerici proposti studiando l'influenza della sequenza di laminazione sulla distribuzione tridimensionale dei parametri di frattura. Infine, dopo aver derivato un modello per sensori piezoelettrici prismatici, si studia una configurazione drop-ply, rappresentativa di una analisi Global/Local di delaminazione tra skin e stiffener, con un array di sensori incollati alla flangia dell'irrigidimento. L'obiettivo è quello di investigare la sensibilità del sensore piezoelettrico alla delaminazione e quindi la sua efficacia in un sistema aeronautico di monitoraggio della salute strutturale. L'obiettivo di questa tesi è quindi dare una approfondita conoscenza sull'efficacia di un nuovo strumento di analisi numerica di strutture aeronautiche in composito in presenza di delaminazioni. La ricerca bibliografica ha inoltre evidenziato come lo studio numerico dei problemi di frattura all'interfaccia tra materiali anisotropi sia condotto esclusivamente per mezzo del metodo degli elementi finiti, mentre il metodo degli elementi al contorno è usato solo per problemi piani e in ambito 3D solo per cricche all'interfaccia tra materiali isotropi o trasversalmente isotropi. Quindi, il presente lavoro è anche sviluppato per 'colmare la lacuna' nell'ambito della ricerca BEM sulla caratterizzazione 3D di cricche all'interfaccia tra materiali anisotropi. Inoltre, l'uso dell'integrale di chiusura di Irwin unitamente ad una soluzione agli elementi al contorno permette di modellare i problemi di frattura in mezzi omogenei usando semplicemente elementi al contorno regolari, poichè il prodotto tra tensioni e spostamenti risulta finito anche al fronte di cricca. Quanto detto, unitamente all'ipotesi che il carattere oscillatorio dei campi di tensione e spostamento al fronte di cricca possa ritenersi trascurabile quando la discontinuità elastica è solo dovuta al diverso orientamento delle fibre suggerisce la possibilità di usare elementi regolari anche per modellare le delaminazioni. Ulteriore obiettivo del lavoro è quindi quello di verificare l'attendibilità dei risultati forniti usando il BEM 3D ad elementi regolari congiuntamente all'integrale di Irwin per il calcolo dei parametri di frattura per delaminazioni in strutture composite Abstract The high stiffness-to-weight and strength-to-weight ratios of composite materials as well as the path-loads management capability make these materials very suitable in the framework of aerospace structures. The increasing interest in using composite materials also finds its rationale on the advantages derived in manufacturing large size panels with less riveted joints, which leads to a reduction of the overall structural complexity and of the manufacturing and inspection times and costs. However, despite the aforementioned advantages, extreme caution must be taken in employing composite materials in highly loaded structures, especially if cutouts, holes or other geometric discontinuities are present, because these sites are prone to delamination onset and show high notch sensitivity. The behavior of such materials appears very complex, particularly in presence of damage, and therefore accurate and efficient numerical modeling strategies are needed to catch their structural response close to the discontinuities. It is to be stressed, in fact, that unlike fracture problems in homogeneous material, delamination cracks in laminate composites usually involve all three modes of fracture, and the crack tip stress and strain fields also exhibits oscillatory singular behavior which leads to complex stress and displacement fields representation as well as to complex valued stress intensity factors (SIFs). The complex nature of the stress and displacement fields close to the crack front is a mathematical artifact introduced by the linear elastic fracture mechanics theory. Moreover, it gives rise to the problem of the fracture mode phase angles Psi and Phi estimation, since the stresses, displacements and SIFs components based definition of the mode-mix generally lead to non-converging results depending on the entity of the bi-material interface discontinuity. This thesis deals with the modeling of three-dimensional fracture between two anisotropic layers using the boundary element method. The multi-domain technique is implemented to model the layered configuration and cracks occurring at the bi-material interface. For the purpose of characterizing the fracture behavior of the delamination, the crack closure integral is implemented, taking full advantage of the boundary element representation of the stress and displacement fields close to the crack front. The fracture mechanics behavior of bi-material cracks is characterized in terms of the total Strain Energy Release Rate G and the phase angles and , which provide information about the mix between the opening and the sliding and tearing modes of fracture, respectively. The mode mix phase angles are computed as the inverse tangent of the square root of the energy release rates opening GI and shearing, GII and GIII, components ratio, because the effect of the material discontinuity at the interface is considered negligible when it is only due to fibers orientation mismatch. A double cantilever beam (DCB) specimen is then modeled and studied for various orthotropic material and delamination length in order to validate the proposed approach and to investigate the influence of interface lay-up on the three-dimensional distribution of the fracture mechanic parameters. Last, a piezoelectric patch sensor bonded on a drop-ply configuration, which is representative of a skin-stiffener debonding in a Global/Local fashion, is analyzed. The study has the aim of assessing the piezoelectric device sensitiveness to delamination and its effectiveness for a typical aeronautical structural health monitoring system implementation. The main objective of this thesis is to develop a new effective numerical tool for obtaining a deeper insight in the fracture behavior of delaminated aircraft composite structures. Moreover numerical studies in the framework of interface crack between dissimilar anisotropic materials have been mostly addressed by using the finite element method. Contributions to this research using the boundary element method remains rare, particularly in the area of three-dimensional problem simulation. The BEM has been used to model 2D interface crack problems between distinct anisotropic material, whereas 3D BEM has been used to analyze interface crack by distinct isotropic materials or transversely isotropic materials. Thus, the present work is also developed to 'fill the gap' in the BEM research area on the subject of three-dimensional characterization of interface crack between distinct anisotropic materials. It is proved that the use of the Irwin's Crack Closure Integral in conjunction with the BEM solutions allows to model the crack front in a homogeneous medium simply by means of regular elements, since the product of stress and displacements represents energy which is finite everywhere in the body including the region next to the crack front. Moreover, the oscillatory behavior characterizing stress and displacements fields near the front of an interface crack can be assumed to be negligible when interface material mismatch is introduced by ply orientation only. Hence, another objective of the work is to validate as effective the proposed regular 3D Crack Closure BEM to compute fracture parameters for a delamination occurring in laminated composite structures

boundary element modeling and analysis of adhesive bonded structural joints

In this paper, a boundary element technique for modeling and analysis of adhesive bonded structural joints is presented. The formulation is developed in the framework of the anisotropic elasticity and attention is focused on the application to composite structural joints built with the splicing concept technique. To model and analyze composite bonded joints a multidomain implementation of the boundary element method has been used. It has been proven well suited and very effective for the characterization of the mechanical behavior of spliced joints, allowing the analysis of the high gradient stress and strain fields near the splice lines as well as the prediction of the overall distribution of the interlaminar tractions. Numerical results show good agreement with analytical solution and finite element analyses

Symmetric Galerkin boundary element method.

This review concerns a methodology for solving numerically, to engineering purposes, boundary and initial-boundary value roblems 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, computer implementations. Areas and aspects which at present require further research are dentified and comparative assessments are attempted with respect to traditional boundary integral-element methods

Efficient spectral element methods for partial differential equations

In this thesis we applied a spectral element approximation to some elliptic partial differential equations. We demonstrated the difficulties related to the approximation of a discontinuous function in which the discontinuity is not fitted to the computational mesh. Such a situation gives rise to the Gibbs phenomenon. A h− p spectral element equivalent of the eXtended Finite Element Method (XFEM), which we termed the eXtended Spectral Element Method (XSEM) was developed. This was applied to some model problems. XSEM removes some of the oscillations caused by Gibbs phenomenon. We then explained that when approximating a discontinuous function, XSEM is able to capture the discontinuity precisely. We derive spectral element error estimates. The convergence of the approximations is studied. We have introduced several enrichment functions with the purpose of improving the approximation of discontinuous functions. In particular we have considered the twodimensional Poisson equation. Unfortunately, this implementation of XSEM was not able to recover spectral convergence. An alternative idea in which the discontinuity is constrained within a spectral element produces accurate SEM approximation. The Stokes problem was considered and solved using SEM coupled with an iterative PCG method. The zero volume condition on the pressure is satisfied identicaly using an alternative formulation of the continuity equation. Furthermore, we investigated the dependence of the accurency of the spectral element approximation on the weighting factor as well as the convergence properties of the preconditioner. An efficient and robust preconditioner is constructed for the Stokes problem. Exponential convergence was attained

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized. © 2022, The Author(s)