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Sarego, Giulia (2016) Structural material damage: novel methods of analysis. [Tesi di dottorato]

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Abstract (inglese)

In the classical continuum theory of solid mechanics, the mathematical framework
involves partial derivatives to represent the state of deformation of a solid
body. A significant drawback due to derivatives is related to the unphysical results
given near the discontinuities, because they are undefined wherever a continuous
field of displacements is not verified, such as in the presence of dislocations, voids,
cracks, interfaces between different phases within the same body and grain boundaries.
Various techniques were employed for overcoming this incapability of the classical
theory in describing material behavior in such conditions; in fact, spontaneous
formation and growth of discontinuities are of great importance in solid mechanics:
they lead to fractures and failures of systems that must be avoided, especially
in aerospace structures, primarily, for safety reasons and, secondly, for economic
purposes.
One of these new approaches concerns employing nonlocal theories, based on integral
formulations (more precisely integro-differential formulations), defined even
when non-derivable displacement fields are involved. Peridynamics is one of these
theories: it was suggested by Stewart Silling in 2000 [1] in order to adopt a consistent
formulation describing material behavior not only when a continuous displacement
field is provided, but also whenever discontinuities are present, avoiding
partial differential equations or pre-setting of conditions which can influence the
results. There are two versions of peridynamic models: bond-based, which was
introduced first (see [1, 2]) and state-based. In the bond-based version, forces
between two material points depend solely on their relative displacement, their
relative initial position, and material properties. Due to its simplicity compared to the state-based version, most of the peridynamic applications have employed
bond-based Peridynamics. However, bond-based models result in several limitations
(the same of other atomistic or molecular dynamics models [3], although this
is a continuum theory, not a discrete one), the most important of these is the fixed
value of Poisson’s ratio: 1/4 in 3D or 2D plane strain, and 1/3 in 2D plane stress
(see e.g. [1, 4]). This peculiarity implies other restrictions, such as the impossibility
of reproducing plastic incompressibility in an accurate way. Nevertheless, for
many purposes, bond-based Peridynamics fits the requirements and gives satisfying
results. State-based peridynamic models remove these restrictions by allowing
the interaction (“bond”) between a pair of points to potentially depend on all other
bonds connected to the two points.
Moreover, there are two types of state-based peridynamic formulations: ordi-
nary and non-ordinary [2, 5, 6]. In the former, the forces between two material
points act along the vector connecting the points in the deformed configuration. In
the latter, such characteristic is not present. The ordinary state-based formulation
requires specific derivation of constitutive models, see examples of viscoelasticity
and plasticity models in [7, 8]. For non-ordinary state-based formulation, two
approaches have been proposed: the development of an explicit model for the peridynamic
force state [2] and the development of a map thanks to which classical
mechanics constitutive relations are incorporated to indirectly establish the relationship
between the interaction force and the deformation. The latter approach
is called correspondence model [2].
The purpose of this thesis has been the investigation of possible advantages and
drawbacks of this new and unexplored theory, so to identify some guidelines for
choosing parameters fundamental for the analyses and the development of models
for particular structural analyses.
In the first year of the PhD course, the state of the art of this theory was studied
and the bond-based linear and nonlinear static solvers developed in Matlabr were
analyzed, employed and improved.
During the second year of PhD course, the author of this thesis has focused
her attention on the second version of the theory, based on concepts of advanced mathematics. She has become familiar with it, thanks to the functional analysis
course that she had attended in the first year.
One of the main original contributions of the present work to the existing
literature is the development of the 2D linearization of the state-based “linear
peridynamic solid” model in the state-based formulation. These models are useful
whenever simplifying assumptions of plane stress and plane strain can be adopted
for the simulation of a system, which, otherwise, would be described by a 3D model
requiring high computational resources (time and memory). Particular attention is
paid to this aspect, because, being a nonlocal model, implementing a peridynamic
code is, in general, more computationally expensive than a code based on a local
approach. The study of the state-based version started before going abroad and
the development of the 2D models was completed during the six month stay at the
University of Nebraska-Lincoln in USA. Both static and dynamic codes have been
developed and the relevant parameters of these models have been analyzed. These
linearized models are described in chapter 1.2.2.
The study of failure criteria in state-based Peridynamics and the improvement
of the algorithms in Matlabr to accelerate the codes and to optimize memory
resources have been the main issues of the third year research. Some failure criteria,
presented in section 1.2.3, have been proposed for brittle homogeneous linear elastic
materials. They are criteria based on the maximum admissible stretch: a given
bond fails at a critical stretch obtained by the work required to break that bond
and this work is related to the fracture energy of the material. The results are
compared to experimental data both for static and for dynamic cases, in bondbased
and in state-based formulations. The detailed description of the algorithms
can be found in chapter 3, while the results are illustrated in chapters 4 and 5.

Abstract (italiano)

Nella teoria classica della meccanica dei solidi, la formulazione matematica
include derivate parziali, grazie alle quali si possono rappresentare stati di deformazione
come funzioni degli spostamenti relativi dei nodi in cui è discretizzato il
sistema continuo. Una carenza rilevante dovuto all’utilizzo delle derivate è legato ai
risultati privi di significato fisico ottenuti in prossimità delle discontinuità perché le
derivate non sono definite laddove manca un campo di spostamenti continuo, come
può capitare in presenza di dislocazioni, vuoti, cricche, interfacce tra fasi differenti
nello stesso corpo e bordi dei grani.
Dato che la formazione spontanea e la crescita di discontinuità sono di grande
importanza in meccanica dei solidi, diverse tecniche sono state utilizzate per superare
questa incapacità della teoria di descrivere il comportamento dei materiali
in tali condizioni, perché situazioni in cui le strutture sono incapaci di continuare
a svolgere la propria funzione devono essere evitate, specialmente per strutture
aerospaziali, in primo luogo, per ragioni di sicurezza ed, in secondo luogo, per
motivi economici.
Uno di questi nuovi approcci riguarda l’utilizzo di teorie non locali basate su
formulazioni integrali (più precisamente formulazioni integro-differenziali), definite
anche quando campi di spostamento non derivabili sono presenti. La teoria “Peridynamics”
è una di queste teorie: è stata proposta da Stewart Silling nel 2000
[1] così da adottare una formulazione unica e coerente capace di descrivere i comportamenti
dei materiali in corpi sia continui che discontinui, evitando l’uso di
equazioni alle derivate parziali o la definizione a priori di alcune condizioni che
possono influenzare (e in un certo senso favorire) dei risultati. Ci sono due versioni
di modelli peridinamici: la state-based, e un suo caso particolare, la bond-based, che è stata introdotta per prima (vedi [1, 2]). Nella versione bond-based, le forze tra
due punti materiali dependono unicamente dal loro spostamento relativo e dalla
loro posizione relativa iniziale, oltre che dalle proprietà del materiale. Vista la sua
semplicità a confronto con la seconda versione, la maggior parte delle applicazioni e
degli articoli sulla Peridynamica ha adottato la formulazione bond-based. Tuttavia,
i modelli nella formulazione bond-based sono caratterizzati da alcune limitazioni
(le stesse dei modelli di altre teorie atomistiche e dei modelli di dinamica molecolare
[3], anche se la Peridinamica è una teoria del continuo, non discreta), la più
notevole di queste è il modulo di Poisson fisso: 1/4 nelle simulazioni 3D oppure in
caso di deformazione piana 2D, e 1/3 nelle simulazioni in stato di tensione piana
2D (si veda per esempio [1, 4]). Questa particolarità implica altre restrizioni, come
l’impossibilità di riprodurre la condizione di incomprimibilità plastica in maniera
accurata. Tuttavia, per la maggior parte degli scopi, la formulazione bond-based
è sufficiente e fornisce risultati approssimati soddisfacenti.
I modelli della versione state-based rimuovono queste restrizioni, permettendo
che le interazioni tra due punti possano dipendere da tutte le interazioni (i “bond”)
connessi ad almeno uno dei due punti, tramite delle mappe avanzate chiamate
“states”. Inoltre, ci sono due tipi di formulazioni state-based: la ordinary e la
non-ordinary [2, 5, 6]. Nella formulazione ordinary, le forze tra due punti materiali
agiscono lungo la congiungente i due punti nella configurazione deformata, mentre
nella formulazione non-ordinary, questa caratteristica non è più vera. La formulazione
ordinary della state-based necessita di modelli costitutivi appositamente
derivati, come per esempio i modelli di viscoelasticità e platicità in [7, 8]. Per la
formulazione non-ordinary della state-based, due approcci sono stati proposti: lo
sviluppo di un modello esplicito per l’espressione dello state della forza peridinamica
[2] e lo sviluppo di una mappa grazie alla quale le relazioni costitutive della
meccanica classica sono incorporate per stabilire indirettamente la relazione tra la
forza d’interazione e la deformazione. I modelli derivanti dal secondo approccio
sono chiamati modelli correspondence [2].
L’argomento di questa tesi è lo sviluppo di modelli per particolari tipi di analisi e
la ricerca di possibili vantaggi e inconvenienti di questa teoria nuova ed inesplorata, così da identificare alcune linee guida per la scelta di parametri fondamentali per
le analisi.
Durante il primo anno del corso di dottorato, lo stato dell’arte relativo a questa
teoria è stato studiato e i solutori statici lineari e non lineari nella formulazione
bond-based sviluppati precedentemente in ambiente Matlabr sono stati analizzati,
usati e migliorati.
Durante il secondo anno, l’autrice di questa tesi si è concentrata sulla seconda
versione, basata su concetti di matematica avanzata con cui ha preso dimestichezza
grazie al corso di analisi funzionale seguito il primo anno. Uno dei principali contributi
originali alla letteratura esistente presenti in questa tesi è lo sviluppo dei
modelli linearizzati 2D del modello solido lineare nella formulazione state-based.
Questi modelli sono particolarmente utili quando semplificazioni di stato piano di
tensione o di deformazione possono essere assunte per la simulazione di un sistema
tridimensionale, che altrimenti verrebbe descritto da un modello 3D che necessiterebbe
di risorse computazionali più elevate (in termini di tempo e memoria).
Una particolare attenzione è richiesta per quest’aspetto, perché, essendo un approccio
non locale, implementare un codice basato sulla teoria peridinamica richiede in
generale più risorse computazionali di un codice basato su un approccio locale. Lo
studio della versione state-based è iniziato prima di andare all’estero e lo sviluppo
dei modelli 2D si è poi completato durante il soggiorno di sei mesi alla University
of Nebraska-Lincoln negli Stati Uniti. Sono stati sviluppati sia un codice dinamico
che uno statico. I parametri principali di questi modelli sono stati analizzati e i
modelli linearizzati si possono trovare descritti nel capitolo 1.2.2.
Lo studio dei criteri di frattura adottabili nella formulazione state-based e il
miglioramento degli algoritmi in Matlabr per accelerare i codici e ottimizzare le
risorse di memoria e gestione dei dati sono stati gli argomenti principali del terzo
anno. Alcuni criteri di frattura, presentati nel capitolo 1.2.3, sono stati proposti per
materiali lineari elastici omogenei e caratterizzati da frattura fragile. Sono criteri
basati sul massimo allungamento: un’interazione non locale (“bond”) viene meno
quando un valore critico di allungamento è raggiunto; questo valore di allungamento
critico è calcolato dal lavoro richiesto per rompere il bond e questo lavoro è a sua volta legato all’energia di frattura. I risultati ottenuti sono stati confrontati con
dati sperimentali per casi sia statici che dinamici, sia nella formulazione bondbased
che in quella state-based. La descrizione dettagliata degli algoritmi si trova
nel capitolo 3, mentre i risultati sono riportati nei capitoli 4 e 5.

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Tipo di EPrint:Tesi di dottorato
Relatore:Galvanetto, Ugo
Correlatore:Zaccariotto, Mirco
Dottorato (corsi e scuole):Ciclo 28 > Scuole 28 > SCIENZE TECNOLOGIE E MISURE SPAZIALI > SCIENZE E TECNOLOGIE PER APPLICAZIONI SATELLITARI E AERONAUTICHE
Data di deposito della tesi:01 Febbraio 2016
Anno di Pubblicazione:01 Febbraio 2016
Parole chiave (italiano / inglese):Peridynamics, nonlocal, nonlinear, static, dynamic, fracture mechanics
Settori scientifico-disciplinari MIUR:Area 09 - Ingegneria industriale e dell'informazione > ING-IND/04 Costruzioni e strutture aerospaziali
Struttura di riferimento:Centri > Centro Interdipartimentale di ricerca di Studi e attività  spaziali "G. Colombo" (CISAS)
Codice ID:9558
Depositato il:24 Ott 2016 16:02
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