Issue 18

Anno V Numero 18 Ottobre 2011

Rivista Internazionale Ufficiale del Gruppo Italiano Frattura Fondata nel 2007

Editor-in-chief:

Francesco Iacoviello

ISSN 1971-8993

Associate Editor:

Luca Susmel

Editorial Advisory Board:

Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Robert O. Ritchie Cetin Morris Sonsino Ramesh Talreja David Taylor

Frattura ed integrità strutturale The International Journal of the Italian Group of Fracture

www.gruppofrattura.it

Frattura ed Integrità Strutturale, 18 (2011); Rivista Ufficiale del Gruppo Italiano Frattura

G. Del Piero, G. Lancioni, R. March The variational theory of fracture: diffuse cohesive energy and elastic-plastic rupture ............................... 5 F. Felli, D. Pilone, A. Scicutelli Fatigue behaviour of titanium dental endosseous implants ……………………………………...… 14 S. Marfia, E. Sacco, J. Toti A coupled interface-body nonlocal damage model for the analysis of FRP strengthening detachment from cohesive material .. ................................................................................................................... 23 G. Ferro, J.-M. Tulliani, S. Musso Carbon nanotubes cement composites …………………………………………………………... 34 V. Di Cocco, C. Maletta, S. Natali Structural transitions in a NiTi alloy: a multistage loading-unload cycle .. .......................................... 45 D. Firrao, P. Matteis Tenacità a frattura e resistenza a fatica di acciai bonificati con microstrutture derivanti da tempra incompleta e riflessi sulle norme tecniche di impiego Fracture toughness and fatigue resistance of quenched and tempered steels with microstructures deriving from a slant quench. Consequences on technical standards ………………………………………... 54 F. Iacoviello Notiziario IGF ……………………………………………….... .......................................... 69

Segreteria rivista presso: Francesco Iacoviello Università di Cassino – Di.M.S.A.T. Via G. Di Biasio 43, 03043 Cassino (FR) Italia http://www.gruppofrattura.it iacoviello@unicas.it

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Frattura ed Integrità Strutturale, 18 (2011); ISSN 1971-9883

Editor-in-Chief Francesco Iacoviello

(Università di Cassino, Italy)

Associate Editor Luca Susmel

(University of Sheffield, UK)

Advisory Editorial Board Alberto Carpinteri

(Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy)

Andrea Carpinteri

Donato Firrao M. Neil James Gary Marquis

(University of Plymouth, United Kingdom) (Helsinki University of Technology, Finland)

Robert O. Ritchie Cetin Morris Sonsino

(University of California, USA) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)

Ramesh Talreja David Taylor

Journal Review Board Stefano Beretta

(Politecnico di Milano, Italy) (Università di Cassino, Italy) (Università di Trieste, Italy) (EADS, Munich, Germany) (IMWS, Wien, Austria) (Politecnico di Torino, Italy) (Politecnico di Milano, Italy) (University of Porto, Portugal) (Politecnico di Torino, Italy) (Università di Parma, Italy)

Nicola Bonora Lajos Borbás Francesca Cosmi

(Budapest University Technology and Economics, Hungary)

Claudio Dalle Donne Josef Eberhardsteiner Giuseppe Ferro Tommaso Ghidini Mario Guagliano

(European Space Agency - ESA-ESRIN)

Lucas Filipe Martins da Silva

Marco Paggi

Alessandro Pirondi Ivatury S. Raju Roberto Roberti Marco Savoia

(NASA Langley Research Center, USA)

(Università di Brescia, Italy) (Università di Bologna, Italy)

Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007

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Frattura ed Integrità Strutturale, 18 (2011); Rivista Ufficiale del Gruppo Italiano Frattura

Descrizione e scopi Frattura ed Integrità Strutturale è la rivista ufficiale del Gruppo Italiano Frattura . E’ una rivista open-access pubblicata on-line con periodicità trimestrale (luglio, ottobre, gennaio, aprile). Frattura ed Integrità Strutturale riguarda l’ampio settore dell’integrità strutturale, basato sulla meccanica della fatica e della frattura, per la valutazione dell’affidabilità e dell’efficacia di componenti strutturali. Scopo della rivista è la promozione di lavori e ricerche sui fenomeni di frattura, nonché lo sviluppo di nuovi materiali e di nuovi standard per la valutazione dell’integrità strutturale. La rivista ha un carattere interdisciplinare e accetta contributi da ingegneri, metallurgisti, scienziati dei materiali, fisici, chimici e matematici. Contributi Frattura ed Integrità Strutturale si prefigge la rapida disseminazione di contributi originali di natura analitica, numerica e/o sperimentale riguardanti la meccanica della frattura e l’integrità strutturale. Si accettano lavori di ricerca che contribuiscano a migliorare la conoscenza del comportamento a frattura di materiali convenzionali ed innovativi. Note tecniche, lettere brevi e recensioni possono essere anche accettati in base alla loro qualità. L’ Editorial Advisory Board sollecita anche la pubblicazione di numeri speciali contenenti articoli estesi presentati in occasione di conferenze e simposia tematici. Istruzioni per l’invio dei manoscritti I manoscritti devono essere scritti in formato word senza necessità di utilizzare un particolare stile e devono essere inviati all'indirizzo iacoviello@unicas.it. Il lavoro proposto può essere in lingua Italiana (con riassunto in inglese di almeno 1000 parole e didascalie bilingue) o Inglese. La conferma della ricezione avverrà entro 48 ore. Il processo di referaggio e pubblicazione on-line si concluderà entro tre mesi dal primo invio. Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (July, October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to iacoviello@unicas.it. The paper may be written in English or Italian (with an English 1000 words abstract). A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission.

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Frattura ed Integrità Strutturale, 18 (2011); ISSN 1971-9883

C

aro Lettore, poche righe per descriverti l’ultimo evento IGF, il XXI Convegno Nazionale che si è svolto a Cassino dal 13 al 15 giugno. Anzitutto l’assemblea Straordinaria che si è svolta il 13 giugno. Fra presenti e deleghe, abbiamo abbondantemente superato il 75% dei Soci partecipanti ed abbiamo potuto procedere a quella discussione che il Consiglio di Presidenza aveva preparato in due anni di lavoro. Il nuovo Statuto è stato approvato all’unanimità!!! Abbiamo quindi un nuovo Statuto, già disponibile nel sito web. Ora abbiamo a disposizione uno strumento di lavoro aggiornato e speriamo ovviamente di farne buon uso. Il Convegno IGF XXI di Cassino 2011 è stato, in poche parole, un successo: oltre quaranta memorie con un numero doppio di partecipanti, con presentazioni sempre interessanti e discussioni stimolanti. Pochissimi i relatori assenti (un paio, un numero assolutamente fisiologico e dovuto a quei molteplici impegni improvvisi che purtroppo ci inseguono nella nostra vita professionale) e quindi un programma sostanzialmente rispettato. Alcuni momenti importanti hanno cadenzato il ritmo del Convegno Nazionale: - l’Assemblea dei Soci Ordinaria, che ha avuto luogo il 14 giugno, durante la quale sono state riassunte le attività svolte nel biennio 2009-2011 ed i risultati ottenuti; - l’approvazione all’unanimità della proposta del Consiglio di Presidenza di nominare Donato Firrao (Politecnico di Torino) socio onorario IGF, il primo di nazionalità italiana; - l’attribuzione a Mauro Madia (Politecnico di Milano) del premio Giovane Ricercatore IGF 2011 , premio attribuito alla migliore memoria presentata con tutti gli autori di età inferiore ai 35 anni; - la rielezione per acclamazione di chi vi sta scrivendo come Presidente IGF per il quadriennio 2011-2015; - l’elezione del nuovo Consiglio di Presidenza. Considerando quest’ultimo punto, è indispensabile sottolineare i meriti del Consiglio di Presidenza uscente che, nei due anni di attività è riuscito ad organizzare numerose iniziative, tutte coronate dal successo. Ritengo doveroso ringraziare di cuore tutti coloro che hanno fatto parte di un gruppo veramente affiatato e che hanno dedicato in questi due anni parte del proprio tempo e del proprio impegno al Gruppo Italiano Frattura. Senza distinguere fra coloro che hanno riconfermato la propria disponibilità a continuare la loro esperienza nel Consiglio di Presidenza IGF e coloro invece che, per i molteplici e pressanti impegni, sono stati costretti a ritirare la propria disponibilità, in ordine rigorosamente alfabetico, desidero ringraziare i colleghi e amici Stefano Beretta, Francesca Cosmi, Giuseppe Ferro, Angelo Finelli, Domenico Gentile, Marco Paggi, Alessandro Pirondi, Luca Susmel. Grazie a tutti per il vostro contributo sempre fattivo ed entusiasta!! Un ciclo è quindi terminato, un nuovo ciclo si riapre. Molte sono le iniziative in cantiere, a partire dalla sessione IGF nell’ormai prossimo Convegno AIPnD. Continuate a seguirci ed a sostenerci. Il successo dell’associazione e di questa rivista dipende esclusivamente da nostro impegno e dal vostro sostegno. Tanti cari saluti, Francesco Iacoviello Presidente IGF Direttore Frattura ed Integrità Strutturale

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

The variational theory of fracture: diffuse cohesive energy and elastic-plastic rupture

Gianpietro Del Piero Dipartimento di Ingegneria, Università di Ferrara, Ferrara (Italy) dlpgpt@unife.it Giovanni Lancioni Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, Ancona (Italy) Riccardo March Istituto per le Applicazioni del Calcolo, CNR, Roma (Italy)

A BSTRACT . This communication anticipates some results of a work in progress [1], addressed to explore the efficiency of the diffuse cohesive energy model for describing the phenomena of fracture and yielding . A first local model is partially successful, but fails to reproduce the strain softening regime. A more robust non-local model , obtained by adding an energy term depending on the deformation gradient, describes many typical features of the inelastic response observed in experiments, including strain localization and necking. Fracture occurs as the result of extreme strain localization. The model predicts different fracture modes, brittle and ductile, depending on the analytical form of the cohesive energy function. S OMMARIO . Nella presente comunicazione si anticipano alcuni risultati del lavoro [1], in preparazione, in cui si analizza l’efficienza del modello dell’ energia coesiva diffusa nel descrivere i fenomeni di plasticizzazione e rottura. Un primo modello locale dà risultati parzialmente soddisfacenti, ma si rivela incapace di descrivere il regime di strain softening . Un più robusto modello non locale, ottenuto aggiungendo un termine energetico dipendente dal gradiente della deformazione, riesce a descrivere molte peculiarità della risposta anelastica osservate sperimentalmente, incluse la localizzazione della deformazione e il necking . La frattura avviene per estrema localizzazione. Il modello prevede la possibilità di diversi modi di frattura, duttile o fragile, a seconda della forma analitica ipotizzata per l’energia coesiva.

K EYWORDS . Fracture mechanics; Energy methods; Elastic-plastic rupture; Variational theory of fracture.

I NTRODUCTION

leading idea in Fracture Mechanics is that the growth of a fracture surface is decided by the competition between the strain energy lost in, and the amount of fracture energy required to, the creation of a new fracture surface [2]. However, it was soon clear that this idea is appropriate to the description of brittle fracture, typically exhibited in large-size bodies, but is unfit to describe the ductile fracture modes which prevail in small-size bodies. This size effect is captured by the cohesive energy models [3, 4]. A

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

Other basic problems, such as the formation of a fracture in an initially unfractured body, were neglected for long time, and ony recently came to the attention of the scientific community. The use of variational techniques for energy minimization was introduced first for Griffith’s fracture model [5], and extended later to cohesive energy models [6, 7]. From the engineering side, it became clear that an appropriate choice of the analytic form of the cohesive energy may lead to a unified description of the phenomena of yielding, damage, and fracture [8, 9]. A very recent research trend consists in questioning the surfacic character of the fracture energy, and in exploring the alternative possibility of an energy diffused over the volume. An example is the paper [10], in which, in the context of a damage model, a rather detailed description of the process zone preceding the final rupture is obtained. In this communication a diffuse cohesive energy is considered, and fracture is regarded as the extreme stage of the localization of inelastic deformation. In a one-dimensional context, we assume that at every point of the bar the energy density is the sum of an elastic and a cohesive part. An incremental energy minimization is performed, under the only assumption that the cohesive energy is totally dissipative. With this simple model, a rather accurate description of the inelastic response of the bar, from the onset of the inelastic regime up to rupture, is obtained. The strain-softening case is not adequately described, but this inconvenience is repaired with the introduction of a non-local energy term of the gradient type. The traditional elements of elastic-plastic response, such as the yield condition, the flow law, the hardening law, the elastic unloading, come out of the model as necessary conditions for an energy minimum. We also find that a convex shape of the cohesive energy favorizes a uniform distribution of inelastic strain and a work-hardening response, while a concave shape is responsible of the localization of the inelastic strain, which, in turn, produces the phenomena of strain softening and necking . Rupture takes place when the localization becomes extreme. If the cohesive energy is approximated by a piecewise polynomial function, its expression becomes a function of a small number of material parameters. The first results of a series of numerical simulations show that an appropriate choice of these parameters provides a good approximation of the experimental response curves. This seems to confirm the efficiency and flexibility of the proposed model. onsider a bar of length l , with constant cross section, free of external loads, and subject to the axial displacements u (0) = 0 , u ( l ) =  l , (1) at the endpoints x  0 and x  l . The bar’s deformation is measured by the derivative u  of the axial displacement u . We assume that, at every point x of the bar’s axis, the deformation u  ( x ) can be split into the sum of an elastic part  ( x ) and of an inelastic part  ( x ): u  ( x ) =  ( x ) +  ( x ) , (2) and that the total energy of the bar has the form    l dx x x w E 0 ))) (( )) (( ( ) ,(      (3) where w and  are the volume densities of the elastic strain energy and of the cohesive energy , respectively. We also assume that w can be recovered, while  is totally dissipated. The last assumption requires that the cohesive power   (  ( x,t )) ),( tx   be non-negative for all x and at all instants t , and if we assume that  is an increasing function of  we get the dissipation inequality 0 ),(  tx   (4) to be satisfied for all x and at all t . In the spirit of the Calculus of Variations, the equilibrium configurations of the bar are identified with the stationary points of the energy C T HE LOCAL MODEL

l

) , , ,( 0   

E

)( )) ((' (  x x x w   

 dx x

0 )) ( )) (('













(5)

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

with the inequality due to the presence of the dissipation inequality (4), which requires the restriction   ( x )  0 (6) on the perturbations  . The analysis of the first variation leads to the following characterization of the equilibrium configurations [DLM] :  The elastic deformation  ( x ) and the axial force    = w  (  ( x )) (7) are constant all over the bar,   The axial force cannot exceed a limit depending on the current inelastic deformation:     (  ( x )) . (8)  The equilibrium condition (8) is in fact a yield condition . It is remarkable that it has not been postulated, but deduced from the variational procedure. The energy minimization is insufficient to determine the evolution of the deformation under a loading process  =  ( t ). To do this, it is necessary to formulate an incremental equilibrium problem , in which at any time t the deformations  ( t ) and  ( x,t ) are supposed to be known, and the unknowns are the deformation increments ),( ), ( tx t     , produced by a given load increment )( t   . One considers the expansion ) ( )) ( ), (( )) ( ), (( )) ( ), (( )) ( ), (( 2 2              o t t E t t E t t E t t E         (9) and minimizes for  sufficiently small to legitimate the truncation at the second-order term. The term of order zero being known, a first-order approximation involves the minimization of the function  dx x t tx t t l t t E l ))( )) ( ),((' ( )( )( )) ( ), (( 0                (10) In it, everything is known except )( x   . Then E  is an affine function of   , and a proper use of the dissipation inequality leads to the Kuhn-Tucker conditions 0 )(  x   , 0 )) (('     x , 0 )( ) )) ((' (   x x      (11) as necessary conditions for a minimum. The first two conditions are (4) and (8), respectively. The third condition, the complementarity condition, states that when   is positive the force  must lie on the border of the yield surface, and that when  lies at the interior of the yield surface then   must be zero. Again, it is remarkable that these distinctive aspects of plastic response have not been postulated, but deduced from incremental energy minimization. Conditions (11) are still insufficient to determine the evolution of the deformation. This is done by minimizing the second-order term of (9) 2 1 which is a quadratic function of   . The new minimization provides a second set of Kuhn-Tucker conditions 0 )(  x   , 0 )( )) ((''        x x , 0 )( ) )( )) (('' (   x x x         ,  x (13) to be satisfied at the portion  of the bar at which (8) is satisfied as an equality. Indeed, we already know that   = 0 out of  . In (13), the first condition is again the dissipation inequality (4). The second condition provides a relation between the increments of the force and of the inelastic deformation. By the complementarity condition (13) 3 , inequality (13) 2 is satisfied as an equality when   > 0. This equality provides the flow rule for the inelastic strain rate in the regime of inelastic deformation. Condition (13) 3 is the consistency condition , which says that the inelastic deformation cannot increase when a dx x t ))( )( )( )),(('' ( x tx   t t l )( )( )) ( ), (( 0        t t E  l          (12)

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

point x tends to re-enter the elastic range (8). This condition determines the property of elastic unloading . Again, these typical properties of plastic response are not assumed, but come as results of the second-order incremental minimization. Together with the necessary condition   (  ( x ))  0 (14)  imposed by the non-negativeness of the second variation, conditions (11) and (13) determine the solution (   ,   ) of the incremental problem. Here we summarize some properties of the continuations. For a more detailed analysis we refer to the main paper [1].  If at the time t = 0 we start with  (0)=0 from the natural configuration  ( x ) =  ( x ) = 0, from the constitutive assumption   (0) > 0 we have that, initially, inequality (8) is strict. Then   is zero, and a purely elastic evolution occurs. This elastic regime ends when, with growing  , the force  reaches the limit value  =   (0), corresponding to the boundary of the elastic range. This event marks the onset of the inelastic regime .  The response in the inelastic regime strongly depends on the sign of   (0). If   (0) is positive, the deformation is homogeneous, that is, both  and  are constant all over the bar. If 0    , the inelastic deformation is given by the solution of the differential equation

w

t

t

)) ( )( (''











t

)( )) ( )( ('' t 

)( 



(15)

t w t 

t

)) (('' 









joined with an appropriate initial condition. The slope of the force-elongation response curve is given by

t w t t w t  

t

)) ((''

)( )) ( )( ('' )) ( )( ('' t t  









 

t

)( 

(16)

)) (('' 









0    , elastic unloading takes

The slope is positive, that is, the inelastic regime starts with a work-hardening response. If place.   If  (0) is zero, the initial response is perfectly plastic. The force is constant in time,

0 )(  t   , the total inelastic

)( t l   , while the punctual distribution of

)( t   remains

deformation rate is equal to the total elongation rate

undetermined.  Finally, if  (0) is negative the necessary condition for equilibrium (14) is violated. The deformation tends to concentrate on a very small portion of the bar, and the slope )( /)( t t     tends to  . This is the model’s representation of the catastrophic failure of the bar. The work-hardening regime ends if, with growing inelastic deformation,   (  ( t )) becomes equal to zero. Of course, this may or may not be possible, according to the shape assumed for the function   

Figure 1 : Force-elongation response curves at the onset of the inelastic regime, for different signs of   (0)  The main result of the local model is that the initial part of the inelastic response is determined by the sign of the second derivative   (0). That is, by the initial convexity or concavity of the cohesive energy. The predictions of the local model

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

at the onset of the inelastic regime are summarized in Fig.1. The subsequent part of the response curve depends on the analytic expression of the cohesive energy  away from the origin. Anyway, the picture shows a major defect of the model, that is, the incapability of reproducing the regime of strain softening , in which the response curve exhibits a negative slope. Indeed, catastrophic failure occurs as soon as   (0) takes a negative value. 

T HE NON - LOCAL MODEL

T

he reproduction of strain softening becomes possible in the non-local model obtained by adding to the energy (3) a term proportional to the square of the gradient of the inelastic deformation    l dx x x w E 0 ))) (( )) (( ( ) ,(      +  l dx x 0 2 2 1 )('   (17) where  is a small positive constant. The addition of the new term brings additional terms to the yield condition (8)      (  ( x ))    ( x ) (18)  and to the Kuhn-Tucker conditions (11) 0 )(  x   , 0 )('' )) (('   x x     , 0 )( )) ('' )) ((' (   x x x       (19) 0 )(  x   ,  x (20) It also adds a positive term to the second variation. Then the necessary condition (14) is relaxed, and this is the reason why a description of strain-softening becomes possible. At the onset of the inelastic regime, due to the additional boundary conditions required by the supplementary gradient term, the inelastic deformation is not anymore constant all over the bar. The choice of the additional boundary conditions is a delicate point. Our choice of taking   ( l ) =  (0) = 0 (21)  0 )('' ,  )( )) ((''  x x x           0 )( )) ('' , )( )) (('' (  x x       x x      and (13)

is discussed in detail in the paper [1]. With these conditions, for the inelastic strain rate we get the expression

l

x

 



2 

  

  



x

xk ,

)( 

tanh

tanh

sinh





(22)

)0(''

2



with  =(   (0) /  ) 1/2 , if   (0) > 0, and 

 



x

) x lx

)( 

(





(23)

2



if   (0) = 0. In both cases, the added energy term has the effect of increasing the slope of the force-elongation response curve. In particular, for   (0) = 0 a hardening response takes the place of the perfectly plastic response predicted by the local model. For   (0) < 0, there are two types of solutions: the full-size solution

xk

lk

 

  

  



x

xk sin 2 tan 2 tan )0('' ,

)( 





(24)



with k = (    (0) /  ) 1/2 , if kl < 2  , and the localized solution  )) ( cos 1( )0('' )( axk x         ,

(25)

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

if kl > 2  .  The latter are concentrated on an interval ( a, a + l y response curve is positive for kl <  , and negative for kl >  . It is usual to consider the material constant l i =  / k

) of length l y

= 2  / k < l . The initial slope of the inelastic

(26) as an internal length of the material. From the preceding analysis it follows that the inelastic regime starts with  a full-size solution and a work-hardening response if l < l i ,  a full-size solution and a strain-softening response if l i < l < 2 l i ,  a localized solution and a strain-softening response if l > 2 l i . Catastrophic failure again occurs when the slope of the response curve becomes  . This never happens for l < l i . For l i < l < 2 l i , this happens when

2/ 2/ tan 1 )0(''    w kl kl

    

0 ) (''



c 



(27)

where  c is the elongation at the onset of the inelastic regime. For l > 2 l i

, this happens when

l

y w l

)0('' 

0 ) (''





c 

(28)

The occurrence of catastrophic rupture at the onset is the response predicted by Griffth’s theory of brittle fracture. For all remaining situations, the subsequent response depends on the behavior of  away from the origin. Depending on the form assumed for   as a function of  , it may happen that an initially full-size solution localizes for some  >  c , and in some cases it may also happen that a localized solution becomes full-size. Catastrophic fracture may take place in both localized and full-size solutions. 

N UMERICAL SIMULATIONS

W

ith an appropriate choice of the expression of  , the non-local model gives the possibility of describing many experimental situations in which rupture is preceded by more or less extended regimes of inelastic deformation, as well as many intermediate situations between the extreme cases of a totally brittle and a totally ductile response. However, the non-homogeneous character of the solutions in the non-local model makes hopeless any attempt of describing the evolution of the inelastic deformation by a simple differential equation, like Eq. (15) in the case of the local model. For this reason, we made a series of numerical simulations. Here we present the current status of our study, which is still in progress. The purpose of our simulations was to determine the shape of the function  giving the best reproducion of the response curves of two experimental tests, one on a steel bar and one on a concrete specimen. For  we chose a piecewise polynomial C 2 representation. This was obtained by subdividing the domain {   0 } into N intervals, in each of which  was taken to be a third-order polynomial, and by imposing the continuity of the function and of its first and second derivatives at the nodal points of the subdivision. The number and position of the nodal points determine the accuracy of the approximation. The advantage of a better approximation obtained with large N is paid with the disadvantage of working with a larger number of material constants. For the steel bar we made two series of simulations, one with N = 3 and one with N = 6. The results of the first series are shown in Fig. 2. In it, we see the response curves obtained numerically for a bar length l equal to 40, 80, 160 mm. They show that a short bar is more ductile than a long bar, since for short bars catastrophic failure takes place for larger  . This is a manifestation of the size effect . In the same figure, the curve for l = 80 mm is compared with the experimental curve, represented by the dotted line, for a bar of the same length. We see a very good agreement, except for the initial horizontal plateau exhibited by the experimental curve and not reproduced in the simulation. The second series of simulations was done with the purpose of eliminating this discrepancy, by increasing the number N of parameters. The result is shown in Fig. 3, where we see an almost perfect agreement between experiment and simulation. 

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01



Figure 2 : Response curves for a steel bar in numerical simulations with N = 3 and different values of l , and experimental response curve (the dotted line) for l = 80 mm.



 Figure 3 : Force-elongation response curve for a steel bar in a numerical simulation with N = 6 and l = 80 mm, compared with the experimental response curve (the dotted line) (a). Detail of the plateau at the onset of the inelastic regime (b) For the concrete specimen, at present the simulations are still in progress. A first result is shown in Fig. 4. While the general trend of the response is well reproduced, just after the onset of the inelastic deformation a sharp change of the slope occurs in the simulation but not in the experiment. A larger N is probably necessary to eliminate this discrepancy. The distribution of the inelastic deformation  ( x ) along the bar’s axis is shown in Fig. 5 for different values of  . The resemblance with the measurements by Miklowitz made many decades ago [11], is impressive. Both diagrams show a progressive concentration of inelastic deformation at the central zone of the bar. If we imagine that, by a sort of Poisson effect, axial elongation is accompanied by a proportional transversal contraction, we have that a concentration of axial deformation is accompanied by the necking of the cross section. Therefore, the proposed model provides a good description of the phenomenon of necking observed in bars subjected to a tensile load.

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G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01

 Figure 4 : Force-elongation response curve for a concrete specimen in a numerical simulation with N = 4 compared with the experimental curve (the dotted line) 

Figure 5 : The inelastic deformation  as a function of x , for different values of the total elongation  . The first diagram shows the results of the numerical simulations, and the second shows the experimental measurements reported in [11].

R EFERENCES

[1] G. Del Piero, G. Lancioni, R. March, A diffuse cohesive energy approach to fracture and plasticity: the one- dimensional case. In preparation (2011).

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[2] A.A. Griffith, Phil. Trans. Roy. Soc. A221, 163-198 (1920). [3] D. Dugdale, J. Mech. Phys. Solids, 8 (1960)100. [4] G.I. Barenblatt, Adv. Appl. Mech. , 7 (1962) 55.

[5] G.A. Francfort, J.-J. Marigo, J. Mech Phys. Solids, 46 (1998) 1319. [6] J.-J. Marigo, L. Truskinovsky, Cont. Mech. Thermodyn, 16 (2004) 391. [7] M. Charlotte, J. Laverne, J.-J. Marigo, Eur. J. Mech., A/25 (2006) 649. [8] G. Del Piero, L. Truskinovsky, Cont. Mech. Thermodynamics, 21 (2009) 141. [9] L. Freddi, G. Royer-Carfagni, J. Mech. Phys. Solids, 58 (2010) 1154. [10] K. Pham, H. Amor, J.-J. Marigo, C. Maurini, Int. J. Damage Mechanics, 20 (2011) 618. [11] J. Miklowitz, In: Proc. Nat. Meeting of the Applied Mechanics Division, ASME, Ann Harbor, Mich, (1949) 159.

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F. Felli et alii, Frattura ed Integrità Strutturale, 18 (2011) 14-22; DOI: 10.3221/IGF-ESIS.18.02

Fatigue behaviour of titanium dental endosseous implants

Ferdinando Felli, Daniela Pilone, Alessandro Scicutelli Sapienza Università di Roma, Dip. Ingegneria Chimica Materiali e Ambiente, Via Eudossiana 18, 00184 Roma, Italy ferdinando.felli@uniroma1.it

R IASSUNTO . Tra gli impianti dentali quelli endossei sono i più diffusi. Un impianto dentale endosseo si inserisce direttamente nell’osso a cui si fissa per tutta la sua lunghezza. Può avere forme diverse anche se attualmente gli studi scientifici riportano una percentuale di successi nettamente più elevata per gli impianti a vite e a cilindro. Nel corso degli anni sono state sviluppate varie tipologie di impianti dentali permanenti che sono poi stati modificati in base all’esperienza applicativa, alle numerose ricerche e ai test eseguiti ottenendo sostanziali miglioramenti in termini di durabilità, resistenza, migliore osteointegrazione ecc. In questo lavoro sono stati analizzati due tipi di impianti per valutarne la resistenza meccanica. Un impianto concepito e prodotto alla fine degli anni ‘90 ed ancora in uso in quei pazienti sottoposti ad implantologia fino a circa 10 anni fa, e uno di concezione più recente. Tali impianti sono stati testati a fatica con un sistema appositamente progettato e realizzato in modo da simulare le sollecitazioni reali agenti su di essi durante la masticazione. I risultati di tali test sono stati riportati sotto forma di diagrammi tipo Wohler che forniscono il limite di fatica al di sotto del quale le protesi potrebbero resistere indefinitamente. Lo studio condotto ha voluto anche valutare l’influenza della presenza di un eventuale mezzo corrosivo (NaCl) e l’analisi delle deformazioni e delle superfici di frattura dei campioni. I test condotti hanno evidenziato che il comportamento a fatica non sembra essere influenzato dalla presenza di un ambiente aggressivo quale quello costituito da soluzioni saline. Molto più influente è risultato invece il disegno dell’impianto. L’analisi delle superfici di frattura ha permesso di individuare le aree di innesco della frattura ed i meccanismi di propagazione. In tutti gli impianti esaminati l’innesco si localizza nelle aree in cui l’intensificazione degli sforzi è maggiore. I risultati dell’analisi agli elementi finiti, condotta su uno dei campioni, è in accordo con il tipo di frattura osservato dopo i test. L’analisi SEM delle superfici di frattura, caratterizzate delle tipiche striature di fatica, hanno mostrato inoltre una chiara similitudine tra il tipo di frattura intercorso nei provini testati in laboratorio e quello osservato sugli impianti rimossi da alcuni pazienti perché avevano ceduto dopo un certo periodo di servizio. A BSTRACT . In this work two different titanium dental implants are analyzed in order to evaluate their mechanical strength. An ad-hoc designed experimental apparatus is prepared to test against fatigue these implants in a way that approximates as much as possible the actual stresses occurring during mastication motion. The results of these endurance tests are summarized in the form of Wohler-type diagrams showing the duration of a specific implant for different applied loads. These plots show a fatigue limit below which the implants could resist indefinitely. Other aspects of this research concern the influence of a potentially corrosive medium and the analysis of the deformation and failure of the specimens. During fatigue cycling, the titanium implants do not seem to be affected by a more aggressive environment, such as a saline solution. The analysis of the broken specimen allowed the crack initiation sites and the type of fracture propagation to be investigated in depth. In all the considered implants fatigue cracks were seen to initiate preferentially from sites in which the tensile stress concentration is the highest. The results of a finite element analysis performed on one of the specimens is in good agreement with the failure mode observed after the tests. The SEM fracture surface analysis shows a clear

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F. Felli et alii, Frattura ed Integrità Strutturale, 18 (2011) 14-22; DOI: 10.3221/IGF-ESIS.18.02

similarity between the fracture mode of the tested implants and of the actual implants broken after a certain operating period. K EYWORDS . Dental implants; Fatigue resistance; Titanium alloys.

I NTRODUCTION

N

owadays dental prosthetic-implant systems are widely used for treatment of partial and total edentulism. This wide diffusion has been promoted by numerous studies which have demonstrated the "biologic" effectiveness of the used materials (in particular Titanium and some of its alloys) [1-4]. In spite of this diffusion some aspects have not been deeply investigated, like the mechanical resistance properties of implants and the stability of the connection between implants and prosthesis elements [5-6]. Although it is rather unusual to observe a clinical failure due to the biological reaction of the bone and/or gingival tissues after the installation of a dental implant, some mechanical problems frequently arise, like the fracturing of fixture or other prosthesis components (abutment, the implant-abutment connecting screws) [7-13]. Another commonly observed problem is represented by the progressive loosening of the screws connecting the implant with the abutment, this event often leads to the fracture of such screws, or even worse, to the fracture of the whole implant [6, 14]. Thus, the study of the mechanical characteristic of dental prosthesis implant systems is important to achieve long-term clinical success of implant-supported prosthesis. This kind of evaluations can be executed on a certain implant both "in vivo" on an installed prosthesis, and by using a mechanical system to accurately simulate the mastication effect. Carrying out "in vivo" studies has many technical difficulties, besides understandable ethic issues. First of all, forces exerted on natural and prosthetic dental elements during physiological functions (mastication, deglutition, phonation) and pathological functions (bruxism and other para-functions) have characteristics of intensity, direction and frequency which are extremely variable depending on the individual. This variability, although included in a well studied and well known range of values, makes difficult to extend results obtained on a limited sample of individuals to all possible clinical situations. To increase the validity of such tests, a much wider number of samples should be studied. To obtain reliable results, it should be considered the long duration of such testing: for a "normal" dental functionality, it would be necessary months, or most likely years [15]. To overcome those difficulties, mechanical systems have been designed and then improved in order to simulate the mastication effect [16-17]. Such systems are able to accurately reproduce: - set of mechanical loads (compression, bending, tension, torsion loads) acting on a prosthesis implant system during functional and para-functional activities. To achieve that, the developed occlusion forces depending on occlusal plane morphology, tooth position in the dental arch, size and hardness of food are schematized and reproduced. - connection between implant and bone tissue. As it is known the biological relationship between the chemically pure surface of dental materials, like titanium, and the living bone tissue is defined as "osseointegration" [18-23]. The contact area between bone and tissue varies according to several parameters such as the permanence time of the fixture in the tissue after the installation, the bone density (variable according to histologic type, cortical or spongy, and anatomic position of the implant in the dental arch) and kind and history of applied loads. - functionality of the prosthesis-implant system, which is the capability of the system to maintain a stable osseointegration (and then the capability of efficiently transmitting the applied load to the bone tissue) and the correct and stable connection between the implant and the prosthetic elements. This work concerns with the study of some clinically failed implants, which fractured at different times after the clinical installation: this study is of interest because those implants are still used from many patients. After observing the fracture surface morphology, a mechanical system has been setup to reproduce the mastication loading conditions to perform more accurate evaluations on implants similar to the ones on which the clinical failures have occurred. Although several methods are described in literature for fatigue testing [24], an specifically designed equipment has been set up in this work to assess the implants in terms of test loads and cycles. This paper analyses some clinically failed implants, which are of the bore cylinder type, and tries to investigate the failure mode. SEM analyses highlighted that fatigue failures occurred in such implants with initiations in the areas near to the bores, whose function is to promote osseointegration. Thus, since fatigue failure is known to initiate and propagate under tensile stress conditions, it is clear that the retrieved implants were subjected to traction load in their service life. So in this

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F. Felli et alii, Frattura ed Integrità Strutturale, 18 (2011) 14-22; DOI: 10.3221/IGF-ESIS.18.02

work the authors tried to replicate this loading condition to verify possible behaviour similarities between the retrieved implants and the laboratory tested samples. After the experiments performed with the bore cylinder type, which demonstrated a similarity of failure mode between retrieved and tested implants, another kind of dental prosthetic implant has been tested. For these implants, more recent than the previous ones, there was no availability of implants broken "in situ". Thus, a finite element analysis was required to assess the most stressed area of the implant under mastication-like loading scheme. After fatigue testing, aim of this work was to verify whether fatigue failures occurred in the regions suggested by FEA. Further studies on these implants concerned other aspects, involved in the component failure, such as loosening of screws and flexural deformation. This paper, for the first implant, aims at comparing service failure and simulated failure behaviour to build up a valid testing method and to estimate the fatigue limit under which the implant could resist indefinitely. As far as the second implant is concerned the aim of the work is to provide information on its fatigue behaviour in order to provide a possible fatigue limit under loading conditions similar to those involved in the actual mastication.

M ATERIALS AND METHODS

Implants T

he tested implants belong to two different types: - Titanium (grade 2, according to the ASTM F-67 standards) hollow cylinder with holes on the side surface to promote osseointegration with the bone tissue (Bonefit hollow cylinder, ITI). - Screw implant (Branemark system, Nobelbiocare) with threaded fixture in titanium (grade 2) with chemically etched and anodic passivated surface; titanium abutment and connecting screw, Cr-Co alloy dental capsule with hemispherical profile, coated with dental ceramic. In Figs. 1 and 2 the two types of implants and their dimensions expressed in millimeters are reported.

Figure 1 : Hollow cylinder type implant.

Figure 2 : Screw type implant.

Mechanical stress Implants are tested under fatigue loading in different ways. The first implant (hollow cylinder) is subjected to tensile stress while the second one (screw type) is dynamically loaded with a complex stress system of compression-bending-torsion loads similar to the one that takes place in the actual mastication action.

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F. Felli et alii, Frattura ed Integrità Strutturale, 18 (2011) 14-22; DOI: 10.3221/IGF-ESIS.18.02

It is known that the load stress regime on the implant during service cannot be simply reduced to the elementary compressive stress. Upon mandible closing, horizontal forces arise. These forces depend both on the dental cusps profiles, differently inclined relatively to the occlusive force which is substantially vertical, and on the mastication motion including side and protrusion movements of the mandible. Moreover, the vector sum of vertical forces acting on a tooth is not directed along the longitudinal axis of the tooth itself, thus bending moments arise. These bending moments added to those resulting from the horizontal forces, determine a flexural stress which is not negligible. Since the horizontal force components generally do not lay on a plane formed by their points of application and the symmetry axis of the implant (the tooth longitudinal axis) a torsion stress arises in addition to existing compressive and flexural stresses. In the cyclic fatigue regime, bending moments determine the necessary conditions for the initiation and subsequent propagation of fatigue cracks in the portion of the tooth which is in tension. Torsion moments, besides contributing to overall tension stress, can lead implant to mobilizing problems due to the decrease of the screw clamping torque and shear actions on the fixture-bone interface. Tests description The fatigue tests performed on the two dental implants are executed using a hydraulic-drive fatigue test machine, while the fracture surface analyses are carried out by a scanning electron microscope (SEM). The first implant (hollow and bored cylinder) is incorporated in an aluminium cylinder for a depth corresponding to the "implantantion-depth" (that is the same depth at which the implant should be inserted in the bone when in service) using an epoxy resin. The aluminium cylinder allows the connection with the test machine grippers. The adhesion between implant and cylinder is also increased by means of steel wires connected to the implant and immersed in the interface epoxy resin which is, specifically, a double-component epoxy resin Mecaprex MA. Firstly, this implant system is tested in air; three fatigue tests are performed using sinusoidal tension loads having increasing maximum values equal to 500, 750, 1000 N. The tests are repeated 3 times in order to achieve an acceptable statistical value considering the limited number of samples available. The selected maximum tension loads take into account the real average stress in the mastication which is, on the most stressed teeth (molar and pre-molar), about 770 N. The minimum value of the sinusoidal load is fixed and constant for all the performed tests and it is equal to 100 N. The loading frequency, fixed at 1 Hz, is comparable with the real average mastication frequency. Then, other three tests are performed using the same set-up and loading system, but in a different environment; the implant works in aqueous solution containing 3%wt NaCl in order to simulate, to a certain extent , the corrosive environment established in the mouth during the mastication. The tension load is selected in order to simulate the stress regime that occurs in the tension area of an implant when subjected to bending moments. As far as the second type of implant (screw type) is concerned, the complex stress regime to which it is subjected is firstly analysed by means of a finite elements solver provided by a software tool (ANSYS 7.0) for the Branemark type implant; the loading scheme and the results are shown in Figs. 3, 4, and 5. In the left part of Fig. 3, the 3D-mesh used for the finite elements calculation is shown. On the right part it is possible to observe the loading scheme with the punch having the contact plane, transmitting the load, inclined of 30° on the horizontal, on which is applied a 1000 N force. The surface in light blue is the part of the implant (fixture) that is constrained by the bone.

Figure 3 : 3D mesh and loading scheme.

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