Issue 17

Anno V Numero 17 Luglio 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, 17 (2011); Rivista Ufficiale del Gruppo Italiano Frattura

M. Paggi, P. Wriggers Numerical modelling of intergranular fracture in polycrystalline materials and grain size effects ............... 5 B. Atzori, G. Meneghetti, M. Ricotta Analysis of the fatigue strength under two load levels of a stainless steel based on energy dissipation …… 15 E. Benvenuti, R.Tovo, P. Livieri A brittle fracture criterion for PMMA V-notches tensile specimens based on a length-enriched eXtended Finite Element approach .......................................................................................................... 23 V. Crupi, G. Epasto, E. Guglielmino Computed Tomography analysis of damage in composites subjected to impact loading ………………... 32

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, 17 (2011); ISSN 1971-9883

Editor-in-Chief Francesco Iacoviello

(Università di Cassino, Italy)

Associate Editor Luca Susmel

(Università di Ferrara, Italy)

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, 17 (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, 17 (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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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

Numerical modelling of intergranular fracture in polycrystalline materials and grain size effects

M. Paggi Department of Structural and Geotechnical Engineering, Politecnico di Torino, Torino (Italy) marco.paggi@polito.it P. Wriggers Institut für Kontinuumsmechanik, Leibniz Universität Hannover, Hannover (Germany) wriggers@ikm.uni-hannover.de

A BSTRACT . In this paper, the phenomenon of intergranular fracture in polycrystalline materials is investigated using a nonlinear fracture mechanics approach. The nonlocal cohesive zone model (CZM) for finite thickness interfaces recently proposed by the present authors is used to describe the phenomenon of grain boundary separation. From the modelling point of view, considering the dependency of the grain boundary thickness on the grain size observed in polycrystals, a distribution of interface thicknesses is obtained. Since the shape and the parameters of the nonlocal CZM depend on the interface thickness, a distribution of interface fracture energies is obtained as a consequence of the randomness of the material microstructure. Using these data, fracture mechanics simulations are performed and the homogenized stress-strain curves of 2D representative volume elements (RVEs) are computed. Failure is the result of a diffuse microcrack pattern leading to a main macroscopic crack after coalescence, in good agreement with the experimental observation. Finally, testing microstructures characterized by different average grain sizes, the computed peak stresses are found to be dependent on the grain size, in agreement with the trend expected according to the Hall-Petch law. S OMMARIO . In questo articolo, il fenomeno della frattura intergranulare nei material policristallini è studiato mediante un approccio di meccanica della frattura non lineare. Il modello non locale di frattura coesiva per interfacce con spessore finito recentemente proposto dai presenti autori è impiegato per descrivere il fenomeno di separazione ai bordi di grano. Da un punto di vista modellistico, considerando la dipendenza dello spessore dei bordi di grano dalla dimensione del grano stesso, si è ottenuta una distribuzione delle proprietà meccaniche delle interfacce. Essendo la forma ed i parametri del modello non locale della frattura coesiva dipendenti dallo spessore dell'interfaccia, si ottiene una distribuzione di energie di frattura come conseguenza della variabilità statistica della microstruttura del materiale. Usando tali dati si conducono simulazioni di meccanica della frattura su elementi di volumi rappresentativi (RVE) in 2D e si determinano le rispettive curve di tensione deformazione. La frattura è il risultato di un insieme di microfessure diffuse che danno luogo alla propagazione di una fessura macroscopica principale, in ottimo accordo con quanto osservato sperimentalmente. Infine, testando microstrutture dotate di diversi diametri medi dei grani, si osserva come le tensioni di picco siano dipendenti dal diametro del grano, secondo un trend in accordo con la legge di Hall e Petch. K EYWORDS . Nonlocal cohesive zone model; Nonlinear and stochastic fracture mechanics; Finite thickness interfaces; Finite elements; Polycrystalline materials.

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

I NTRODUCTION

P

olycrystalline materials present heterogeneous microstructures where polyhedral grains are separated by interfaces. A typical scanning electron microscope (SEM) image of these microstructures is shown in Fig. 1(a). The different colours of the crystals indicate their different crystallographic orientations. The mechanical behaviour is strongly affected by the grain boundaries which govern the strength, the toughness and the ductility of the material. These properties are of paramount importance in forming processes and for the design of super-hard materials. As a general trend, the smaller the grain, the higher the material strength and the hardness. Interfaces are also important in conduction processes. In this case, however, they should be viewed as defects and the material conductivity of single crystalline materials is usually much higher than that of their polycrystalline counterparts. Fracture in polycrystalline materials can be schematically classified according to two different types: intergranular and transgranular . The former mode of fracture corresponds to the decohesion of the grains along the interfaces. Correspondingly, the material response is quite brittle. The latter is characterized by a propagation of cracks into the grains, with the occurrence of high plastic deformations leading to a much more ductile response. In spite of the fact that failure is often the result of a combination of these two modes of fracture, it is instructive to investigate each mode separately and understand the underlying mechanisms. In the present study, attention is paid to intergranular fracture, which is typically observed in brittle polycrystals. During a tensile test, microcracks develop at the grain boundaries (see Fig. 1(b) taken from [1]). At a certain deformation level, the microcracks coalescence and lead to the final failure with the propagation of a single main crack. Correspondingly, the stress-strain curve reaches a maximum and a sudden loss of load carrying capacity takes place. In order to investigate the effect of interfaces on the mechanical response of polycrystalline materials, a nonlinear fracture mechanics model is herein proposed. Intergranular fracture is depicted as a phenomenon of progressive separation at the grain boundaries governed by a nonlinear traction-separation law, or cohesive zone model (CZM). In this context, the finite thickness properties of interfaces are suitably taken into account by using the nonlocal CZM recently proposed in [2, 3] and briefly summarized in the next section. Virtual tensile tests of representative volume elements (RVEs) of material microstructures are carried out in order to simulate the phenomena of crack nucleation, coalescence and strain localization. Finally, grain size effects are investigated by changing the average grain size of the polycrystalline material and computing the peak stress of the simulated stress-strain curves. As it will be shown, numerical results are in agreement with the trend expected by the Hall-Petch law. This result confirms that fracture mechanics of interfaces is one of the most important factors governing the strength of polycrystalline materials.

(a) SEM image of the microstructure (b) Microcracks observed during a tensile test Figure 1 : SEM images of brittle polycrystalline materials showing microcracks (courtesy of Dr. Ing. M. Schaper [1]).

A NONLOCAL COHESIVE ZONE MODEL FOR INTERFACE FRACTURE

P

olycrystals are an important example of a material with finite thickness interfaces. In particular, examining the materials science literature [4], a power-law dependency of the interface thickness on the grain diameter has been noticed. Figure 2 shows the relation proposed in [4], where the grain diameter is computed as the mean diameter

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

of two grains sharing the same interface. A criterion of equivalence of cross-sectional areas is used to compute the diameter d of a polyhedral grain. Although the interface thickness l 2 is approximately one order of magnitude smaller than the grain diameter, as shown in Fig. 2(b), its simplification as a zero-thickness region, also called boundary layer , is not always possible. Numerical simulations considering an elasto-plastic behaviour of the interfaces suggest that the ductility of the material is strongly affected by these finite thickness regions [4].

l 2

d

d = grain diameter l 2 = interface thickness

Figure 2 : Dependency of the interface thickness on the grain diameter according to the relation proposed in [4].

In order to simplify the real material microstructure by considering a finite element discretization with zero-thickness interfaces, a suitable interface constitutive law has to be used (see [5-16] for a wide range of problems modelled using CZMs). Here, we consider the nonlocal CZM recently proposed in [2] for finite thickness interfaces. The traction separation relations that describe the nonlinear response of the interface are the following: 1 T e e gD D      (1a) 1 N e e gD D      (1b) where  and  are the tangential and normal cohesive tractions. The parameters T g and N g denote, respectively, the tangential and normal anelastic displacements evaluated at the boundaries of the finite thickness interface. Finally, e  and e  are the threshold values of the cohesive tractions for the onset of damage that correspond to the global tangential and normal displacements e  and e  of the interface region. The damage variable D (0 1) D   is computed as follows:

/2



2                      2 c c w u w u

D



(2)

where and c u are material parameters analogous to the critical opening and sliding displacements Nc l and Tc l used in standard CZMs [9] and  is a free parameter. The displacements u and w are given by: c w

l

2

u g

T e    



(3a)

G

2

l

2

w g

N e    



(3b)

E

2

where 2 2 G are the initially undamaged normal and tangential elastic moduli of the interface material. Changing the parameter  , different shapes of the CZM can be obtained, as shown in Fig. 3 for a l is the thickness of the interface, 2 E and

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

Mode I problem. Similar considerations apply for Mode Mixity. For more details about the calibration of the model parameters using molecular dynamics simulations, the readers are referred to [2].

Figure 3 : Shape of the Mode I nonlocal CZM as a function of  . This nonlocal CZM has been implemented in the FE code FEAP [17] using zero-thickness interface elements, see [3] for more details about the computational aspects. The anelastic relative displacements N g and T g are the main input of the element subroutine and are obtained as the difference between the normal and tangential displacements of the nodes of the finite elements of the continuum opposite to the shared interface. The residual vector and the tangent stiffness matrix of the interface element are computed by linearizing the corresponding weak form using a Newton-Raphson algorithm. Due to the implicit form of the CZM in Eq. (1), since the damage variable D on the r.h.s. of Eq. (1) is a function of the unknown cohesive tractions through Eqs. (2) and (3), a nested Newton-Raphson iterative scheme is used in to compute the cohesive tractions. A quadratic convergence is achieved, as shown in Fig. 4 for a Mode I problem.

Figure 4 : Quadratic convergence of the Newton-Raphson method used for the computation of the cohesive tractions, for three different values of / N c g w and for / 0 T c g u  .

A PPLICATIONS TO POLYCRYSTALLINE MATERIALS

he proposed nonlocal CZM for finite thickness interfaces is applied to the polycrystalline material microstructure of Copper analyzed in [1] and depicted in Fig. 5. From this input geometry, the grain size distribution, shown in Fig. 6(a), can be computed. The average grain diameter is 1 μm and its r.m.s. deviation is 0.26 μm. The interface thickness distribution is also computed and shown in Fig. 6(b). T

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

Figure 5 : Material microstructure of polycrystalline Copper numerically analyzed in this work.

 1 μm d m

(a) (b) Figure 6 : (a) Grain size distribution and (b) interface thickness distribution of the microstructure shown in Fig. 5. According to the thickness-dependent nonlocal CZM summarized in the previous section, the distribution of the Mode I interface fracture energy, represented by the area below the Mode I traction-separation curve, is obtained and shown in Fig. 7. It is interesting to note that the distribution of Mode I interface fracture energies for the present case (dots in Fig. 7) is better approximated by a Gaussian than by a Weibull distribution (see the probability plots in Fig. 7(a) and 7(b), where the Gaussian and Weibull distributions computed from the sample population are depicted with dashed-dotted lines). This is in general agreement with ductility of the material microstructure herein examined. Incidentally, we note that the Weibull modulus for these data is equal to 7.7, which is in agreement with the typical range of variation between 5 and 10 found in polycrystals [16]. This can be considered as an indirect experimental confirmation of the fact that the interface thickness distribution is responsible for the interface fracture energy distribution.

(a) Gaussian probability plot (b) Weibull probability plot Figure 7 : Mode I interface fracture energy distribution and comparison with the Gaussian and the Weibull distributions.

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

The parameters of the average Mode I traction-separation curve, corresponding to an average value of l 2 =0.15 μm, were selected in order to obtain an average Mode I fracture energy of 0.18 N/mm and a peak stress of 500 N/mm 2 , as suggested in [6] for polycrystalline interfaces (see the corresponding curve in Fig. 8 with solid line). More specifically, 0.53 c c     μm, 500 e c     N/mm 2 , 3 2 110 10 E   N/mm 2 were set for the grains, and the parameter α was selected as 0.0035. Alternatively, the parameters of the nonlocal CZM can be tuned to fit MD simulations, as illustrated in [2]. Interestingly, the shape of the nonlocal CZM can be matched with a good approximation with the standard CZM proposed by Tvergaard [9]. It is superimposed to Fig. 8 with dashed line and has the same Mode I fracture energy as our model.

Figure 8 : Shape of the nonlocal CZM (average curve) used for the fracture simulations and that of the Tvergaard [9] model with the same average fracture energy. To quantify the effect of using the nonlocal CZM with random properties instead of the Tvergaard CZM with the same properties for all the interfaces, we consider an elastic modulus of the grains equal to 3 1 110 10 E   N/mm 2 and a Poisson ratio equal to ν=0.3 for all the grains. The material microstructure shown in Fig. 5 is first simplified by augmenting the size of the grains, according to the procedure outlined in [2]. After this preliminary operation, each grain is meshed with constant strain triangular elements according to a Delauney triangulation. Then, zero-thickness interface elements governed by the thickness-dependent nonlocal CZM are placed along the grain boundaries. The size of the finite elements used for discretizing the continuum has been chosen in order to be one order of magnitude smaller than the process zone size, estimated according to the method suggested in [16]. The Dirichlet boundary conditions imposed on the model are selected to reproduce a tensile test, i.e., the nodes pertaining to the left vertical boundary are restrained to the horizontal displacements, whereas horizontal displacement are imposed on the nodes of the vertical boundary on the right. A Newton-Raphson solution scheme is adopted to solve the nonlinear boundary value problem at each step. The tolerance for the internal Newton-Raphson loop used to compute the residual and the tangent stiffness matrix of the interface elements is chosen as 1×10 −12 . In Fig. 9, the evolution of the crack pattern using the Tvergaard CZM with the same parameters for all the interfaces (pictures at the top) is compared with that obtained using the nonlocal CZM and random fracture properties (pictures at the bottom), for three different deformation levels,  =0.100, 0.115 and 0.124. The last deformation level corresponds to the final failure of the samples. Dashed lines correspond to fictitious cracks, i.e., microcracks where cohesive tractions are still acting. Solid lines correspond to the interfaces with D =1, i.e., real stress-free microcracks. The evolution of cohesive microcracks is widely distributed in both cases. At a certain point, when the microcracks coalescence into a single rough macrocrack, a phenomenon of strain localization takes place. The cohesive microcracks far from the main crack experience a stress relief, whereas the deformation accumulates on the main crack. This is well evidenced by the fact that microcracks (dashed segments) almost disappear at the deformation level of 0.124. The final crack pattern in case of uniform interface fracture properties appears to be characterized by a single main crack. On the contrary, using the nonlocal CZM with thickness dependent fracture properties, we obtain a separation of some grains and a more diffuse crack pattern, which is often found in experiments. The homogenized stress-strain responses of the composite cell are compared in Fig. 10. The homogenized stress is computed by summing the horizontal reactions of the constrained nodes on the vertical boundary on the right, and dividing it for its length. The use of the Tvergaard model (local CZM with uniform interface fracture properties) leads to a higher peak stress than using the proposed nonlocal CZM. This is mainly due to the prevalence of subvertical microcracks

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

subjected to pure Mode I. In both cases, the final failure of the sample is characterized by a sudden stress drop in the stress-strain diagram, probably due to a snap-back instability, typical of cohesive solids.

Local CZM

- - Cohesive

microcracks

 =0.115

 =0.124

 =0.100

— Stress-free cracks

Nonlocal CZM

Figure 9 : Crack patterns (cohesive microcracks with dashed line and stress-free cracks with solid line) for different strain levels.

Figure 10 : Homogenized stress-strain curve. The nonlocal CZM response (with stochastic distribution of interface properties) is compared with the prediction of the local CZM by Tvergaard with the same fracture energy for all the interfaces. The proposed nonlocal CZM is also able to capture the grain-size effects on the tensile strength. In polycrystalline materials, the tensile strength significantly depends on the grain size. An empirical correlation was proposed by Hall [18] and Petch [19], suggesting that the tensile strength is in general proportional to the inverse of the square root of the grain size at the microscale. To assess the capability of the proposed nonlocal CZM to capture this effect, we consider different material microstructures, replicas of that in Fig. 5, obtained by rescaling the diameters of the grains. Since the interface thicknesses depend on the grain size according to the power-law relation displayed in Fig. 2(b), the rescaled geometries are not self similar. As a consequence, the distribution of the interface fracture parameters will also depend on the grain size. The Mode I fracture energy distributions corresponding to microstructures with average grain sizes of 0.1μm, 1μm and 10μm are shown in Fig. 11. The shapes of the CZM for the three cases are similar to that shown in Fig. 8. In particular, the maximum cohesive stress of the curves changes, whereas the critical separation remains the same. The average fracture

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

energy for the three cases is equal to 0.8974 N/mm, 0.1794 N/mm and 0.0361 N/mm, for d m respectively. The r.m.s values are equal to 0.122 N/mm, 0.024 N/mm and 0.005 N/mm, respectively.

=0.1μm, 1μm and 10μm,

=0.1  m

d m

=1  m

d m

=10  m

d m

Figure 11 : Distribution of interface fracture energies for three different average grain sizes.

The peak stress of the stress-strain curves, obtained from tensile test simulations on the different cases, considering also d m =0.5μm, 2μm and 5μm in addition to 0.1μm, 1μm and 10μm, are shown in Fig. 12 vs. the grain diameter. The FE simulations lead to a peak stress which is a decreasing function of the grain size, in general agreement with the Hall-Petch relation that is superimposed to the same data in Fig. 12 by the dashed line. Therefore, the nonlocal CZM is fully able to reproduce the scaling of the tensile strength through the variation of fracture mechanics parameters connected to the variation of the interface thicknesses with the grain size. These numerical results imply that thicker interfaces are weaker than the thinner ones.

FE results

Hall-Petch law

d m

[  m]

Figure 12 : Numerically predicted vs. experimentally obtained peak stress vs. average grain size.

C ONCLUSIONS

n this paper, intergranular fracture in polycrystalline materials has been numerically investigated using finite elements. The main novelty with respect to previous contributions based on CZMs is represented by the use of a more sophisticated CZM whose properties (shape, fracture energy, peak stress) depend on the finite thickness of the interface. This is particularly suitable for polycrystalline materials in the micro-scale range, where the grain boundary thickness is not negligible and has an important role. The proposed nonlocal CZM is based on continuum damage I

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M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01

mechanics through the introduction of a damage variable that reduces the elastic modulus of the interface material. This approach allows us to perform an upscaling of the complex nonlinear mechanisms occurring in the interface region. The parameters entering the damage formulation can be tuned according to simple axial and shear tests to be performed on RVEs of the interface microstructure. This strategy has the great advantage of avoiding computationally expensive multiscale simulations based on the FE 2 method which requires, for each Gauss point, a micromechanical computation of the response of a lower scale RVE with a complete description of its constitutive nonlinearities [20-22]. Possible applications to fiber-reinforced interfaces and polymeric interfaces are therefore envisaged, with a tuning of the damage evolution law depending on the actual forms of nonlinearities present in those materials. Finally, regarding the grain size effects on the tensile strength of polycrystals, it has to be remarked that an inversion of the trend suggested by Hall and Petch has been observed at the nanoscale. Although a different description of the material has probably to be invoked, using molecular dynamics simulations instead of continuum mechanics, our proposed model may provide an insight into this debated problem. The results obtained in the present study show that the thinner the interface, the higher its fracture energy. As a result, the tensile strength of the material increases by refining the material microstructure. This suggests that an inversion of the Hall-Petch law may occur in case of thicker interfaces at the nanoscale. Experimental results seem to confirm this predicted trend. In fact, data in [23] show that the grain boundary thickness tends to vanish at the nanoscale. However, the percentage of atoms at the grain vertices, the so called triple junctions , drastically increases. As a result, the volumetric content of the all interface atoms (the sum of the atoms belonging to triple junctions and those belonging to grain boundaries), is much higher than that suggested by the scaling law holding at the microscale and shown in Fig. 2.

A CKNOWLEDGEMENTS

T

he support of AIT, MIUR and DAAD to the Vigoni 2011-2012 project "3D Modelling of fracture in polycrystalline materials" is gratefully acknowledged. MP would also like to thank the Alexander von Humboldt Foundation for supporting his research fellowship at the Institut für Kontinuumsmechanik, Leibniz Universität Hannover (Hannover, Germany) from February 1, 2010, to January 31, 2011.

R EFERENCES

[1] P. Kustra, A. Milenin, M. Schaper, A. Gridin, Computer Methods in Materials Science, 9 (2009) 207. [2] M. Paggi, P. Wriggers, Computational Materials Science, 50 (2011) 1625. [3] M. Paggi, P. Wriggers, Computational Materials Science, 50 (2011) 1634. [4] D.J. Benson, H.-H. Fu, M.A. Meyers, Materials Science and Engineering A, 319–321 (2001) 854. [5] Yan-Qing Wu, Hui-Ji Shi, Ke-Shi Zhang, Hsien-Yang Yeh, International Journal of Solids and Structures, 43 (2006) 4546. [6] T. Luther, C. Könke, Engineering Fracture Mechanics, 76 (2009) 2332. [7] P.D. Zavattieri, P.V. Raghuram, H.D. Espinosa, Journal of the Mechanics and Physics of Solids, 49 (2001) 27. [8] G. Beer, International Journal for Numerical Methods in Engineering, 21 (1985) 585. [9] V. Tvergaard, Material Science and Engineering A, 107 (1990) 23. [10] N. Point, E. Sacco, International Journal of Fracture, 79 (1996) 225. [11] M. Ortiz, A. Pandolfi, International Journal for Numerical Methods in Engineering, 44 (1999) 1267. [12] J. Segurado, J. Llorca, International Journal of Solids and Structures, 41 (2005) 2977. [13] S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Composites Science and Technology, 65 (2005) 281. [14] C. Leppin, P. Wriggers, Computers & Structures, 61 (1996) 1169. [15] J.C.J. Schellekens, R. de Borst, International Journal for Numerical Methods in Engineering, 36 (1993) 43. [16] H.D. Espinosa, P.D. Zavattieri, Mechanics of Materials, 35 (2003) 365. [17] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, 5th ed., Butterworth-Heinemann, Oxford and Boston, (2000). [18] E.O. Hall, Proceedings of the Physical Society of London B, 64 (1951) 747. [19] N.J. Petch, Journal of the Iron Steel Institute of London, 173 (1953) 25. [20] C.B. Hirschberger, S. Ricker, P. Steinmann, N. Sukumar, Engineering Fracture Mechanics, 76 (2009) 793.

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[21] K. Matous, M.G. Kulkarni, P.H. Geubelle, Journal of the Mechanics and Physics of Solids, 56 (2008) 1511. [22] M.G. Kulkarni, K. Matous, P.H. Geubelle, International Journal for Numerical Methods in Engineering, 84 (2010) 916. [23] Y. Zhou, U. Erb, K.T. Aust, G. Palumbo, Scripta Materialia, 48 (2003) 825.

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Analysis of the fatigue strength under two load levels of a stainless steel based on energy dissipation

B. Atzori, G. Meneghetti, M. Ricotta University of Padova, Department of Mechanical Engineering, 35131 Padova, Italy giovanni.meneghetti@unipd.it Published in Proceedings of 14 th International Conference on Experimental Mechanics ICEM14, Poitiers, France, 4-9 July, 2010 The European Physical Journal EPJ Web of Conferences volume 6-2010 ISBN: 978-2-7598-0565-5

R IASSUNTO . In questo lavoro è stato analizzato il comportamento a fatica di un acciaio inossidabile AISI 304L. Nella prima parte del lavoro sono presentati i risultati ottenuti da prove ad ampiezza di sollecitazione costante sintetizzati sia in ampiezza di tensione sia in termini di densità di energia dissipata dal materiale per ciclo, Q . Successivamente alcuni provini sono stati sollecitati ad un livello di carico superiore al limite di fatica per circa il 70% della presunta vita e poi ad un livello di tensione inferiore al limite di fatica ad ampiezza costante precedentemente determinato ed è stata confrontata l’energia dissipata nella seconda parte della prova con quella trovata in una prova ad ampiezza costante, allo stesso livello di tensione. Il confronto ha mostrato come per il materiale analizzato il parametro Q sia sensibile al danneggiamento precedentemente subito. A BSTRACT . In this paper the fatigue behaviour of a stainless steel AISI 304L is analysed. In the first part of the work the results obtained under constant amplitude fatigue are presented and synthesised in terms of both stress amplitude and energy released to the surroundings as heat by a unit volume of material per cycle, Q . Then some specimens have been fatigued in variable amplitude, two different load level tests: the first level was set higher while the second was lower than the constant amplitude fatigue limit. The Q values, evaluated during the second part of the fatigue test, have been compared with those calculated under constant amplitude fatigue at the same load level. The comparison allowed us to notice that the Q parameter is sensitive to the fatigue damage accumulated by the material during the first part of the fatigue test. K EYWORDS . AISI 304L; Dissipated energy; Fatigue; Stainless steel; Two load levels; Miner rule. he evaluation of fatigue limit of materials based on the experimental measurements of thermal increments is an experimental procedure well documented in the literature [1-3]. Recently one of the Authors suggested to adopt the energy released to the surroundings as heat by a unit volume of material per cycle, Q , as a fatigue damage indicator [4]. In view of this, a particular strategy was conceived in order to derive the Q parameter from measurements of the material surface temperature. Parameter Q was able to correlate the fatigue strength of smooth and notched specimens made of stainless steel, fatigued under constant amplitude stress. To the authors’ knowledge, the material behaviour in terms of Q in the case of variable amplitude fatigue load was never investigated. In this paper two load level fatigue tests T I NTRODUCTION

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B. Atzori et alii, Frattura ed Integrità Strutturale, 17 (2011) 15-22; DOI: 10.3221/IGF-ESIS.17.02

were carried out in order to investigate the sensitivity of Q parameter as damage indicator in the case of variable amplitude fatigue.

T HEORETICAL MODEL TO ESTIMATE THE Q PARAMETER he experimental technique used to evaluate the Q parameter is based on a theoretical model presented elsewhere [4], so that only the main features will be presented here. Let us consider a control volume dV of material under fatigue loading conditions, as shown in Fig. 1. The first law of thermodynamics applied to the control volume can be written in terms of power as:

 p T W dV H H H dV c E t                      cd cv ir

dV

(1)

where W is the expended mechanical power in a unit volume; H cd , H cv , H ir represent the thermal power dissipated in a unit volume due to conduction, convection and radiation, respectively; the last term in the second member is the rate of variation of the internal energy, which is related to the material density  , the specific heat c, the time variation of the temperature T and to the time variation of energy absorbed by the material p E  . The term p E  represents the rate of accumulation of plastic hysteresis energy, i.e. fatigue damage.

W

dV

U

Q

Figure 1 : First law of thermodynamics applied to a control volume of material undergoing a fatigue test.

Usually the surface temperature of material rapidly increases during the first part of fatigue test and then reaches a stationary value which depends on the applied stress level [3, 4]. In steady state conditions Eq. (1) becomes   cd cv ir p W H H H E      (2) By considering a sudden stop of fatigue test, the terms W and p E  become zero and then from Eq. (1):   cd cv ir T c H H H t          (3) is possible to evaluate the thermal power H dissipated in steady state conditions (Eq.(2)) by measuring the time derivative of temperature (see Eq.(3)). Finally, the energy released to the surroundings as heat by a unit volume of material per cycle, Q , can be calculated as: HQ f  (4) where f is the test frequency. M ATERIAL , SPECIMEN GEOMETRY AND TEST PROCEDURE he experimental tests were carried out on specimens prepared from 6-mm-thick AISI 304L stainless steel sheets. The specimen geometries used for static and fatigue tests are shown in Fig. 2a and Fig. 2b, respectively. Tests were carried out at room temperature on a Schenck Hydropuls PSA 100 servo-hydraulic test machine, T

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B. Atzori et alii, Frattura ed Integrità Strutturale, 17 (2011) 15-22; DOI: 10.3221/IGF-ESIS.17.02

equipped with a 100 kN load cell. The static behaviour was investigated by means of tensile tests under displacement control with a crosshead speed equal to 2 mm/min. During the static test the axial strain was measured by means of a MTS extensometer having a gauge length of 25 mm. The fatigue tests were carried out under load control, with a sinusoidal wave, nominal load ratio R (R =  min /  max ) equal to -1 and a test frequency variable in the range of 2-36 Hz depending on the applied stress level. To investigate the material fatigue behaviour, constant amplitude fatigue tests were carried out up to the specimen failure. Concerning the two load level fatigue tests, some specimens were fatigued at a stress level higher than the fatigue limit for a significant fraction of fatigue life. Then the stress level was decreased lower than the fatigue limit and kept constant up to 10 millions of cycles or specimen failure. Temperature increments were monitored by means of an AGEMA THV 900 LW/ST infrared camera able to detect infrared radiation in the range of wave lengths between 8 and 12  m with a resolution of 0.1 °C. The thermal images were post processed by using the dedicated software AGEMA research 2.1.

48

50 50

R30

30

12

205

30

R30

7

10

113

(a)

30 (b) Figure 2 : Specimen geometry for static (a) and fatigue (b) tests.

E XPERIMENTAL RESULTS n order to characterise the material static behaviour, five tests were carried out. The mean value of elastic modulus E, engineering tensile strength  R , proof strength  p0,2 and true fracture strain A% are summarised in Tab. 1. By means of an electrolytic etching (stainless steel anode and cathode, voltage 1.2 V, current 0.2 A) on a 60% nitric acid solution the microstructure was analysed. A typical example is shown in Fig. 3: the white matrix represents the austenitic phase while the dark zones inside the grains are ferrite, which represents the 1% of the volume. E [MPa]  p0,2 [MPa]  R [MPa] A% 194750 315 699 59% Table 1 : Material properties of AISI 304L stainless steel I

60  m

60  m

(a)

(b) Figure 3 : Example of microstructure observed in the cross section: mid-thickness (a) and below surface (b) .

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B. Atzori et alii, Frattura ed Integrità Strutturale, 17 (2011) 15-22; DOI: 10.3221/IGF-ESIS.17.02

The grain size was assessed according to ASTM E112 standard [5]. Two specimens were analysed and three measurements were done on each of them: in particular, two measurements were performed in the cross-section (one in the mid thickness and one below specimen’s surface) and one on the surface of the sheet. The mean values are listed in Tab. 2. One can note that the grains are bigger in the external surface, as it can be expected in the case of rolling sheets. Finally, measurements of Vickers micro-hardness (applied load 0.2 kg) were carried out. Tab. 3 shows the results obtained in the middle of the thickness and at 0.2 mm below the surface.

Specimen 1 [  m]

Specimen 2 [  m]

Below surface Mid-thickness

37 30 41

27 27 35

Surface

Table 2 : Experimentally measured grain size.

Specimen 1

Specimen 2

175 163 151 169 162 162

155 157 151

Below surface

170 163 167 Table 3 : Measured values of Vickers micro-hardness (applied load 0.2 kg).

Mid-thickness

Synthesis of experimental data in terms of stress amplitude The material fatigue limit was evaluated according to a shortened stair-case procedure according to Dixon’s rule [6], which involved 7 specimens. Tests were stopped after 10 million cycles. As a results, the fatigue limit in terms of stress amplitude  A  ,-1 resulted equal to 217 MPa. The Wöhler curve, shown in Fig. 4, was evaluated via statistical analysis of the available fatigue data, by assuming log normal distribution of the number of cycles to failure. In the same figure the fatigue limit  A  ,-1 , the 10-90% scatter band, the value of the inverse slope k of Wöhler curve, the scatter index T  (T  =  A  ,-1,10% /  A  ,-1,90% ), the scatter index T N,  (T N,  = T  k ) and the number of cycles N A , which corresponds to the knee point of the curve, are listed.

300

10%

 a [MPa]

90%

200

R=-1

T 

=1.20

broken run-out broken

k=12.9  A 

T N, 

=10.5

Stair case procedure

= 217 MPa

N A

=165800

140

10 4

10 5

10 6

10 7

10 8

N. cycles

Figure 4 : Wöhler curve and 10%-90% scatter band of the AISI 304 L stainless steel.

Synthesis of experimental data in terms Q parameter As already discussed, the evaluation of Q parameter is based on the measurement of the cooling rate just after a sudden interruption of the fatigue test, according to Eqs. (3) and (4). The material density, experimentally measured by using Archimedes method and a Sartorius 1801 balance, with a resolution of 10 -7 kg, was 7940 kg/m 3 . By using a calorimeter, the specific heat c was determined equal to 507 J·kg -1 ·K -1 .

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The cooling rate after a sudden interruption of the fatigue test was measured by using the infrared camera and by considering the maximum value of the temperature inside on area encompassing the reduced section of the specimen. The maximum sampling frequency of the thermal images allowed by the available measuring system was 7 Hz. Fig. 5a shows an example of an infrared image: the rectangle identifies the area where the maximum surface temperature was detected. A typical example of the recorded temperature trend is plotted in Fig. 5b: t 0 indicates the time when the fatigue test was stopped while in the y-axis the temperature variation with respect to that stabilized before the test stop is shown. After evaluating the cooling gradient, Q was derived according to Eqs. (3) and (4).

(a)

(b) Figure 5 : Example of control area surrounding the specimen where the maximum temperature was analysed (a) and typical maximum temperature signal measured after a sudden interruption of fatigue test (b) (  a =210 MPa, run out). In order to evaluate the evolution of Q parameter, each fatigue test was interrupted several times. Fig. 6 shows the Q values plotted versus the number of cycles normalised with respect to the number of cycles to failure or, in the case of run-out specimens, with respect to 10 millions. It can be noted that the value of Q reached a constant value after about 50% of the total fatigue life. Moreover, the plotted curves show as soon as the stress amplitude is increased above the fatigue limit (  a > 220 MPa) then the stabilised values of Q increase of a factor 7 (from  100 kJ/(m 3 ·cycle) to  700 kJ/(m 3 ·cycle)). The fatigue data were analysed in terms of the stabilised energy parameter found during each fatigue test, by assuming a log-normal distribution of the number of cycles to failure, according to the following equation: cos k Q N t   (5) where N represents the number of cycles to failure and k is the inverse slope of the new fatigue curve.

1400

1200

1000

800

400 Q [kJ/(m 3 cycle)] 600

260 MPa 240 MPa 230 MPa 220 MPa 210 MPa 190 MPa

200

0

0

0.2

0.4

0.6

0.8

1

N/N f

Figure 6 : Evolution of the specific energy loss versus the normalised fatigue life.

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