F Frraattttuurraa · Fracture Mechanics [2-3]. The presented examples will be concerned with the...

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Anno III Numero 10

Ottobre 2009

Rivista Ufficiale del Gruppo Italiano Frattura Fondata nel 2007

ISSN 1971-8993

www.gruppofrattura.it

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Frattura ed Integrità Strutturale, 10 (2009)

Rivista Ufficiale del Gruppo Italiano Frattura; ISSN 1971-8993 Reg. Trib. Cassino n. 729/07 del 30/07/2007

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Complexity: a new paradigm for fracture mechanics A. Carpinteri, S. Puzzi …………………………………………………....…………………………. 3

The theory of critical distances applied to problems in fracture and fatigue of bone D. Taylor, S. Kasiri, E. Brazel ………………...….………..…………………………………….………… 12

Failure prediction of T-peel adhesive joints by different cohesive laws and modelling approaches A. Pirondi …………………... ....…………………………………………….…………………………... 21

Sismabeton: a new frontier for ductile concrete B. Chiaia, A. P. Fantilli, P. Vallini …………………………………………….…………………………... 29

Numerical analysis of soil bearing capacity by changing soil characteristics A. Namdar, M.K. Pelko ……………….....…………………………………….…………………………... 38

A dimensional analysis approach to fatigue in quasi-brittle materials M. Paggi ………………………….…...………………………………..…….…………………………... 43

Mechanical characterization of metal-ceramic composites G. Bolzon, M. Bocciarelli, E. J. Chiarullo ….…………………………………….…………………………... 56

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 [email protected] Direttore Responsabile: Francesco Iacoviello, Università di Cassino Comitato Scientifico: Goffredo De Portu, CNR - ISTEC Andrea Pavan, Politecnico di Milano Nicola Bonora, Università di Cassino Angelo Finelli, ENEA Centro Ricerche Faenza Alberto Carpinteri, Politecnico di Torino Domenico Gentile, Università di CassinoMartino Labanti, Enea Centro Ricerche Faenza Giuseppe Ferro, Politecnico di Torino David Taylor, University of DublinGiovanna Gabetta, ENI E&P Division Donato Firrao, Politecnico di Torino Luca Susmel, Università di Ferrara Stefano Beretta, Politecnico di Milano Marco Paggi, Politecnico di Torino Andrea Carpinteri, Università di ParmaMario Guagliano, Politecnico di Milano Marco Savoia, Università di Bologna Alessandro Pirondi, Università di Parma Giulio Mayer, Politecnico di Milano Roberto Roberti, Università di Brescia Vincenzo Maria Sglavo, Università di TrentoRoberto Frassine, Politecnico di Milano Franco Furgiuele, Università della Calabria Francesca Cosmi, Università di Trieste



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Frattura ed Integrità Strutturale, 10 (2009)

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lla fine di giugno si è svolto a Torino il Convegno Nazionale IGF. Ricco di presentazioni interessanti, che trovate nel sito sia nel volume degli atti che nella sezione WebTV, ha visto anche un parziale avvicendamento nelle cariche sociali. Il nuovo Consiglio di Presidenza risulta ora costituito da: Stefano Beretta, Francesca Cosmi,

Giuseppe Ferro, Angelo Finelli, Domenico Gentile, Marco Paggi, Alessandro Pirondi e Luca Susmel. E’ doveroso ringraziare i Colleghi uscenti che in questi anni hanno consentito all’IGF di raggiungere l’attuale livello di attività e, speriamo, di apprezzamento. Il loro contributo alla vita dell’IGF comunque non termina, restando a far parte del Comitato Scientifico della Rivista IGF. Come potete osservare dalla firma del presente editoriale, c’è stato anche un avvicendamento nella Presidenza IGF. E’ durante la presidenza di Giuseppe Ferro, per molti Beppe, che il sito IGF ha intrapreso il suo cammino, che la rivista Frattura ed Integrità Strutturale ha iniziato i suoi primi passi e che tante iniziative hanno preso vita trovando un crescente apprezzamento. Il Presidente uscente comunque continuerà a partecipare attivamente alla vita dell’IGF rimando a far parte del Consiglio di Presidenza, ma, in ogni caso… Grazie Beppe!! Come nuovo Presidente, non posso che fare i miei migliori auguri al nuovo Consiglio per un proficuo lavoro, augurandomi di riuscire nel difficile compito di superare i risultati ottenuti in questi tre decenni di vita dell’IGF. Dal 9 all’11 settembre 2009 si è svolto sempre a Torino il convegno AIAS (Associazione Italiana Analisi Sollecitazioni), che ha visto l’organizzazione di due sessioni IGF coordinate da Franco Furgiuele. Alcune di queste sono inserite nella sezione WebTV del sito IGF ed i lavori in formato pdf saranno presto disponibili nel sito AIAS (http://www.aiasonline.org/). Inoltre, nelle prossime settimane è prevista una sessione tematica all’interno del prossimo convegno AIPnD (Associazione Italiana Prove non Distruttive; Roma dal 15 al 17 ottobre 2009). Il coordinatore di questa sessione sarà Stefano Beretta ([email protected]). Concludendo, vi anticipo che per il 2010 abbiamo previsto un calendario di attività veramente molto intenso… non mancate!!!

Francesco Iacoviello

Presidente IGF

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A. Carpinteri et alii, Frattura ed Integrità Strutturale, 10 (2009) 3-11; DOI: 10.3221/IGF-ESIS.10.01

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Complexity: a new paradigm for fracture mechanics

A. Carpinteri, S. Puzzi Politecnico di Torino, Department of Structural and Geotechnical Engineering, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, [email protected]; [email protected]

RIASSUNTO. Le cosiddette Scienze della Complessità sono un argomento di interesse in forte crescita all'interno della Comunità Scientifica. In realtà, i ricercatori non sono ancora giunti ad un’unica definizione di Complessità, per il fatto che essa si manifesta attraverso svariate forme [1]. Questo campo d’indagine, infatti, non è rappresentato da una singola disciplina, ma piuttosto da un insieme eterogeneo costituito da differenti tecniche matematiche e da diversi ambiti della scienza. Sotto l’allocuzione di Scienze della Complessità comprendiamo una grande varietà di approcci: la dinamica non lineare, la teoria del caos deterministico, la termodinamica del non-equilibrio, la geometria frattale, l’asintoticità intermedia, l’autosomiglianza completa ed incompleta, la teoria del gruppo di rinormalizzazione, la teoria delle catastrofi, la criticalità auto-organizzata, le reti neurali, gli automi cellulari, la logica sfumata (fuzzy logic), etc. Scopo del presente lavoro è quello di approfondire il ruolo della Complessità nel campo della Scienza dei Materiali e della Meccanica della Frattura [2-3]. Gli esempi presentati riguarderanno il fenomeno instabile dello snap-back nel comportamento di strutture composite (Carpinteri [4-6]), l’insorgere di pattern frattali e dell’autosomiglianza nella deformazione e nel danneggiamento dei materiali eterogenei, oltre agli effetti di scala sulle proprietà meccaniche nominali dei materiali disordinati (Carpinteri [7,8]). Ulteriori esempi si occuperanno dell’interpretazione dei fenomeni critici e degli effetti di scala temporale sulla vita ultima delle strutture per mezzo della Emissione Acustica (Carpinteri et al.[9]). Infine, verranno presentati risultati sulla transizione verso il caos nel comportamento dinamico di travi fessurate (Carpinteri and Pugno [10,11]). ABSTRACT. The so-called Complexity Sciences are a topic of fast growing interest inside the scientific community. Actually, researchers did not come to a definition of complexity, since it manifests itself in so many different ways [1]. This field itself is not a single discipline, but rather a heterogeneous amalgam of different techniques of mathematics and science. In fact, under the label of Complexity Sciences we comprehend a large variety of approaches: nonlinear dynamics, deterministic chaos theory, nonequilibrium thermodynamics, fractal geometry, intermediate asymptotics, complete and incomplete similarity, renormalization group theory, catastrophe theory, self-organized criticality, neural networks, cellular automata, fuzzy logic, etc. Aim of this paper is at providing insight into the role of complexity in the field of Materials Science and Fracture Mechanics [2-3]. The presented examples will be concerned with the snap-back instabilities in the structural behaviour of composite structures (Carpinteri [4-6]), the occurrence of fractal patterns and self-similarity in material damage and deformation of heterogeneous materials, and the apparent scaling on the nominal mechanical properties of disordered materials (Carpinteri [7,8]). Further examples will deal with criticality in the acoustic emissions of damaged structures and with scaling in the time-to-failure (Carpinteri et al. [9]). Eventually, results on the transition towards chaos in the dynamics of cracked beams will be reported (Carpinteri and Pugno [10,11]). KEYWORDS. Catastrophe Theory, Fractal Geometry, Scaling of Material Properties, Self-Organized Criticality, Deterministic Chaos.

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INTRODUCTION

omplexity, as a discipline, generally refers to the study of large-scale systems with many interacting components, in which the overall system behaviour is qualitatively different from (and not encoded in) the behaviour of its components. Complex systems lie somehow in between perfect order and complete randomness –the two

extreme conditions that occur only very seldom in nature– and exhibit one or more common characteristics, such as: sensitivity to initial conditions, pattern formation, spontaneous self-organization, emergence of cooperation, hierarchical or multiscale structure, collective properties beyond those directly contained in the parts, scale effects. Complexity has two distinct and almost opposite meanings: the first goes back to Kolmogorov's reformulation of probability and his algorithmic theory of randomness via a measure of complexity, now referred to as Kolmogorov Complexity [1]; the second to the Shannon's studies of communication channels via his notion of information. In both cases, complexity is a synonym of disorder and lack of a structure: the more random a process is, the more complex it results to be. The second meaning of complexity refers instead to how intricate, hierarchical, structured and sophisticated a process is. Associated with these two almost opposite meanings, are two natural trends of composite systems, and two corresponding questions: how does order and structure emerge from large, complicated systems? And, conversely, how do randomness and chaos arise from systems with only simple constituents, whose behaviour does not directly encode randomness? The former case is typical of all those phenomena which could be described through the concepts of scale invariance, phase transition, and with the use of power laws. The latter case is that of instability and bifurcations and of dynamical systems showing chaotic attractors and transition to chaos. In this paper, several fracture mechanics applications will be shown, in which both trends are present. THE NONLINEAR COHESIVE CRACK MODEL: SNAP-BACK INSTABILITY AS A CUSP CATASTROPHE

he first example dates back to the 1980's, when the senior author [4-6] approached the snap-back instability of cracked bodies with a Cohesive Crack model, which can be interpreted in the general framework of Catastrophe Theory (Thom [12]). This first section is thus devoted to a brief review of the ductile-to-brittle transition in the

mechanical behaviour of cracked solids, described by means of the Cohesive Crack model. The Cohesive Crack Model was initially proposed by Barenblatt [13] and Dugdale [14]. Subsequently, Dugdale's model was reconsidered by several other Authors (for a review see [15]); Hillerborg et al. [16] proposed the Fictitious Crack Model in order to study crack propagation in concrete. The cohesive crack model is based on the following assumptions ([4,15]): 1. The cohesive fracture zone (plastic or process zone) begins to develop when the maximum principal stress achieves the ultimate tensile strength u. 2. The material in the process zone is partially damaged but still able to transfer stress. Such a stress is dependent on the crack opening displacement w. The energy GF necessary to produce a unit crack surface is given by the area under the w diagram. The real crack tip is defined as the point where the distance between the crack surfaces is equal to the critical value of crack opening displacement wc and the normal stress vanishes. On the other hand, the fictitious crack tip is defined as the point where the normal stress attains the maximum value and the crack opening vanishes (Fig. 1). With some modifications, the cohesive crack model has been applied to model a wide range of materials and fracture mechanisms, most prominently concrete. Regarding this material, there is a very large literature; for a review, the reader is referred to the review papers by Carpinteri and co-workers [15,17]. Now, let us quantify the ductile-to-brittle transition by showing synthetically the numerical results for concrete elements in Mode I conditions (Three Point Bending Test - TPBT), based on the cohesive model, obtained using the Finite Element Code FR.ANA. (FRacture ANAlysis Carpinteri [5,18,19]). Extensive series of analyses were carried out from 1984 to 1989 by A. Carpinteri and co-workers. The experimental results can be found in the RILEM report [20]. The cases described in the reference papers regard three slenderness ratios, and four initial crack depths, and a concrete-like material. Fig. 2a refers to the case of an initially uncracked beam, whilst Fig. 2b reports results for the case of an initially cracked beam with relative crack depth equal to 0.5. For each ratio, the response was analyzed for different values of the brittleness number, SE [4]. As can be seen from the diagrams, by increasing SE the behaviour of the element changes from brittle to ductile. Generally speaking, the specimen behaviour is brittle (snap-back) for low SE numbers, i.e., for low fracture toughness, GF, high tensile strengths, u, and/or large sizes, h. In particular, in the case of uncracked beam, for SE 10.45x10-5, the P–δ curve presents positive slope in the

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softening branch and a catastrophical event occurs if the loading process is deflection-controlled. Such indenting branch is not virtual only if the loading process is controlled by a monotonically increasing function of time (Biolzi et al. [21]).

Figure 1: Constitutive laws of the cohesive crack model: (a) undamaged material; (b) process zone.

In the case of the cracked beam, on the contrary, the initial crack makes the specimen behaviour more ductile; for the set of SE numbers considered in Fig. 2b, the snap-back does not occur. By varying the initial crack depth, it is possible to describe the gradual transition from simple fold catastrophe (softening) to bifurcation or cusp catastrophe (snap-back instability), generating an entire equilibrium surface, or the catastrophe manifold.

Figure 2: Dimensionless load vs. deflection diagrams by varying the brittleness number SE, initially uncracked (a) and cracked (b) specimen

THE FRACTAL INTERPRETATION OF THE SIZE-SCALE EFFECT

he second topic is concerned with the size-scale effects on the mechanical properties of heterogeneous disordered materials that can be interpreted synthetically through the use of fractal sets. Fractal sets are characterized by non-integer dimensions (Mandelbrot [22]). For instance, the dimension α of a fractal set in the plane can vary between

0 and 2. Accordingly, increasing the measure resolution, its length tends to zero if its dimension is smaller than 1 or tends to infinity if it is larger. In these cases, the length is a nominal, useless quantity, since it diverges or vanishes as the measure resolution increases. A finite measure can be achieved only using noninteger units, such as meters raised to αl. Fractals sets can be profitably used to describe the size-scale effects on the parameters of the cohesive crack model. As shown in the previous section, this model captures the ductile-brittle transition occurring by increasing the size of the structure. On the other hand, uniaxial tensile tests on dog-bone shaped specimens [23,24] have shown that the three material parameters defining the cohesive law are size dependent: increasing the specimen size, the tensile strength u, tends to decrease, whilst the fracture energy GF and the critical displacement wc increase. In order to overcome the original cohesive crack model drawbacks, a scale-independent (fractal) cohesive crack model has been proposed recently by the first Author [25]. This model is based on the assumption of a fractal-like damage localization, suggested by experimental evidence [26,27].

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Let us consider fractal geometries for both the resistant cross section at maximum load (Fig. 3a) and the dissipation domain (Fig. 3c) [25]. Hence we can compute the maximum load F, the critical displacement wc and the total dissipated energy W as:

* *u 0 u resA AF (1a)

ε1-*ε ε dc c cw b b (1b)

*F 0 F disA A*W G G (1c)

These quantities are size-dependent. The true scale-independent quantities are the right hand side ones, i.e. the fractal strength u*, the fractal critical strain εc* and the fractal fracture energy GF*. They show non-integer physical dimensions: [F][L]–(2–dσ) for u*, [L](dε) for wc and [FL][L] –(2+dG) for GF*. Because of the measure of the resistant cross section Ares and the dissipation domain Adis, from Eqs. (1) the scaling laws for strength, critical displacement and fracture energy can be obtained:

σ*u u

db (2a)

ε1-*ε dc cw b (2b)

F F

d*b GG G (2c)

Figure 3: A concrete specimen subjected to tension. Fractal localization of the resistant cross section (a); fractal localization of the strain (b) and the energy dissipation inside the damaged band (c).

The three size effect laws (2) of the cohesive law parameters are not completely independent of each other. In fact, there is a relation among the scaling exponents that must be always satisfied. In order to get this relation, the simplest path is to consider the damage domain in Fig. 3c as the cartesian product of those in Figs. 3a and 3b. As a result, we obtain:

σ ε 1d d d G (3)

According to these definitions, we call the *ε* diagram the fractal or scale-independent cohesive law. Contrarily to the classical cohesive law, which is experimentally sensitive to the structural size, this curve is an exclusive property of the material since it is able to capture the fractal nature of the damage process. The area below the softening fractal stress-strain diagram represents the fractal fracture energy GF*. In order to validate the model, it has been applied to the data obtained in 1994 by Carpinteri and Ferro [23,24] for tensile tests on dog-bone shaped concrete specimens of various sizes under controlled boundary conditions (Fig. 4a). They




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interpreted the size effects on the tensile strength and the fracture energy by fractal geometry. Fitting the experimental results, they found the values dσ= 0.14 and dG= 0.38. Some of the –ε (stress vs. strain) and –w diagrams are reported respectively in Fig. 4b and 4c, where w is the displacement localized in the damaged band. Eq. (3) yields dε= 0.48, so that the fractal cohesive laws can be plotted in Fig. 4d. As expected, all the curves related to the single sizes tend to merge in a unique, scale-independent cohesive law. The overlapping of the cohesive laws for the different sizes proves the soundness of the fractal approach to the interpretation of concrete size effects.

Figure 4: Tensile test on dog-bone shaped specimens (a) by Carpinteri and Ferro [28]; stress-strain diagrams (b), cohesive law diagrams (c), fractal cohesive law diagrams (d).

THE FRACTAL INTERPRETATION OF MULTISCALE CRACKING PHENOMENA

he third topic deals with the criticality of the complex multiscale cracking phenomena in heterogeneous and disordered materials, evaluated by means of the Acoustic Emission (AE) technique. Acoustic Emission (AE) is represented by the class of phenomena whereby transient elastic waves are generated by the rapid release of energy

from localized sources within a material. All materials produce AE during both the generation and propagation of cracks. The elastic waves move through the external solid surface, where they are detected by sensors. In this way, information about the existence and location of possible damage sources is obtained. This is similar to seismicity, where seismic waves reach the station placed on the earth surface (Richter [28]). With regard to the basis of AE research in concrete, the early scientific papers were published in the 1960s. Particularly interesting are the contributions by Rusch [29], L'Hermite [30] and Robinson [31]. They discussed the relation between fracture process and volumetric change in the concrete under uniaxial compression. The most important applications of AE to structural concrete elements started in the late 1970s [32]. Regarding the determination of the defects position and orientation in the material, research has been growing at a fast rate in the last decade (Shah & Zongjing [33] and Ohtsu [34]). In the last few years the AE technique has been applied to identify defects and damage in reinforced concrete structures and masonry buildings (Carpinteri & Lacidogna [35,36]). By means of this technique, a particular methodology has been put forward for crack propagation monitoring and crack stability assessment in structural elements under service conditions. This technique permits to estimate the amount of energy released during fracture propagation and to obtain information on the criticality of the ongoing process [9,37]. Without entering the details, recent developments in fragmentaron theories (Carpinteri & Pugno [38,39]) have shown that the energy dissipation E during microcrack propagation occurs in a fractal domain comprised between a surface and the

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specimen volume V. The fractal criterion predicts a volume-effect on the maximum number of acoustic emission events Nmax, that, in a bilogarithmic diagram, would appear as:

maxlog log log V3AE

DN (4)

with a slope equal to D/3, where ΓAE is the critical value of fractal acoustic emission density and D is the fractal exponent, comprised between 2 and 3 [37]. Experiments carried out by Carpinteri et al. [36] on concrete specimens tested in compression confirm the soundness of the proposed approach. For all the tested specimens, the critical number of acoustic emissions Nmax was evaluated in correspondence to the peak-stress u. The compression tests show an increase in AE cumulative event number by increasing the specimen volume. More in detail, subjecting the average experimental data to a statistical analysis, the parameters D and ΓAE in eq. (4) were quantified. From the best-fitting, reported graphically in Fig. 5, the estimated value of the slope was computed as D/3 = 0.766, so that, as predicted by the fragmentation theories, 2D3. This result is a confirmation of the fact that the energy dissipation, measured by the number of acoustic emissions N, occurs over a fractal domain. Interestingly, the criticality of the cracking phenomena does appear not only in space, but also in time. A scaling relation of the type of eq. (4) can be written for the time t, allowing one to define the damage parameter , which can be expressed [9,37] as a function of different parameters, i.e., stress σ, strain ε or time t:

σ ε tβ β β

max max max max

σ εη

σ ε

N t

N t

(5)

where the exponents β can be obtained from the AE data of a reference specimen. The fractal multiscale criterion of Eq. (5) is a fundamental result, since it allows to predict the damage evolution also in large concrete structural elements. Monitoring the damage evolution by AE, it is therefore possible to evaluate the damage level as well as the time to the final collapse [9].

Figure 5: Volume effect on the maximum number of acoustic emissions.

ROUTE TOWARDS CHAOS IN THE DYNAMICS OF CRACKED BEAMS

he fourth and last topic is concerned with the dynamical behaviour of cracked beams (Carpinteri and Pugno [40,10,11]. Dealing with the presence of a crack in the structure, previous studies have demonstrated that a transverse crack can change its state (from open to closed and vice versa) when the structure, subjected to an

external load, vibrates. As a consequence, a nonlinear dynamic behavior is introduced. This phenomenon has been detected during experimental testing performed by Gudmundson [41], in which the influence of a transverse breathing crack upon the natural frequencies of a cantilever beam was investigated.

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Several models have been proposed in the past for dealing with cracked vibrating beams [42-44], but, in all these models, the main assumption has been that the crack can be either fully open or fully closed during the vibration. Carpinteri and Pugno [10] recently developed a coupled theoretical and numerical approach to evaluate the nonlinear complex oscillatory behaviour in damaged structures under excitation. In their approach, they have focused their attention on a cantilever beam with several breathing transverse cracks and subjected to harmonic excitation perpendicular to its axis. The method, that is an extension of the super-harmonic analysis carried out by Pugno et al. [45] to subharmonic and zero frequency components, has allowed to capture the complex behavior of the nonlinear system, e.g., the occurrence of period doubling, as experimentally observed by Brandon and Sudraud [46] in cracked beams. A pioneer work on period doubling was written in 1978, when Mitchell Feigenbaum [47] developed a theory to treat the route from ordered to chaotic States. Even if oscillators showing the period doubling can be of different nature, as in mechanical, electrical, or chemical systems, they all share the characteristic of recursiveness. He provided a relationship in which the details of the recursiveness become irrelevant, through a kind of universal parameter, measuring the ratio of the distances between successive period doublings, the so called Feigenbaum's delta. His understanding of the phenomenon was later experimentally confirmed [48], so that today we refer to the so-called Feigenbaum's period doubling cascade. However, even if the period doubling has a long history, only recently it has been experimentally observed in the dynamics of cracked structures [46]. To highlight the influence of the crack on the beam dynamics, let us consider two different numernical examples: a wikely nonlinear structure and a strongly nonlinear one. Only in the latter case the so called period doubling phenomenon clearly appears. Details about the beam geometry and materials can be found in [10]. For each of the two considered structures (Figs. 6a and 6b) the trajectory in the phase space is represented in Figs. 7a and 7b. In a hypothetical linear structure, the structural response is linear by definition with obviously only one harmonic component at the same frequency of the excitation. In the weakly nonlinear structure of Fig. 6a, the response converges and it appears only weakly nonlinear. The trajectory in the phase diagram is close to an ellipse. The diagram is nonsymmetric as the spatial positions of the cracks (placed in the upper part of the beam). The trajectory is an unique closed curve since here the period of the response is equal to the period of the excitation.

Figure 6: Damaged structures: weakly nonlinear (a) and strongly nonlinear (b).

Figure 7: Dimensionless phase diagram of the response (free end displacement): weakly (a) and strongly (b) non linear structure.

In the strongly nonlinear structure of Fig. 6b the nonlinearity increases. The harmonic components in the structural response are the zero one, the superharmonics as well as the subharmonic ones. It should be emphasized that a strong nonlinearity causes the period doubling of the response, i.e., the ω/2 component. The free-end vibrates practically with a period doubled with respect to the excitation. A nonnegligible component at ω/4 is observed too, representing a route to chaos through a period doubling cascade. The corresponding phase diagram clearly evidences this: the trajectory is composed by multiple cycles since here the period of the response is not equal to the period of the excitation. The




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distortions in the trajectory are consequences of the presence of the super- or subharmonics. Also in this case, the diagram is nonsymmetric as the spatial positions of the cracks. This method is able to catch the transition toward deterministic chaos, like the occurrence of a period doubling, as shown in the numerical examples and experimentally observed in the context of cracked beam by Brandon and Sudraud [46]. CONCLUSIONS

he so-called "Complexity Sciences" represent a subject of fast-growing interest in the Scientific Community. They have entered also our more circumscribed Communities of Material Science and Material Strength, as the proposed examples may confirm. The presented topics were concerned with the structural behaviour of

composite structures with snap-back instabilities (an example of cusp catastrophe), the occurrence of fractal patterns and geometrically self-similar morphologies in deformation, damage and fracture of heterogeneous materials, the apparent scaling in the nominal mechanical properties of disordered materials, the acoustic emission criticality in progressive structural collapse, the route towards chaos in the dynamics of cracked structures. As shown in these examples, the most interesting behaviors and phenomena can be synthetically interpreted only through the use of new and refined conceptual tools in the framework of "Complexity Sciences". ACKNOWLEDGEMENTS

he authors would like to gratefully acknowledge the contributions made to this work by all members of the research group led by the senior author at the Department of Structural Éngineering and Geotechnics of the Politécnico di Torino. In particular, the warmest thanks go to Giuseppe Ferro, Nicola Pugno, Pietro Cornetti and

Giuseppe Lacidogna. Support by the European Community is gratefully acknowledged by the authors. Thanks are also due to the Italian Ministry of University and Research (MIUR). REFERENCES [1] M.S. Garrido, R.Vuela Mendes, Complexity in physics and technology, World Scientific, Singapore, (1992). [2] A. Carpinteri, S. Puzzi, Strength, Fracture and Complexity, 4 (2006) 189. [3] A. Carpinteri, S. Puzzi, Strength, Fracture and Complexity, 4 (2006) 201. [4] A. Carpinteri, in Application of Fracture Mechanics to Cementitious Composites, edited by S.P. Shah, Martinus

Nijhoff Publishers, Dordrecht, (1985) 287. [5] A. Carpinteri, J. Mech. Phys. Solids, 37 (1989) 567. [6] A. Carpinteri, Int. J. Frac., 44 (1990) 57. [7] A. Carpinteri, Mech. Mater., 18 (1994) 89. [8] A. Carpinteri, Int. J. Solids Struct., 31 (1994) 291. [9] A. Carpinteri, G. Lacidogna, N. Pugno, Engng. Frac. Mech., 74 (2007) 273. [10] A. Carpinteri, N. Pugno, J. Appl. Mech., 72 (2005) 511. [11] A. Carpinteri, N. Pugno, J. Appl. Mech., 72 (2005) 519. [12] R. Thom, Structural Stability and Morphogenesis: an Outline of a General Theory of Models. Benjamin (1975). [13] G. I. Barenblatt, J. Appl. Math. Mech., 23 (1959) 622. [14] D. S. Dugdale, J. Mech. Phys. Solids, 8 (1960) 100. [15] A. Carpinteri, P. Cornetti, F. Barpi, S. Valente, Engng. Frac. Mech., 70 (2003) 1809. [16] A. Hillerborg, M. Modeer, P.E. Petersson, , Cement Concr. Res., 6 (1976) 773. [17] A. Carpinteri, P. Cornetti, S. Puzzi, Appl. Mech. Rev., 59 (2006) 283. [18] A. Carpinteri, Int. J. Solids Struct., 25 (1989) 407. [19] A. Carpinteri, Engng. Frac. Mech., 32 (1989) 265. [20] Determination of the fracture energy of mortar and concrete by means of three-point bending tests on notched

beams. Technical Report 18, Materials and Structures, R.I.L.E.M. (1985). [21] L. Biolzi, S. Cangiano, G.P. Tognon, A. Carpinteri, Mater. Struct., 22 (1989) 429. [22] B.B. Mandelbrot, The Fractal Geometry of Nature. New York: Freeman (1982).

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[23] A. Carpinteri, G. Ferro, Mater. Struct., 28 (1994) 563. [24] A. Carpinteri, G. Ferro, Mater. Struct., 31 (1998) 303. [25] A. Carpinteri, B. Chiaia, P. Cornetti, Eng. Frac. Mech., 69 (2002) 207. [26] A. Carpinteri, B. Chiaia, K.M. Nemati, Mech. Mater., 26 (1997) 93. [27] A. Carpinteri, B. Chiaia, S. Invernizzi, Theor. Appl. Frac. Mech., 31 (1999) 163. [28] C.F. Richter, , Elementary Seismology, W.H. Freeman & Company, San Francisco and London (1958). [29] H. Rusch, , Zement-Kalk-Gips (Wiesbaden), 12 (1959) 1. [30] R.G. L'Hermite, in Proc. 4th Int. Symp. on Chemistry of Cement, V-3. NBS Monograph 43, NBS, Washington DC,

(1960) 659. [31] G. S. Robinson, in Proc. Int. Conf. on the Structure of Concrete and Its Behavior Under Load, Cement and Concrete

Association, (1965) 131. [32] W. M. McCabe, R. M. Koerner, A. E. Jr. Load, , ACI Journal, 13 (1976) 367. [33] P. Shah, L. Zongjin, , ACI Mater. J., 91 (1994) 372. [34] M. Ohtsu, , Magazine Concr. Res., 48 (1996) 321. [35] A. Carpinteri, G. Lacidogna, in Proc. of STREMAH VII (Bologna, 2001), WIT Press, Southampton, (2001) 327. [36] A. Carpinteri, G. Lacidogna, Italian Patent N. To 2002 A000924, deposited on 23 October 2002. [37] A. Carpinteri, G. Lacidogna, N. Pugno, in Fracture Mechanics of Concrete and Concrete Structures (Proceedings of

the 5th International FraMCoS Conference, Vail, Colorado, USA, (2004), edited by V.C. Li et al., 1 (2004) 31. [38] A. Carpinteri, N. Pugno, Magazine Concr. Res., 54 (2002) 473. [39] A. Carpinteri, N.Pugno, Int. J. Numer. Anal. Methods Geomech., 26 (2002) 499. [40] A. Carpinteri, N. Pugno, Proceedings of the 9th International Congress on Sound and Vibration, Orlando, USA, CD-

ROM, (2002) paper N. 114. [41] P. Gudmundson, J. Mech. Phys. Solids, 31 (1983) 329. [42] M.I. Friswell, J. E. T. Penny, in Proc. X Int. Modal Analysis Conf., (1992) 516. [43] W. Ostachowicz, M. Krawczuk, Comput. Struct., 36 (1990) 245. [44] R. Ruotolo, C. Surace, P. Crespo, D. Storer, Comput. Struct., 61 (1996) 1057. [45] N. Pugno, C. Surace, R. Ruotolo, J. Sound Vib., 235 (2000) 749. [46] J. A. Branden, C. Sudraud, , J. Sound Vib., 211 (1998) 555. [47] M. J. Feigenbaum, J. Stat. Phys., 19 (1978) 25. [48] P.S. Linsay, Phys. Rev. Lett., 47 (1981) 1349.



D. Taylor et alii, Frattura ed Integrità Strutturale, 10 (2009) 12-20; DOI: 10.3221/IGF-ESIS.10.02

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The theory of critical distances applied to problems in fracture and fatigue of bone

David Taylor, Saeid Kasiri, Emma Brazel Engineering School, Trinity College, Dublin 2, Ireland [email protected]

ABSTRACT. The theory of critical distances (TCD) has been applied to predict notch-based fracture and fatigue in a wide range of materials and components. The present paper describes a series of projects in which we applied this approach to human bone. Using experimental data from the literature, combined with finite element analysis, we showed that the TCD was able to predict the effect of notches and holes on the strength of bone failing in brittle fracture due to monotonic loading, in different loading regimes. Bone also displays short crack effects, leading to R-curve data for both fracture toughness and fatigue crack propagation thresholds; we showed that the TCD could predict this data. This analysis raised a number of questions for discussion, such as the significance of the L value itself in this and other materials. Finally, we applied the TCD to a practical problem in orthopaedic surgery: the management of bone defects, showing that predictions could be made which would enable surgeons to decide on whether a bone graft material would be needed to repair a defect, and to specify what mechanical properties this material should have. KEYWORDS. Bone; Fracture; Fatigue; Critical Distance. INTRODUCTION

he critical distance approach is now well established as a method for the prediction of fatigue and fracture, and is being used extensively both in research and in engineering design. A recent book [1] describes the approach in detail. It is applicable for predicting failure in bodies containing notches or other stress concentrations, in

situations where the mechanism of failure is one involving cracking. It has been employed by many workers for the solution of problems which can be described as essentially linear-elastic, i.e. problems in which any non-linear material behaviour (due to plasticity or damage) is localised in a small process zone: in this respect it has been used to predict brittle fracture and fatigue in all types of materials: metals, polymers, ceramics and composites. The history of this type of use goes back more than fifty years: more recent work has shown that the approach can also be applied to problems involving more extensive plasticity, such as low and medium-cycle fatigue and the static fracture of tough metallic materials. The present paper is concerned with the application of these methods, hereafter referred to as the Theory of Critical Distances (TCD), to the prediction of a number of fracture problems in a particular material which is of interest to us all: human bone. What follows is essentially a summary of work conducted in our research group over the last four years, published previously in a number of journal articles. We hypothesised that the TCD could be applied to human bone, because bone is a quasi-brittle, fibrous composite material whose mechanical behaviour has many similarities with that of two well-known classes of engineering materials, namely fibre reinforced polymers and concrete. The TCD has previously been applied successfully to both of these types of materials [2, 3]. The mechanism of failure in bone always involves cracking, and the failure process is accompanied by both plasticity (of a limited but significant extent) and damage (in the form of microcracks, delaminations etc). We attempted to use the TCD to predict experimental data, taken from the literature, on the monotonic fracture of bone samples containing cracks, notches and holes, and on the fatigue behaviour

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D.Taylor et alii, Frattura ed Integrità Strutturale, 10 (2009) 12-20; DOI: 10.3221/IGF-ESIS.10.02

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of short cracks. Following the success of this work, we then applied the approach to some problems of clinical significance. One example of this type of work is described: the surgical management of bone defects. THE THEORY OF CRITICAL DISTANCES: A BRIEF INTRODUCTION

hat follows is a very brief introduction to the TCD: further details are available in [1] and in many other recent publications. The TCD recognises the fact that, in order to predict failure arising from a stress concentration feature such as a notch, it is not sufficient to know the stress and strain at the notch surface, at the maximum

stress point, often known as the “hot spot”. Rather, it is essential to have information about the stress field in the vicinity of the notch, because fracture processes that involve crack initiation and propagation are strongly influenced by aspects of the stress field in this region, such as the gradient of stress or, to put it another way, the absolute volume of material which is experiencing high stress. This recognises the fact that cracking-type failures require, in general, a solution of the type which is now generally referred to as a “non-local approach”, characterised by a physical mechanism of failure involving a process zone in the vicinity of the crack tip in which failure, deformation and damage processes occur. A variety of methods, more or less complex, have been devised to make predictions using stress and strain information in this critical region. In our most strict definition of the TCD, it consists of a group of methods which have the following two features in common. Firstly, the use of a linear, elastic material model for the stress analysis. Secondly, the use of a material parameter which has the units of length, known as the critical distance, L. The value of L cannot be known a priori; it can only be found by processing data from samples containing stress concentrations, tested to failure in the particular failure mode of interest. It is taken to represent a critical dimension in the material over which relevant failure processes occur. For example, in many cases it is found to be related to critical microstructural parameters such as grain size, which are known to control the material’s strength and toughness: this relationship will be discussed further below. Having stated this strict definition, it is important to point out that exceptions do occur, in which the TCD is used in cases where these conditions are violated. For example it may be appropriate to use a non-linear material model, and we have indeed done so ourselves as will be discussed below. Also, some realisations of the theory make use of a value of L which is not a material constant [4, 5], though these will not be considered in the present paper. We can define two different types of TCD methods. In the first type, predictions are made using information about the stress field, specifically the stress as a function of distance from the hot spot, on a line (known as the focus path) along which crack growth is expected to occur. The simplest example of this approach is the so-called Point Method, which uses only the stress at a given point, located a distance L/2 from the hot spot. Failure is predicted to occur if the stress at this point exceeds a critical value. A variant of this approach is the Line Method, in which the stress parameter is the average stress along the line, over a distance from zero to 2L from the hot spot. Area and volume averages have also been used, though these more complex methods do not seem to confer any more accuracy than the simple point and line methods. The second type of TCD method involves a modification of fracture mechanics, whereby the critical distance appears as the length of an imaginary crack located at the notch, or, alternatively, as the magnitude of finite crack growth increments [6]. Once such a modification is accepted, normal linear-elastic fracture mechanics approaches can be used. In what follows we will use approaches of the first type, i.e. stress-based methods, making use of finite element analysis (FEA) to obtain the appropriate stress fields. INITIAL VALIDATION: NOTCH FRACTURE DATA

e obtained from the published literature three extensive sets of data on the effect of notches on brittle fracture in bone. All three involved tests in which monotonically-increasing loads were applied until failure occurred. One publication [7] was concerned with the effect of notch length for sharp notches machined in bone

samples, whilst the other two [8, 9] reported the results of tests conducted on whole bones, loaded in bending and torsion respectively, containing circular holes of various sizes. We found that the TCD was able to predict all this data. Fig.1 shows an example: the effect of hole size on failure load for bones loaded in torsion. Further details can be found in a recent publication [10]. At this point it may be worth pointing out that the TCD can be used with any type of applied loading, including multiaxial load cases, though an appropriate multiaxial failure criterion should be used. In the present study our criterion was simply the maximum principal stress: we have reported the use of other multiaxial criteria to predict fatigue and fracture in various materials, in an extensive series of previous publications (e.g. [11-13] ).

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Note in particular from Fig. 1 that the TCD was able to predict the fact that small holes (hole diameter = 0.1 x bone diameter) have no effect on strength, a very useful finding for clinicians and one that was not predicted by other approaches. In the TCD approach this finding arises because if a notch is very small, the critical distance (being constant) becomes effectively much larger than the hole, so the stress at the critical point is similar to the nominal applied stress: effectively the hole has become “invisible” as far as this approach is concerned. A particularly encouraging aspect of this validation exercise was the fact that the appropriate value of the critical distance was found to be almost constant across the three sets of data. It is well known that other mechanical properties of bone, such as stiffness and strength, vary considerably, so we had expected that L would also vary, but this seems not to be the case: a value of 0.32-0.38mm was able to give good predictions throughout.

Figure 1: The effect of hole diameter (normalized by bone diameter) on fracture torque (normalized by fracture torque for bones containing no hole), for whole bones tested in torsion, containing single transcortical

holes of various diameters. Predictions using the TCD and two other theories.

The same situation arises in fibre composite materials, which are also known to have only a small range of L values [14]. In those materials, strength and toughness are roughly proportional to each other over quite a wide range of values, so that an increase in strength (for example by increasing the proportion of fibres) confers a similar proportional increase in toughness. An equation can be derived which links three material constants used in the TCD: L, Kc and the critical stress for failure o, as follows:

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o

cKL

(1)

Since L is related to the ratio of strength to toughness, it stays constant if these two properties change in a proportionate manner. The critical stress parameter defining failure in bone, o, was found to be slightly larger than the material’s tensile strength, u as measured from tests conducted on plain, unnotched samples. We found that accurate predictions could be made using a critical stress of Tu where T had a constant of value 1.33. This finding is in line with our investigations of other materials, in which the value of T has been found to take values close to 1.0 for brittle ceramics and composites [15], in the range 1.4-3 for polymers [1] and values typically greater than 3 for metals [16]. A precise interpretation of the meaning of the T parameter is still unclear. In considering the significance of this value it is worth noting the link between the three parameters of toughness, strength and L, as shown in equation 1 above. If two of these constants are known, the third can be calculated, which implies that only two of these three constants are of fundamental significance. In my personal opinion, the two fundamental parameters are L and Kc. The value of o differs from that of u, in my view, because of two assumptions which we make in this analysis. Firstly, we assume that the material is linear and elastic, which of course it is not. It is significant that values of T become larger in materials and fracture processes involving more plasticity. Secondly, we assume that the mechanism of failure in a plain specimen is the same as that in a notched specimen. This is clearly not the case in some materials: plain specimens may fail differently due to, for example, plastic instability (necking) in ductile materials or the presence of pre-existing defects in brittle materials. It is interesting to note that, when we




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conducted a different analysis of bone, to predict indentation fracture, for which a non-linear material model was needed, we found that T=1 [13]. SHORT CRACK BEHAVIOUR IN FATIGUE AND BRITTLE FRACTURE

t is well known than short cracks often display behaviour which does not conform to linear elastic fracture mechanics. For example, the values of toughness (Kc) and fatigue threshold (Kth) for short cracks are often smaller than the material-constant values measured from long cracks. Data in which the measured Kc (or Kth) is plotted as a

function of crack length are known as resistance curves, or R-curves. Figs 2 and 3 show R-curve data for bone, for brittle fracture and fatigue respectively, along with predictions using the TCD. In this case the analysis can be made very easily using the Line Method, because predictions for the case of a small crack (length a) in an infinite body can be expressed using the following simple equation:

La

a

K

K

c

ca

(2)

Figure 2: Two sets of data showing the variation of measured fracture toughness as a function of

crack length for bone along with TCD predictions. For more details see [17].

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Figure 3: Threshold stress intensity range for fatigue crack growth in bone, defined

at a crack growth rate of 3-6 x 10-8m/cycle; for details see [17]. As can be seen from the figures, the predictions are very satisfactory for both types of failure, even including data for very small crack lengths obtained using nanoindentation experiments [18]. It should be remarked that currently there is considerable controversy in the literature about the validity of measuring toughness using indentation, a technique which has been used for brittle material for many years but which is now being seriously questioned. The interested reader may wish to refer to recent letters in the Journal of Biomechanics arising from the publication by Mullins et al [18]. Currently, indentation is one of the few options available for estimating the toughness of materials at very small length scales, a subject which is of increasing interest given the advent of micro and nano scale materials and devices. The data in Figs 2 and 3 here all refer to crack growth in the transverse direction: bone is highly anisotropic so further work is needed to explore fracture properties in different directions. In making these predictions we used the same value for L as previously obtained from the predictions of notch fracture behaviour. This implies that L takes the same value in fatigue as in brittle fracture in this material, at least for cracking in the transverse direction. We have previously found significant differences between L values for fatigue and brittle fracture in metallic materials, but similar values for a polymer, PMMA. In the present case the fatigue data available are relatively sparse, so further validation is needed before this conclusion can be stated with confidence. It is perhaps worth considering at this stage why bone has this particular value of L. As noted above, L values for many materials are often related to the size of microstructural features which control fracture behaviour. Bone has a hierarchical structure, displaying features at a range of size scales, especially nanometres (the thickness of reinforcing crystals of hydroxyapatite), microns (the thickness of lamellae consisting of crystals and collagen fibres in a composite structure) and hundreds of microns (the size and spacing of structural units known as osteons). A number of mechanisms operating at the hundred-micron scale have been identified, notably uncracked ligaments bridging the crack faces [19] and the role of the osteon boundary in crack arrest (O'Brien et al., 2005), in a manner similar to the grain boundary in metals. Figs 4 and 5 show examples of these mechanisms. Ritchie and co-workers have investigated these mechanisms in some detail and have laid particular emphasis on the role of uncracked ligaments. They showed a definite relationship between the rising R-curve for a given crack and the increasing number of uncracked ligaments observed as the crack extended [20]. In our studies on high-cycle fatigue in bone we have placed emphasis on the role of the osteon boundary, showing that the great majority of fatigue cracks become non-propagating when they reach the first boundary and developing relationships between crack length, growth rate and the proximity of this boundary [21, 22]. All of these various observations imply that L takes a value equal to a few hundred microns because this corresponds to the size scale on which important toughening mechanisms operate in this material. In fact, this turns out to be the case for many different materials. Fig.6 shows the value of L for various different classes of materials, plotted against the relevant structural parameter d. In some cases there are very clear and demonstrable relationships between L and d: for example we showed that L takes values very close to d in steels failing by brittle cleavage fracture at low temperatures [23]. In other cases the relationship is less clear but for most materials it seems that L falls between d and 10d in magnitude. There are, however, some important exceptions: for example amorphous polymers such as PMMA have no microstructure as such, and yet have L values of the order of 0.1mm. This coincides with the typical size of crazes in the material.

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It is perhaps not surprising that there should be a relationship between L and d, since in most materials the microstructure plays a strong role in determining properties related to crack growth, such as toughness and fatigue behaviour. Thus, knowing a value of L for a particular material may shed light on the physical mechanism of failure and may give hints about how changing microstructural parameters could affect performance.

(a) (b) Figure 4: Two SEM images showing cracks in bone which display bridges consisting of uncracked ligaments across the crack faces.

Photo (a) from tests conducted in our laboratories by Stewart Mahoney; photo (b) from [19].

Figure 5: Image taken using optical fluorescence microscopy of a transverse section of bone, showing a crack (C), of length approximately 100m, whose left-hand tip has stopped growing on reaching

the boundary of an osteon (O). From [21].

Figure 6: Values of L and d in various classes of materials.

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tica

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ista

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, L

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PRACTICAL APPLICATIONS: THE MANAGEMENT OF BONE DEFECTS

great advantage of the TCD is that it can be applied very easily to practical problems; in this respect the stress-based methods are particularly attractive because they can be used in any situation where a stress analysis can be conducted using FEA or similar numerical techniques.

Stress concentrations frequently arise in bone as a result of disease or clinical intervention. Surgeons use the terminology “bone defect” to refer to any hole which occurs in a bone, i.e. any part of the bone cortex or internal cancellous structure which is missing. Defects occur for various reasons, for example they may also arise following a complex fracture: when the broken parts of a bone are reassembled there may be some pieces missing. Also, holes may be deliberately drilled to take samples for biopsy or for the fixation of fracture plates which may be later removed. One method for the replacement of the anterior cruciate ligament in the knee involves taking a piece of bone from the patella of the other knee, often leaving a square hole with sharp corners. In a previous study we showed that the impact energy of this patella was significantly reduced by the presence of the hole, and that the situation could be considerably improved by cutting a hole with round corners [24]. This is a good example of how a concept which is very obvious to the mechanical engineer can have immediate benefits in the field of medicine. If the hole is considered to confer significant risk of failure, the surgeon may fill it using a bone graft material. Various types of materials are used, including the patient’s own bone (taken from some other site and ground into a powder) and various artificial materials. Over a period of time, the patient’s own natural healing processes will cause the hole to be filled with new, living bone, so the bone graft material is intended only as a temporary substitute, required to last for a few months at the most. Artificial bone graft materials are designed to provide a scaffold for the rapid ingrowth of bone, and recently there has been much interest in the use of tissue engineering techniques for the development of these materials. Scaffolds have been made from a wide variety of materials, including porous metals, ceramics and hydrogels. There is great interest in the use of resorbable materials which can gradually dissolve, aiding the development of new bone, but current versions of these materials are relatively weak, increasing the risk of fracture in the critical period just after surgery. A major problem is the lack of a predictive model to aid surgeons in deciding what to do about a given defect, whether to use a bone graft material and, if so, what the properties of that material should be. Such a predictive model, presented in the form of a computer simulation of the defective bone, could greatly aid in the planning of surgical operations. As an initial step towards developing such a tool, we carried out some simple simulations of the behaviour of bone defects. Fig 7 shows the geometry used for the finite element model: the bone is envisaged as a simple tube, containing a defect: we modelled square and circular holes of various sizes. A complete description of the methodology and results can be found in a recent publication [25]. In brief, we used a damage mechanics approach to predict the increase of fatigue damage due to cyclic loading in normal daily activities. The TCD was incorporated by performing all the damage calculations at the critical point, i.e. a distance L/2 from the hole, rather than at the hot spot. The capacity of bone to repair itself was included in the model as a constant, negative damage rate, following earlier work [26]. The use of different bone graft materials was modelled by filling the hole with a material of given Young’s modulus, Eo. Bone ingrowth was included in the simulation by allowing the Young’s modulus of the graft material to gradually increase over time, from Eo to a value typical for normal cortical bone (17GPa). Fig. 7 shows an example of the results of the simulation. If repair and ingrowth are not modelled, damage increases rapidly. Incorporating ingrowth causes damage to level out to a plateau value, and the additional incorporation of repair allows the damage to return to normal levels after peaking. The value of the peak is of course the critical one: provided this is less than unity we predict that no failure will occur.As Fig.8 shows, there is a very strong effect arising from the value of Eo, the stiffness of the bone graft material. This occurs because the stress concentrating effect of the hole is greatly reduced, even when the material in the hole has much less stiffness than the surrounding bone. This analysis enabled us to make a specification for a safe value of Eo,as a function of hole size as shown in Fig.8. Obviously the result also depends on the shape of the hole, and on the assumed daily loading, i.e. the activity level of the person. These predictions show that small holes (in this case less than 5mm diameter) do not need to be filled in with graft material: this finding is in agreement with the current practice of surgeons who regard such small holes as innocuous. Larger holes do require filling, but here we predict that the material needed can have an Eo value which is considerably smaller than that of normal bone: this finding is original and potentially of great value to researchers and manufacturers who are developing new types of bone graft materials. This work is very preliminary in nature, but has the potential to be developed to a greater level of sophistication, for example incorporating the changing behaviour of resorbable materials and the effect of different postoperative activity levels, such as walking with the support of a crutch or cane or carrying out more strenuous exercise.

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Figure 7: The geometry used to study defects of various shapes and sizes in a typical long bone – the focus path is the line on which TCD calculations are carried out. Typical predictions showing the effect on damage

evolution of including ingrowth of bone into the defect, and bone repair processes.

(a) (b)

Figure 8: (a) Variation of peak damage amount with Young’s modulus of the bone graft material; (b) Specification for the safe value of Eo (i.e. the value above which failure will not occur) as a function of hole size.

CONCLUDING REMARKS

his work has shown that the TCD can be used to study fracture and fatigue problems in bone. Classic problems which the TCD has been able to solve in other materials, such as notch-initiated fracture and fatigue and the short crack problem, have been successfully addressed in this material. Even though bone shows large variations in its

mechanical properties, it seems that L remains approximately constant, of the order of 0.3-0.4mm, which is very convenient when making predictions. This value reflects the role of osteons and other microstructural features in impeding crack growth and thus controlling toughness and fatigue. Current work has been limited to cases where crack growth occurs across the bone, i.e. in the transverse direction: longitudinal crack growth requires separate study. We have also limited ourselves to cortical bone: the failure of spongy, cancellous bone is also of great interest and merits further attention. The TCD can be employed as part of a practical software tool to aid orthopaedic surgeons in the planning of operations and of post-operative treatments. ACKNOWLEDGEMENTS

e are grateful to the Higher Education Authority of Ireland for provision of funding for part of the work described above, which was conducted in collaboration with the Institute of Technology, Sligo, Ireland.

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REFERENCE LIST [1] D. Taylor, The Theory of Critical Distances: A New Perspective in Fracture Mechanics. Elsevier, Oxford, UK (2007). [2] J. M. Whitney, R.J. Nuismer, Journal of Composite Materials, 8 (1974) 253. [3] P. Cornetti, N. Pugno, D. Taylor, Proceedings of the 11th International Conference on Fracture ESIS, Turin, Italy

(2005) 73. [4] P. Cornetti, N. Pugno, A. Carpinteri, D. Taylor, Engineering Fracture Mechanics, 73 (2006) 2021. [5] D. Leguillon, European Journal of Mechanics A/Solids, 21 (2002) 61. [6] D. Taylor, P. Cornetti, N. Pugno, Engineering Fracture Mechanics, 72 (2005) 1021. [7] W. Bonfield, P.K. Datta, Journal of Biomechanics, 9 (1976) 131. [8] R.J. McBroom, E.J. Cheal, W.C. Hayes, Journal of Orthopaedic Research, 6 (1988) 369. [9] J.A. Hipp, B.C. Edgerton, K.N. An, W.C. Hayes, Journal of Biomechanics, 23 (1990) 1261. [10] S. Kasiri, D. Taylor, Journal of Biomechanics 41 (2008) 603-609. [11] L. Susmel, Fatigue and Fracture of Engineering Materials and Structures, 27 (2004) 391. [12] F. Pessot, L. Susmel, D. Taylor, in Crack Paths Conference Parma, Italy (2006). [13] S. Kasiri, G. Reilly, D. Taylor, WIT Transactions on Biomedicine and Health, 12 (2007) 113. [14] J. Awerbuch, M. S. Madhukar, Journal of Reinforced Plastics and Composites, 4 (1985) 3. [15] D. Taylor, Engineering Fracture Mechanics, 71 (2004) 2407. [16] D. Taylor, Structural Integrity and Durability, 1 (2006) 145. [17] D. Taylor, S. Kasiri, in Proc ASME Summer Bioengineering Conference ASME, USA (2008). [18] L. P. Mullins, M. S. Bruzzi, P. E. McHugh, Journal of Biomechanics, 40 (2007) 3285. [19] Nalla,R.K., Kinney,J.H., and Ritchie,R.O. (2003) Mechanistic fracture criteria for the failure of human cortical bone.

Nature Materials 2, 164-168. [20] R. K. Nalla, J. S. lken, J. H. Kinney, R. O. Ritchie, Journal of Biomechanics, 38 (2005) 1517. [21] F. J. O'Brien, D. Taylor, T.C. Lee, Journal of Orthopaedic Research, 23 (2005) 475. [22] D. Taylor, F. O'Brien, T. C. Lee, Meccanica, 37 (2002) 397. [23] D. Taylor, Microstructural parameters in the theory of critical distances (2008). [24] K. Moholkar, D. Taylor, M. O'Reagan, G. Fenelon, Journal of Bone and Joint Surgery, 84A (2002) 1782. [25] E. Brazel, D. Taylor, Predicting the Structural Integrity of Bone Defects Repaired Using Bone Graft Materials;

Computer Methods in Biomechanics and Biomedical Engineering, in press. [26] P. J. Prendergast, D. Taylor, Journal of Biomechanics, 27 (1994) 1067.



A. Pirondi, Frattura ed Integrità Strutturale, 10 (2009) 21-28; DOI: 10.3221/IGF-ESIS.10.03

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Failure prediction of T-peel adhesive joints by different cohesive laws and modelling approaches

Alessandro Pirondi University of Parma, Department of Industrial Engineering, Parco Area delle Scienze, 181/A - 43100 Parma, Italy [email protected]

RIASSUNTO. In questo articolo si è simulato mediante il modello di zona coesiva il cedimento di un giunto T-peel incollato. Gli aderendi sono lamiere di acciaio Fe360 e sono unite mediante l’adesivo strutturale Loctite Multibond 330. I parametri del modello di zona coesiva sono stati calibrati sulla base di esperimenti di frattura condotti in precedenza su provini Double Cantilever Beam (DCB) incollati con il medesimo adesivo. La simulazione è stata condotta utilizzando il software di analisi ad elementi finiti ABAQUS, sviluppando modelli 2-D. Il cedimento avviene in uno strato modellato utilizzando elementi di tipo coesivo disponibili nel software. L’analisi è volta ad individuare l’influenza di: i) differenti formulazioni della legge coesiva, ii) la modellazione o meno dello strato di adesivo con le sue proprietà elasto-plastiche. ABSTRACT. In this work, Cohesive Zone Modelling (CZM) was used to simulate failure of T-peel bonded joints with 1.5mm thick adherends, respectively, bonded toghether with Loctite Multibond 330 adhesive. The fracture toughness and load-opening behaviour recorded in previous experiments on bonded Double Cantilever Beam (DCB) specimens were taken as reference to calibrate CZM parameters. Two-dimensional models were analysed using the FE code ABAQUS. The failing interface was modeled with the cohesive elements available in this software. The influence of: i) different cohesive law shapes, ii) modeling the presence of the adhesive layer explicitly, was studied. KEYWORDS. Adhesive joints, Fracture, Cohesive zone modeling. INTRODUCTION

he use of adhesive joining in civil, aerospace and mechanical constructions has considerably increased in the last decades thanks to the advantages over traditional joining techniques such as: i) ability to join dissimilar materials, ii) stress distribution over a wider area, iii) potentially lower weight. However, joint fabrication procedures and

component service loads may introduce or initiate defects, whose evolution will control the performance and the reliability of the bonded joint. In those cases, Fracture Mechanics (FM) can be used to assess the structural integrity of a bonded joint [1]. The FM approach consists in the comparison of a parameter, function of load and geometry of the cracked body (for example the strain energy release rate, G), with the fracture resistance (Gc). The simulation of fracture therefore requires to implement a criterion that triggers propagation when G=Gc. An attractive way to simulate the effect of a defect on joint strength is to incorporate a model of the rupture process (i.e. the criterion to trigger propagation). In particular, the fracture of bonded joints has been successfully simulated using the Cohesive Zone Model (CZM) in [2-7]. According to this approach, the zone in front of the physical crack tip opens and then tears progressively apart following a given traction-separation behaviour. Although straightforward methods to evaluate experimentally CZM parameters in adhesive joints have been recently presented [8], questions on the physical meaning or, in other words, on the transferability of the parameters to joint geometries different from the one from they were extracted is still an open issue. In particular, studies have been carried

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out about the influence of the thickness of the adhesive layer and of the adherends [9-11] that can be seen also as the evaluation of different degrees of constraint. The constraint effect in homogenous materials is well known to arise from the out-of-plane width and from the in-plane interaction of the crack with the boundaries. In the case of a crack in an adhesive joint, the adhesive layer thickness defines the distance of the boundaries (i.e. the adherends) from the crack tip. The flexibility of the boundaries plays of course also an important role in the constraint imparted to the crack. In a previous work [12], 2D CZM analyses were carried out to simulate failure of T-peel bonded joints) with 1.5 and 3mm thick adherends, respectively. A trapezoidal cohesive law was used and a CZ was introduced using nonlinear springs. The fracture toughness and load-opening behaviour recorded in experiments on bonded DCB specimens [13] were taken as reference to calibrate CZM parameters. This work is aimed at extending those results. The failing interface is modeled with the cohesive elements available in this software. The influence of: i) different cohesive law shapes, ii) modelling the presence of the adhesive layer explicitly, is studied. EXPERIMENTAL Materials

he DCB aluminum adherends (E=70GPa, =0.3) were bonded with Loctite 330, a modified methacrilate ester supplied as viscous paste plus activator. Loctite 330 is a structural adhesive suitable for bonding of metals, wood, ceramics and plastics. Its advantage with respect to other structural adhesives is that it does not require specific

surface preparation (only degreasing) and the joint can be handled after five minutes, allowing for higher production rates. Shear and tensile strength declared by the supplier are about 20MPa. The tensile behavior of the adhesive was modeled as a power law:

E 0 (1) 00n 0 (1bis)

where E = 878MPa, = 4.8MPa, n = 0.44 and = /E were determined by tensile testing of bulk adhesive [13]. The Poisson's ratio was 0.15. The T-peel joints were made of italian standard UNI Fe 360 unalloyed steel, whose properties are typically E=210GPa, =0.3, u=360MPa, =240MPa. DCB tests The specimen dimensions are shown in Fig. 1. A four-step procedure was used for surface preparation of the substrates before bonding: i) polishing with sandpaper, ii) degreasing with acetone, iii) rinsing in hot water, iv) drying. The procedure was applied twice, initially using a coarse sandpaper and then a fine sandpaper. The recommended bondline thickness for Multibond 330 ranges between 0.05 to 0.5 mm. In all the experiments reported here, a thickness of 0.25 mm was adopted. Bond thickness control during specimen preparation was achieved placing two controlled-thickness copper shims between the adherends (see Fig. 1a). A 0.05-mm-thick Teflon tape was placed just before bonding at mid-thickness of the adhesive layer to obtain a cohesive crack-like defect. The test were performed after at least a 24-hrs curing at room temperature, that is the minimum time to fully develop the mechanical strength according to the datasheet of the adhesive manufacturer.

h

h t

a W b

a0 = 40 mm W = 120 mm h = 15 mm t = 0.3 mm b = 30 mm

calibrated shims teflon

Figure 1: Outline of the DCB joint.

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To account for the elasticity of the adhesive, the DCB specimen can be modeled as a beam on an elastic foundation, where the foundation modulus, k, will depend on the elastic constants of the adhesive (i.e. Ea and a) and on the bondline thickness, t. The solution developed in [14] neglecting shear contributions was therefore considered. The load line compliance is given by the following equation:

32'a

a32a2a21

bEt2

PC (2)

where E’a=Ea/(1-a2) is plane strain modulus of the adhesive and E'E

th6

EIk4 a

34 . The strain energy release rate G is

given by 2222

a11

bEIaP

aC

b2PG

(3)

For the present DCB specimen the term (a)-1 is about 0.25. T-peel tests A series of tensile tests was conducted at the Polytechnic of Turin [15] on UNI Fe360 construction steel T-peel joints (Fig. 2).

Figure 2: Outline of half of the 1.5mm-thick T-peel joint tested in [15].

A metal sheet thickness of 1.5mm was investigated. The adhesive used is again the Loctite 330. A bondline thickness of 0.1mm was obtained in this case. Displacement-controlled tests were conducted up to partial or complete separation of the two halves. MODELLING Fe models

he 2D FE models corresponding to DCB and T-peel joints are shown in Figs. 3a-b. Symmetry conditions with respect to the bondline have been applied. Plane stress, eight-noded isoparametric elements have been used to simulate the metal adherend.

The cohesive zone has been modelled with the cohesive elements available in the FE code ABAQUS. A convergence analysis was preliminarly conducted to determine the appropriate element size. The presence of the adhesive layer was either included in the cohesive zone (Adherend + Cohesive Zone, ACZ) or explicitly modelled with four-noded, hybrid, plane strain elements (Adherend + Adhesive + Cohesive Zone, AACZ), as done for example in [11]. The number of elements along the thickness of the bond was defined so far according to the indications given in [16]. In the case of the T-peel joint, different extensions of the adhesive layer at the fillet were simulated, since they were not known in detail.

T

150

1.5

50

20

= =

10

8

R5




24

Cohesive law Three different kind of cohesive laws were accounted for: i) trapezoidal, ii) triangular and iii) exponential as shown qualitatively in Fig. 4. The relationship between the parameters of the laws is:

i) 0 2 11 12 m c c c (4)

ii) 0

12 m c (5)

iii) 0 11 11

1m ce

(6)

where c1 = 1/c e c2 = 2/c. The cohesive energy 0 was taken equal to the fracture toughness measured in [11], that is 0 = GIc = 550J/m2 in the case of the ACZ kind of models. The parameters of the different laws were tuned until the peak load of the DCB test was achieved, while keeping also a good correspondance with the overall behaviour.

(a)

F,

F,

CZ or ACZ

(b)

F,

F,

CZ or ACZ

Figure 3: FE model of the DCB (a) and of the T-peel (b) joints.

()

c 1 2

()

m

Figure 4: Outline of the cohesive zone traction-separation laws (Eqns. 4 - 6).




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RESULTS Tuning of CZ parameters

he first step was the tuning of the DCB-ACZ model on the related experiment performed in [13]. Some parameters were pre-calibrated based on the results shown in a previous work [12]. Therefore, a value of m=5MPa was adopted in all of the models and in the case of the trapezoidal law, c1=0.2 and c2=0.5 were taken,

as tipically found in the literature. It is visible in Fig. 5 that with such a cohesive law the propagation phase is well matched and the peak load is only slightly overestimated, but initial slope is quite different. Much better results are obtained instead with a triangular law with c1=0.01. A similar agreement (not shown here for the sake of clarity) was found with the trapezoidal law using c1=0.01 and c2=0.02, in fact an almost triangular law. The exponential law was taken with a 1 equal to the 1 defined for the triangular law in order to keep the same initial slope. Anyway, this choice resulted in a lower peak load and a different post-failure trend with respect to the triangular law. Specifically, it resembles the results that would be obtained using a lower value of m. All of the following simulation were therefore conducted using the triangular law, which incidentally is also easier to handle.

Figure 5: Tuning of ACZ model on DCB experiment. The value of m=5MPa outcome from the ACZ tuning is only a bit higher than the yield strength of the adhesive (4.8MPa). The calibration of AACZ parameters was attempted using couples of (0, m) lower and higher than the values obtained for the ACZ model, respectively, as shown in [11]. Anyway, since the value of m=5MPa outcome from the ACZ tuning is only a bit higher than the yield strength of the adhesive (4.8MPa), plasticity within the adhesive layer is not significant and the AACZ model did not significantly differ from the ACZ. Simulation of T-peel tests The simulation of T-peel tests was conducted using an ACZ model with m=12.5 MPa to take into account the higher adesive layer stiffness of the T-peel joints with respect to the DCB due to the different thickness (i.e. 0.1mm against 0.25mm) and m=20.6 MPa, to take into account also the higher strength when joining steel (T-peel) instead of aluminum (DCB), as described in the technical datasheet of the adhesive. The results shown in Fig. 6 in the case of a 1.5mmthick T-peel arm exhibit a good correlation with the experiments in the propagation phase but a very poor match in the loading phase, regarding both the slope and the peak load. On the other hand, it is known [11] that in very thin adhesive layers as in the present case plasticity is strongly confined and due to this, the value of m of an ACZ model increases with decresing adhesive layer thickness.

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Figure 6: Comparison between experiment and T-peel simulations (ACZ model).

The simulations done with the AACZ kind of model are shown in Fig. 7 in the case of 1.5mm-thick peel arm. The use of the parameters determined from DCB (m=5MPa, 0=550J/m2), where ACZ and AACZ were coincident, resulted in a large difference with the experimental behavior.

Figure 7: Comparison between experiment and T-peel simulations (AACZ model). From these analyses, it is evident that CZ parameters may not be the same for the nucleation and propagation of a crack. Besides discrepancies that may arise between the properties of bulk adhesive and joint, and between different joint geometries due to the bonding process, where adhesive and activator cannot be pre-mixed in a fixed ratio, it has been demonstrated that the tearing process is influenced by the local constraint [11]. Therefore, a further analysis has been attempted using AACZ modeling where:

i) the adhesive was extended at the fillet, with the extension length limited by the maximum adhesive thickness that can be polimerized according to the supplier datasheet;

ii) the cohesive strength m has been defined as the value of stress in the adhesive normal to the plane of the joint, at the point in the analysis where the uniaxial failure strain was met locally. The distribution of cohesive strength obtained in this way is shown in Fig.8.

The results of the simulation with the cohesive law input from Fig. 8 and 0=550J/m2 reported in Fig. 9 show a neat increase of the agreement with the experiment concerning the phase up to the maximum load, while the crack propagation phase is matched well as in the previous analyses. This fact confirms that the crack propagation may be reproduced quite easily tuning the CZ parameters on standard fracture experiments, while the nucleation requires specific




27

parameters, possibly dependent on the local stress state. In the specific case of the T-peel test it is important also to know the real fillet dimensions.

Figure 8: Distribution of cohesive strength (AACZ model) obtained as the value of stress in the adhesive normal to the plane of the joint, at the point in the analysis where the uniaxial failure strain was met locally.

CONCLUSIONS

he fracture of adhesively bonded T-peel joints was simulated using a Cohesive Zone Model. The CZ parameters were calibrated on fracture tests conducted on a DCB joint bonded with the same adhesive as the T-peels. The explicit modelling of the adhesive layer (AACZ) did not give in this case any significant variation of the parameters

with respect to the simpler model where the adhesive was included in the CZ behaviour (ACZ). The T-peel test was simulated first using the set of parameters determined from DCB and the ACZ approach, after a partial recalibration to account for the different thickness of the bondline and for the different metal joined. The results showed a good agreement between experiments and simulations with respect to the propagation phase but the peak load is underestimated and the stiffness is dependent on the modelling of the adhesive at the arm fillet. The simulation with the AACZ approach did not give better results using the same set of parameters. A much better agreement was found using a cohesive strength m defined as the value of stress in the adhesive normal to the plane of the joint, at the point in the analysis where the uniaxial failure strain was met locally. This means that the crack propagation may be reproduced quite easily tuning the CZ parameters on standard fracture experiments, while the nucleation requires specific parameters, possibly dependent on the local stress state. ACKNOWLEDGEMENTS

he author gratefully acknowledge Prof. M. Rossetto, Polytechnic of Turin, Italy, for the experimental data of T-peel joints.

REFERENCES [1] A. J. Kinloch, , Adhesion and Adhesives, Chapman and Hall, London, UK, (1986). [2] J. W. Hutchinson, A. G. Evans, Acta Mater., 48 (2000) 125. [3] I. Mohammed, K.M. Liechti, J. Mech. Phys. Solids, 48 (2000) 735. [4] Q.D. Yang, M.D.Thouless, S.M. Ward, J. Mech. Phys. Solids, 47 (1999) 1337. [5] W. G. Knauss, G. U. Losi, J. Appl. Mech., 60 (1993) 793. [6] H. Hadavinia, A.J. Kinloch, J.G. Williams, in Adv. in Fract. and Damage Mech. II, M. Guagliano and M.H. Aliabadi

eds., Hoggar, Geneva, (2001) 445. [7] B. F. Sorensen, Acta Mater., 50 (2002) 1053. [8] B. F. Sorensen, T. K. Jacobsen, Eng. Fract. Mech., 70 (2003) 1841.

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[9] I. Georgiou, H. Hadavinia, A. Ivankovic, A. J. Kinloch, V. Tropsa, J. G. Williams, J. Adhesion, 79 (2003) 239. [10] B. R. K. Blackman, H. Hadavinia, A. J. Kinloch, J. G. Williams, Int. J. Fract., 119 (2003) 25. [11] T. Pardoen, T. Ferracin, C. M. Landis, F. Delannay, J. Mech. Phys. Solids, 53 (2005) 1951. [12] A. Pirondi, Proc. ECF 15, Stockolm, Sweden (2004). [13] A. Pirondi, G.Nicoletto, Proc. IGF 2000, Bari, Italy, (2000). [14] S. Krenk, , Eng. Fract. Mech., 43-4 (1992) 549. [15] M. Rossetto, Private communication, Polytechnic of Turin, Turin, Italy (2003). [16] K. S. Madhusudhana, R. Narashiman, Eng. Fract. Mech., 69 (2002) 865.



B. Chiaia et alii, Frattura ed Integrità Strutturale, 10 (2009) 29-37; DOI: 10.3221/IGF-ESIS.10.04

29

Sismabeton: a new frontier for ductile concrete

Bernardino Chiaia, Alessandro P. Fantilli, Paolo Vallini Politecnico di Torino, Dep. of Structural and Geotechnical Engineering Corso Duca degli Abruzzi, 24 -10129 Torino, Italy [email protected], [email protected], [email protected]

RIASSUNTO. I calcestruzzi fibrorinforzati ed autocompattanti (definiti Sismabeton) manifestano una elevata duttilità non solo in trazione ma anche in presenza di sforzi compressione. Ciò e messo in evidenza nel presente lavoro attraverso la misura della risposta meccanica, in regime di compressione triassiale, di calcestruzzi ordinari (NC) ed autocompattanti (SC) con e senza fibre. In strutture semplicemente compresse, la presenza del Sismabeton è da sola sufficiente a garantire un confinamento attivo uniforme. ABSTRACT. The high ductility of Fiber Reinforced Self-consolidating concrete (called Sismabeton) can be developed not only in tension but also in compression. This aspect is evidenced in the present paper by measuring the mechanical response of normal concrete (NC), plain self-compacting concrete (SC) and Sismabeton cylindrical specimens under uniaxial and triaxial compression. The post-peak behaviour of these specimens is defined by a non-dimensional function that relates the inelastic displacement and the relative stress during softening. Both for NC and SC, the increase of the fracture toughness with the confinement stress is observed. Conversely, Sismabeton shows, even in absence of confinement, practically the same ductility measured in normal and self-compacting concretes with a confining pressure. Thus, the presence of Sismabeton in compressed columns is itself sufficient to create a sort of active distributed confinement. KEYWORDS. Fiber-reinforced concrete, self-compacting concrete, confining pressure, triaxial tests, fracture toughness. INTRODUCTION

everal reinforced concrete (RC) structures fail via concrete crushing in compressed zones. This is the case, for instance, of over-reinforced concrete beams, like those in four point bending tested by Mansur et al. [1]. When fiber-reinforced, the post-peak behaviour of such members is remarkably more ductile than that observed in beams

having the same geometry, the same steel rebars, and the same bearing capacities, but made of normal concrete (NC) without fiber. Thus, when crushing occurs, the type of concrete rules both the mechanical response and the ductility of RC structures. The experimental campaign conducted by Khayat et al. [2] on highly confined RC columns, subject to concentric compression, also confirms the influence of the cement-based composites on the structural performances. More precisely, for a given cross-section, the load vs. average axial strain diagrams appear more ductile in the case of columns made of self-compacting concrete (SC) than in NC columns. These experimental observations can be usefully applied to designing RC compressed columns in seismic regions. According to Eurocode 8 [3], if a required ductility cannot be attained because concrete strains are larger than 0.35% , a compensation for the loss of resistance due to crushing can be achieved by means of an adequate confinement. Such a confinement, usually provided by transversal steel reinforcement (i.e., stirrups), and indicated by the confining pressure 3 (Fig.1), allows designers to consider a more ductile stress strain (c-c) relationship in compression. For instance, Fig.1 shows the so-called parabola-rectangle diagrams proposed by Eurocode 2 [4] for confined and unconfined concretes.

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30

Short steel fibers randomly dispersed in a cement-based matrix can generate confining pressures comparable with that of stirrups. The experimental campaign of Ganesan and Ramana Murthy [5], performed on short confined columns with and without fibers (Fig.2a), investigates on this aspect. As shown in Fig.2b, the applied load- average strain (P-cm) diagram of RC columns, made with ordinary concrete and a transversal reinforcement percentage equal to s=1.6%, is more or less similar (in terms of strength and ductility) to that of fiber-reinforced (FRC) columns, made with a reduced quantity of stirrups (s =0.6%) and FRC (volume fraction Vf = 1.5%, aspect ratio L/ = 70).

Figure 1: The stress-strain relationship of compressed concrete with and without confinement [4].

Figure 2: The columns tested by Ganesan and Ramana Murthy [5].

Although fiber-reinforcements have been introduced in order to increase the ductility of cement-based composites in tension, they can also provide a sort of confinement, and therefore higher ductility in compression. For this reason, when a better fiber matrix bond can be achieved, like the Fiber-Reinforced Self-compacting Composites [6], higher compressive fracture toughness should be expected. To confirm such a conjecture, the post-peak responses of different cementitious composites under uniaxial and multi-axial compression are here investigated. POST-PEAK RESPONSE OF CONCRETE UNDER COMPRESSION

he stress-strain relationships of concrete and quasi-brittle materials in compression (Fig.3a) can be divided into two parts (Fig.3b). In the first part, when the stress is lower than the strength fc (and c < c1 ), the specimen can be considered undamaged. In the case of plain concrete, the ascending branch of c-c can be defined by the Sargin’s

relationship proposed by CEB-FIP Model Code [7]. As soon as the peak stress is reached, localized damage develops and strain softening begins. In this stage, the progressive sliding of two blocks of the cement-based material is evident. In Fig.3a, the angle between the vertical axis of the specimen and the sliding surfaces is assumed to be =18°. This value, as measured in many tests, can be also obtained through the Mohr-Coulomb failure criterion, if the tensile strength is

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assumed to be 1/10 of compression strength ( fct = 0.1 fc ). The inelastic displacement w of the specimen, and the consequent sliding s of the blocks along the sliding surface, are the parameters governing the average post-peak compressive strain c of the specimen (Fig. 3). Referring to the specimen depicted in Fig. 3a, post peak strains can be defined by the following equation [8]:

H

w

EH

w

c

ccelcc

1, (1)

where, c1 = strain at compressive strength fc ; c = stress decrement after the peak; H = height of the specimen (see Fig. 1b).

Figure 3: The post-peak response of quasi-brittle materials in compression.

According to test measurements [8, 9], the post-peak slope of c-c increases in longer specimens (Fig.3b), due to the w/H ratio involved in the evaluation of c [Eq.(1)]. The stress decrement c can be defined as:

wFff cccc 1 (2)

where, F(w) = non-dimensional function which relates the inelastic displacement w and the relative stress c / fc during softening (Fig.3c); fc = compressive strength (assumed to be positive). Substituting Eq.(2) into Eq.(1), it is possible to obtain a new equation for c :

H

w

E

wFf

c

ccc

11 for c > c1 (3)

Eq.(3), adopted for the post-peak stage of a generic cement-based material in compression, is based on the definition of F(w), which has to be considered as a material property [8-9]. In all cement-based composites, this function should be evaluated experimentally on cylindrical specimens, as performed by Jansen and Shah [9] for plain concrete (Fig.3c). Fig.4a shows the F(w) relationships proposed by Fantilli et al. [10]. It consists of two parabolas and a constant branch:

1 2 wbwaf

wFc

a

bw

20for (4a)

444

1 22

22

w

b

aw

b

a

a

b

fwF

c

a

bw

a

b

2for (4b)

0cf

wF

a

bw for (4c)




32

The parabolas are both defined by the same coefficients a, b and have the same extreme point at w = -0.5 b/a , whereas w = - b/a (i.e. twice the value at extreme point) is considered the maximum inelastic displacement corresponding to F(w) higher than zero.

Figure 4: The stress-inelastic displacement relationship proposed by Fantilli et al. [10].

In the case of the plain concrete specimens, the values a = 0.320 mm-2 and b = -1.12 mm-1 were obtained by means of the least square approximation of several tests [10]. As observed in Fig.4b, the curves defined by Eqs.(4) fall within the range of the data experimentally measured by Jansen and Shah [9]. In the case of multi-axial compression, stress-inelastic displacement relationships, which should reproduce the confined post-peak stage, cannot be found in the existing literature. As is well known, two types of confinement, namely passive and active, can be produced. In compressed columns, passive confinements provided by transversal reinforcement (i.e., stirrups, tubes, strips, spirals, etc.), are only activated by concrete displacements. Thus, to define quantitatively this contribution, it is necessary to know the stress-transversal displacement relationship of concrete. Active confinement is due to external stresses 3 applied by multi-axial compression tests on cubes in two or three directions, or by triaxial tests on cylinders (see the book by van Mier [8] for a review). Only a single campaign of triaxial tests, performed by Jamet et al. [11] on micro-concrete, is reported in the current literature. In that case, the applied confinement was relatively high (3 >3 MPa), if compared to those produced by stirrups in ordinary RC columns. In accordance with Eurocode 2 [4], in columns under concentric compression, transverse reinforcement can develop about 3 = 1MPa [12]. Consequently, with the aim of analyzing the equivalent confing pressures produced by a new Fiber-reinforced Self-consolidating concrete (called Sismabeton), the comparison between the results of new triaxial tests on NC, SC and Sismabeton cylinders under uniaxial compression are reported. EXPERIMENTAL PROGRAM

he post-peak behaviour of cement-based composites under multi-axial compression has been investigated at the Department of Structural and Geotechnical Engineering of Politecnico di Torino (Italy) by means of triaxial tests on concrete cylinders (Fig.5a). The experimental equipment, named HTPA (High Pressure Triaxial Apparatus) and

described by Chiaia et al. [13], is generally used to test cylindrical specimens made of soft rocks. Each triaxial test consists of two stages. A specimen is initially loaded with a hydrostatic pressure σ3 (Fig.5b), then deviatoric loads P are applied along the longitudinal direction with a velocity of 0.037 mm per minute (Fig.5c). During the second stage of loading, the confining pressure 3 = const. is applied to the lateral surface, whereas the longitudinal nominal stress c becomes:

234D

Pc

(5)

where, P = applied deviatoric load; D = diameter of the cross-section. Through a couple of LVDT, local longitudinal displacements, and therefore nominal longitudinal strains c , are also measured (Fig. 5a). Two confining pressures, namely σ3 = 0 MPa and σ3 = 1 MPa (reached in 10 minutes), are applied to the specimens. During the application of hydrostatic loads (Fig.5b), stress increments are electronically recorded every 10 seconds.

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Similarly, in the second stage, when σ3 = const. and P increases, the values of deviatoric load, the relative displacement between the specimen’s ends, and the longitudinal displacement along the lateral surface (taken by the LVDTs of Fig.5a) are measured. Two types of self-consolidating concrete (SC_mix1 and SC_mix2) and a single ordinary concrete (NC) were tested.

Figure 5: The two stages of triaxial tests on cement-based cylinders.

Their compositions and strengths are reported in Tab. 1. Specifically, the self-consolidating concretes have the same unit weight, but different amounts of aggregates. With respect to SC_mix1, in a cube meter of SC_mix2 the content of carbonate filler was increased by 90 N and, contemporarily, the weight of coarse aggregate was reduced by the same quantity. Regarding the Fiber-reinforced Self-consolidated concrete (i.e., Simabeton), two specimens were tested, under uniaxial compression (σ3 = 0). As indicated in Tab. 1, Sismabeton is reinforced with 700 N/m3 of Dramix RC 65/35 BN steel fibers having hooked ends (length L = 35 mm, diameter Φ = 0.55 mm, volume fraction Vf = 0.9%). which were added to the self-consolidating concrete with the higher quantity of filler (i.e., SC_mix2).

NC SC_mix 1 SC_mix 2 Sismabeton

Component N/m3 N/m3 N/m3 N/m3

Water 1770 1770 1770 1770 Superplasticizer

(Addiment Compactcrete 39/T100) - 44 44 44 Superplasticizer

(Addiment Compactcrete 39/T11) 14 - - - Cement

(Buzzi Unicem II/A-LL 42.5 R) 2840 2450 2450 2450 Carbonate filler

(Nicem Carb VG1-2) 0 3240 3730 3730 Fine aggregate (0-4 mm) 8830 8930 8930 8930

Coarse aggregate (6.3-12 mm) 6380 6380 5890 5890 Steel fibers

Dramix RC 65/35 BN - - - 700

Cubic strength -MPa- 30.0 31.1 30.4 33.8

Table 1: Compositions and strengths of NC, SC_mix1, SC_mix2, and Sismabeton.

The specimens of each concrete mixture were cast simultaneously in polystyrene form, then cured for one week under identical laboratory conditions, and finally tested after one month. Three couples of cylinders, with H=140 mm and D=70 mm, were made of NC (NC0 and NC1), SC_mix1 (SC0 and SC1), and SC_mix2 (SC0b and SC1b). The two specimens of




34

these couples were tested, respectively, at σ3 = 0 MPa and σ3 = 1 MPa. Two Sismabeton cylinders (HC0 and HC0b), with H=140 mm and D=70 mm, were tested in uniaxial compression. The properties of each specimen are reported in Tab. 2.

Table 2: Mechanical and geometrical properties of the specimens tested in uniaxial and triaxial compression. TEST RESULTS

ig. 6 reports the stress-strain relationships obtained from the specimens made respectively with Sismabeton (Fig.6a), normal concrete (Fig.6b) and self-consolidating concrete (Fig.6c). The higher the confinement, the higher the values of fc and c1 , which are reported, together with Young’s modulus Ec , in Tab. 3. In all the cases, after the

peak stress fc , a remarkable strain softening branch can be observed in the c-c diagrams. Although Sismabeton is fiber-reinforced, its compressive strength does not differ substantially from those of ordinary and self-consolidating concrete. However, the post peak response of Sismabeton appears more ductile. Only when the confining pressure 3 increases, does the ductility of NC and SC increase. By comparing all the post-peak branches reported in Fig.6, it seems that the post-peak branches of SC and NC specimens in the presence of 3 =1 MPa are more or less the same of Sismabeton without any confinement.

Figure 6: The stress-strain relationships of Sismabeton, NC and SC.

Specimen fc (MPa)

c1

(%) Ec

(MPa) 0NC0 19.4 0.293 24000 0NC1 30.5 0.473 23000 0SC0 20.1 0.479 17000

0SC0b 23.2 0.372 23000 0SC1 36.4 0.604 19000

0SC1b 32.0 0.696 27000 HC0 21.8 0.352 19000

HC0b 22.2 0.534 20000

Table 3: Mechanical properties measured in the tests.

Specimen H

(mm) D

(mm) Type of concrete 3 (MPa)

NC0 140 70 NC 0 NC1 140 70 NC 1 SC0 140 70 SCC mix 1 0 SC1 140 70 SCC mix 1 1

SC0b 140 70 SCC mix 2 0 SC1b 140 70 SCC mix 2 1 HC0 100 50 Sismabeton 0

HC0b 100 50 Sismabeton 0

F




35

However, a direct comparison between the analyzed concretes is not possible in terms of nominal stress and strain, because specimens have different nominal strengths. Post-peak comparison in terms of F(w) A more accurate comparison between the post-peak responses of Sismabeton, NC and SC under compression can be conducted in terms of F(w) (Fig.7). In particular, for a given c c1, the decrease of compressive stress c = fc-c (and F = c / fc ) can be obtained through the c-c diagrams experimentally evaluated (Fig.6), whereas the corresponding w (Fig.3a) can be obtained from Eq.(3) (fc , c1 , Ec and H are known from the tests). The F(w) curves reported in Fig.7 are limited to w = 2mm, when compressive strains c are relatively high although, in some cases, stresses are higher than zero. However, in all the tests the relative stress F = c / fc decreases with w. The dashed curves reported in Fig.7 represent the behaviour of NC and SC as predicted by Eq.(4) in the case of zero confinement. As in the case of 3 = 0 the post-peak responses of the specimens NC0, SC0, SC0b are correctly predicted by Eq.(4), and all the tests can be considered consistent [10]. Both for NC and SC, Fig.7a and Fig7b, respectively show the increase of the compressive fracture toughness (within the range w0-2 mm) with the confining pressure 3. However, this phenomenon is also evident in the case of Sismabeton, which can show, in absence of confinement, more or less the same F(w) obtained for NC and SC when 3 = 1MPa.

Figure 7: The post peak behaviour in terms of F(w).

Figure 8: The active confinement of Sismabeton.

Fig.8a shows the post-peak responses of the specimens HC0 and HC0b, which are closer to those of confined SC and NC (i.e., the range defined by NC1, SC1, SC1b), than to the theoretical F(w) obtained in absence of confinement [10] (the dashed line in Fig.8a). Within the observed range (w0-2 mm), compressive facture toughness of different concretes can be objectively measured by the area AF under the function F(w):

dwwFAF 2

0 (6)




36

In fact, as F(w) is a relative stress normalized with respect to the compressive strength fc , a comparison between all the cement-based composites, under uniaxial and multi-axial compression, is possible. Higher values of AF are attained in concretes capable of maintaining high loads after failure (i.e., in the case of ductile materials). Obviously, the maximum ductility AF,max = 2mm is reached in the case of plastic behaviour [F(w) = 1= const.]. The areas AF computed by Eq.(6) for the tested specimens (Tab. 2) are also reported in the histogram of Fig. 8b. In all cases, AF is between AF,max = 2 mm and the lower limit AF,min = 0.61 mm, corresponding to the normal and self-consolidating concretes without any confinement (Fig.8b). To be more precise, AF,min is obtained by substituting Eqs.(4) (with a = 0.320 mm-2 and b = -1.12 mm-1 ) into Eq.(6). At 3 = 1MPa, for the specimens made of SC and NC (i.e., NC1, SC1, SC1b) the values of AF range between 1.39 mm and 1.46 mm (Fig.8b), and do not differ substantially from those measured for Sismabeton (AF 1.56 mm) without confinement. CONCLUSIONS

rom the results of an experimental campaign performed on NC, SC and Sismabeton cylinders under uniaxial and multi-axial compression, the following conclusion can be drawn:

- In normal and self-consolidating concrete, fracture toughness in compression increases in the presence of an active confinement.

- During the post-peak stage, the ductility of Sismabeton is comparable with that of NC or SC at 1MPa of confining pressure.

- In compression, the performance of fiber-reinforced composites can be quantified by the distributed confining pressure generated by the fibers.

The presence of an active confinement can improve the mechanical behaviour of concrete and, consequently, its durability. Thus, further researches should be developed in order to introduce new sustainability indexes, which take into account fracture toughness, both in tension and compression.

ACKNOWLEDGEMENTS

he authors wish to express their gratitude to the Italian Ministry of University and Research (PRIN 2006) and to Fondazione Cassa di Risparmio di Alessandria for financing this research work, and also to Buzzi Unicem S.p.A. for its technical support.

REFERENCES [1] M. A. Mansur, M. S. Chin, T. H. Wee, ACI Structural Journal, 94-6 (1997) 663. [2] K. H. Khayat, P. Paultre, S. Tremblay, ACI Materials Journal, 98-1 (2001) 371. [3] UNI EN 1998-1:2005. Eurocodice 8 – Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings, (1998) 229 [4] UNI EN 1992-1-1:2005. Eurocodice 2- Design of concrete structures- Part 1-1: General rules and rules for building, (1992) 225. [5] N. Ganesan, J. V. Ramana Murthy, ACI Materials Journal, 87-3 (1990) 221. [6] G. Pons, M. Mouret, M. Alcantara, J. L. Granju, Materials and Structures, 40-2 (2007) 201. [7] CEB (Comite Euro-International du Beton), “CEB-FIP Model Code 1990”, bulletin d'information n°203-205, Thomas Telford, London, UK (1993). [8] J. G. M. van Mier, , Fracture Processes of Concrete: Assessment of Material Parameters for Fracture Models. CRC Press, (1996) 448. [9] D. C. Jansen, S. P. Shah, ASCE Journal of Engineering Mechanics, 123-1 (1997) 25. [10] A. P. Fantilli, H. Mihashi, P. Vallini, ACI Materials Journal, 104-5 (2007) 501. [11] P. Jamet, A. Millard, G. Nahas, Int. conference on concrete under multiaxial conditions, Toulouse (1984) 133. [12] S. J. Foster, J. Liu, S. A. Sheikh, ASCE Journal of Structural Engineering, 124-12 (1998) 1431.

F

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[13] B. Chiaia, A. P. Fantilli, P. Vallini, in 3rd North American Conference on the Design and Use of Self-Consolidating Concrete (SCC), Chicago (2008).



A.Namdar et alii, Frattura ed Integrità Strutturale, 10 (2009) 38-42; DOI: 10.3221/IGF-ESIS.10.05

38

Numerical analysis of soil bearing capacity by changing soil characteristics

Abdoullah Namdar Mysore University, Mysore, India, [email protected] Mehdi Khodashenas Pelko Amirkabir University, Tehran, Iran

ABSTRACT. In this research work by changing different parameters of soil foundation like density, cohesion and foundation depth and width of square foundation at angle of friction of 0 to 50 with increment of 5, numerically safe bearing capacity of soil foundation is calculated and it is attempted to assess economical dimension of foundation as well as understanding variation range of bearing capacity at different degree. It could help of civil engineering in design of foundations at any situation. KEYWORDS. Safe bearing capacity, Density, Cohesion of soil (c), Angle of friction (φ) and Dimension of foundation INTRODUCTION

mprovement of soil foundation is a method to disabling earthquake forces and stabilization of soil foundation. For achieving this aim designing a series of foundations by changing gradually the main parameters of soil is the shortest way to better understanding soil characteristics.

It has been reported the effect of soil confinement on the behavior of a model footing resting on sand under eccentric – inclined load [1]. Rea and Mitchell conducted a series of model plate loading tests on circular footings supported over sand-filled square-shaped paper grid cells to identify different modes of failure and arrive at optimum dimensions of the cell [2]. Mahmoud and Abdrabbo presented an experimental study concerning with a method of improving the bearing capacity of strip footing resting on sand sub-grades utilizing vertical non-extensible reinforcement. They found out this type of reinforcement increases the bearing capacity of sub-grades and modify the load–displacement behavior of the footing [3]. The investigations also made on the laboratory-model tests for the bearing capacity of a strip foundation supported by a sand layer reinforced with layers of geo-grid [4]. Research on the ultimate bearing capacity of strip and square foundations supported by sand reinforced with geo-grid has also been performed [5]. Das and Omar presented the ultimate bearing capacity of surface strip foundations on geo-grid-reinforced sand and un-reinforced sand [6]. Dash et al. investigated the use of vertical reinforcement along with horizontal reinforcement. The reinforcement consisted of a series of interlocking cells, constructed from polymer geo-grids, which contain and confine the soil within its pockets [7]. Schimizu and Inui carried out load tests on a single six-sided cell of geo-textile wall buried in the subsurface of the soft ground [8]. Mandal and Manjunath used geo-grid and bamboo sticks as vertical reinforcement elements and studied their effect on the bearing capacity of a strip footing [9]. Rajagopal et al have studied the strength of confined sand and the influence of geo-cell confinement on the strength and stiffness behavior of granular soils [10]. Dash et al. performed an experimental study on the bearing capacity of a strip footing supported by a sand bed reinforced with a geo-cell mattress [11]. Study of bearing capacity of footing without reinforcement under eccentric inclined loads by many researchers has been carried out [12-15]. Nearly 340 foundations by adopting Terzaghi method have been calculated and the results presented in form of tables and graphs, the result helps better understanding of improving of soil foundation bearing capacity.

I




A. Namdar et alii, Frattura ed Integrità Strutturale, 10 (2009) 38-42; DOI: 10.3221/IGF-ESIS.10.05

39

METHODOLOGY AND EXPERIMENTS

y adopting of Terzaghi method and altering characteristics of soil such density, cohesion, angle of friction of soils and dimension of foundation safe bearing capacity of soil foundation has been evaluated. Here three hundred forty one problems by assuming various soil properties have been solved and results of them in form of table and

graph have been considered. For all models, safe bearing capacity considered to assess soil foundation improvement thorough the interpreting of the suggested results. Formulas for calculation of safe bearing capacity are the following:

qf = 1.3C Nc + γDNq + 0.4 γBNγ (1)

qnf = qf - qnf = qf-γD (2)

qs =(qnf /F)+ γD (3) Also Nq, Nc and Nγ are the general bearing capacity factors and depend upon 1) Depth of footing 2) Shape of footing 3) Φ, have been used from suggestion by the Terzaghi calculation method [16]. RESULTS AND DISCUSSION

mprovement of any site could be possible, if soil characteristics of site were clearly identified [17]. Increasing Cohesion of the soil has more effect on bearing capacity when degree of friction is low (Tab. 1 and Fig. 1). Changing of soil density at any level of friction has constant effect on bearing capacity (Tab. 2 and Fig. 2). Increasing depth of

the foundation has more effect on bearing capacity when degree of friction is low (Tab. 3 and Fig. 3). Increasing width of the foundation has less effect on bearing capacity when friction is low (Tab. 4 and Fig. 4). It could be suggested when a soil has low friction, increasing depth of foundation is the best way of increasing bearing capacity or increasing cohesion of soil by adding clay mineral to the soil foundation, this technique could be done by employment of mixing soil technique. In a soil with high friction, increasing width of foundation could be best way to increase its bearing capacity. In general decreasing depth of foundation leads to a weak bearing capacity, and appeared maximum declination in soil foundation stability. Soil density is acting like gravity force to holding foundation in stable conditions, decreasing that is deduction of holding gravity force of foundation.

Sl. No Φ qs if C=0

(kN/m2) qs if C=1 (kN/m2)

qs if C=2 (kN/m2)

qs if C=3 (kN/m2)

qs if C=4 (kN/m2)

qs if C=5 (KN/m2)

1 0 21.00 23.47 25.94 28.41 30.88 33.35 2 5 27.53 30.70 33.86 37.02 40.19 43.35 3 10 38.50 42.66 46.82 50.98 55.14 59.30 4 15 56.47 62.06 67.65 73.24 78.83 84.42 5 20 89.13 96.80 104.47 112.14 119.81 127.48 6 25 148.17 159.04 169.92 180.80 191.67 202.55 7 30 263.43 279.55 295.67 311.79 327.91 344.03 8 35 501.67 526.71 551.76 576.81 601.85 626.90 9 40 1051.63 1093.10 1134.57 1176.04 1217.51 1258.98 10 45 2615.43 2690.10 2764.76 2839.42 2914.09 2988.75 11 50 8301.30 8451.88 8602.47 8753.05 8903.63 9054.22

Table 1: Safe bearing capacity of soil when cohesion of soil (C) is varied, soil density is 14 kN/m3 and

width and depth of foundation are equal to 1.5 and 2.5 m, respectively.

B

I




40

Figure 1: Safe bearing capacity Vs angle of friction of soil

Sl. No Φ qs if =14

(kN/m3) qs if =15 (kN/m3)

qs if =16 (kN/m3)

qs if =17 (kN/m3)

qs if =18 (kN/m3)

qs if =19 (kN/m3)

qs if =20 (kN/m3)

1 0 21.00 22.50 24.00 25.50 27.00 28.50 30.00 2 5 27.53 29.50 31.47 33.43 35.40 37.37 39.33 3 10 38.50 41.25 44.00 46.75 49.50 52.25 55.00 4 15 56.47 60.50 64.53 68.57 72.60 76.63 80.67 5 20 89.13 95.50 101.87 108.23 114.60 120.97 127.33 6 25 148.17 158.75 169.33 179.92 190.50 201.08 211.67 7 30 263.43 282.25 301.07 319.88 338.70 357.52 376.33 8 35 501.67 537.50 573.33 609.17 645.00 680.83 716.67 9 40 1051.63 1126.75 1201.87 1276.98 1352.10 1427.22 1502.33 10 45 2615.43 2802.25 2989.07 3175.88 3362.70 3549.52 3736.33 11 50 8301.30 8894.25 9487.20 10080.15 10673.10 11266.05 11859.00

Table 2: Safe bearing capacity of soil when soil density () is varied, cohesion is zero and width and depth of foundation are equal to 1.5 and 2.5 m, respectively.

Figure 2: Safe bearing capacity Vs angle of friction of soil.



A. Namdar et alii, Frattura ed Integrità Strutturale, 10 (2009) 38-42; DOI: 10.3221/IGF-ESIS.10.05

41

Sl. No Φ

qs if d=0.7 (m)

qs if d=0.8 (m)

qs if d=0.9 (m)

qs if d=1 (m)

qs if d=1.1 (m)

qs if d=1.2 (m)

qs if d=1.3 (m)

qs if d=1.4 (m)

qs if d=1.5 (m)

1 0 9.80 11.20 12.60 14.00 15.40 16.80 18.20 19.60 21.002 5 14.09 15.77 17.45 19.13 20.81 22.49 24.17 25.85 27.53 3 10 20.95 23.15 25.34 27.53 29.73 31.92 34.11 36.31 38.50 4 15 32.57 35.56 38.55 41.53 44.52 47.51 50.49 53.48 56.475 20 54.04 58.43 62.81 67.20 71.59 75.97 80.36 84.75 89.13 6 25 93.29 100.15 107.01 113.87 120.73 127.59 134.45 141.31 148.17 7 30 171.97 183.40 194.83 206.27 217.70 229.13 240.57 252.00 263.438 35 339.64 359.89 380.15 400.40 420.65 440.91 461.16 481.41 501.67 9 40 740.65 779.52 818.39 857.27 896.14 935.01 973.89 1012.76 1051.6310 45 1960.98 2042.79 2124.59 2206.40 2288.21 2370.01 2451.82 2533.63 2615.4311 50 6744.13 6938.77 7133.42 7328.07 7522.71 7717.36 7912.01 8106.65 8301.30

Table 3: Safe bearing capacity of soil when depth of foundation (d) is varied, cohesion is zero, width of foundation is 2.5 m and soil density () is 14 kN/m3.

Figure 3: Safe bearing capacity Vs angle of friction of soil

Sl. No

Φ qs if

w=1.7 (m)

qs if w=1.8

(m)

qs if w=1.9

(m)

qs if w=2.0

(m)

qs if w=2.1

(m)

qs if w=2.2

(m)

qs if w=2.3

(m)

qs if w=2.4

(m)

qs if w=2.5

(m) 1 0 21.00 21.00 21.00 21.00 21.00 21.00 21.00 21.00 21.00 2 5 26.79 26.88 26.97 27.07 27.16 27.25 27.35 27.44 27.53 3 10 36.71 36.93 37.16 37.38 37.60 37.83 38.05 38.28 38.50 4 15 52.73 53.20 53.67 54.13 54.60 55.07 55.53 56.00 56.47 5 20 81.67 82.60 83.53 84.47 85.40 86.33 87.27 88.20 89.136 25 133.68 135.49 137.30 139.11 140.92 142.73 144.55 146.36 148.17 7 30 234.01 237.69 241.37 245.05 248.72 252.40 256.08 259.76 263.43 8 35 438.35 446.26 454.18 462.09 470.01 477.92 485.84 493.75 501.679 40 901.70 920.44 939.19 957.93 976.67 995.41 1014.15 1032.89 1051.6310 45 2171.17 2226.70 2282.23 2337.77 2393.30 2448.83 2504.37 2559.90 2615.4311 50 6579.19 6794.45 7009.72 7224.98 7440.24 7655.51 7870.77 8086.04 8301.30

Table 4: Safe bearing capacity of soil when width of foundation (w) is varied, cohesion is zero, depth of foundation is 1.5 m and soil density () is 14 kN/m3.




42

Figure 4: Safe bearing capacity Vs angle of friction of soil.

CONCLUSION By understanding all factors, which effected to the soil, could be starting way of the improvement of any soil. Some element have linear and some of them nonlinear effect on soil bearing capacity. Unit weight has linear correlation with bearing capacity at any degree of angle of friction, cohesive of soil and width and depth of foundation have nonlinear correlation with bearing capacity at any angle of friction. Data has been produces in this paper could make clear economical design of any foundation. Increasing of bearing capacity in a soil with low angle of friction is not same, as a soil with high level of angle of friction, at any of them should apply proper technique. REFERENCES [1] V. K. Singh, A. Prasad, R.K. Agrawal, EJGE, 12 E(2007) 1. [2] C. Rea, J.K. Mitchell, Proc. Symposium on Earth Reinforcement, Pittsburg, ASCE (1978) 644. [3] M. A. Mahmoud, F.M. Abdrabbo, Canadian Geotechnical Journal, 26 (1989) 154. [4] K. H. Khing, , B.M. Das, V.K. Puri, E.E. Cook, S.C. Yen, , Geotextiles and Geomembranes, 12 (1993) 351. [5] V. K. Puri, K.H. Khing, B.M. Das, E.E. Cook, S.C. Yen, Geotextile and Geomembrane, 12 (1993) 351-361. [6] B. M. Das, M.T. Omar, Geotechnical and Geological Engineering, 12 (1994) 133. [7] S. Dash, K. Rajagopal, N. Krishnaswamy, Geotextile and Geomembrane, 19 (2001) 529. [8] M. Schimizu, T. Inui, Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products, 1

(1990) 254. [9] J.M. Mandal, V.R. Manjunath, Construction and Building Material, 9-1 (1995) 35. [10] K. Rajagopal, N. Krishnaswamy, G. Latha, Geotextile and Geomembrane, 17 (1999)171. [11] S. Dash, N. Krishnaswamy, K. Rajagopal, Geotextile and Geomembrane, 19 (2001) 535. [12] G. G. Meyerhof, 3rd ICSMFE, Zurich, 1 (1953) 1. [13] G. G. Meyerhof, Canadian Geotechnical Journal, 1-1 (1963) 16. [14] G. G. Meyerhof, J. of SMFD, ASCE, 91 SM2 (1965) 21. [15] S. Prakash, S. Saran, J. of SMFE div, ASCE, SM1 (1971) 95. [16] B. C. Punmia, Soil Mechanics and Foundations, Madras, 1988. [17] A. Namdar, G. S. Gopalakrishna, Modern Applied Science, 3-5 (2009).



M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06

43

A dimensional analysis approach to fatigue in quasi-brittle materials

Marco Paggi Politecnico di Torino, Dep. of Structural and Geotechnical Engineering, Corso Duca degli Abruzzi 24, 10129 Torino, Italy [email protected]

RIASSUNTO. Nel presente lavoro si propone uno studio di analisi dimensionale del fenomeno della fatica nei materiali quasi-fragili. Esso costituisce una generalizzazione della metodologia pionieristica proposta da Barenblatt e Botvina e si prefigge di interpretare le deviazioni dalle leggi di potenza classiche usate per caratterizzare il comportamento a fatica dei materiali. In base a questo approccio teorico, gli effetti dovuti alla dimensione microstrutturale (correlabile al contenuto volumetrico di fibre nei calcestruzzi fibrorinforzati), alla dimensione delle fessure e alla scala strutturale sulla legge di Paris e sulle curve di Wöhler sono discussi in un contesto matematico unificato. Il modello teorico è confermato dal confronto con rilevanti risultati sperimentali disponibili in letteratura, usati per determinare i valori degli esponenti di autosimilarità incompleta. Le informazioni fornite da questa teoria possono essere particolarmente utili per guidare la progettazione di nuovi esperimenti, dal momento che viene chiarito il ruolo delle diverse variabili adimensionalizzate che governano il fenomeno della fatica. ABSTRACT. In this study, a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth is proposed in order to highlight and explain the deviations from the classical power-law equations used to characterize the fatigue behaviour of quasi-brittle materials. According to this theoretical approach, the microstructural-size (related to the volumetric content of fibres in fibre-reinforced concrete), the crack-size, and the size-scale effects on the Paris’ law and the Wöhler equation are presented within a unified mathematical framework. Relevant experimental results taken from the literature are used to confirm the theoretical trends and to determine the values of the incomplete self-similarity exponents. All these information are expected to be useful for the design of experiments, since the role of the different dimensionless numbers governing the phenomenon of fatigue is herein elucidated. KEYWORDS. Fatigue; Quasi-brittle materials; Fibre-reinforced concrete; Dimensional analysis; Incomplete self-similarity. INTRODUCTION

he assessment of the fatigue behaviour of quasi-brittle materials, such as plain or fibre-reinforced concrete (FRC), is particularly important from the engineering point of view. Concrete pavements for highways are in fact subjected to millions of cycles of repeated axial loads of high stress amplitude due to passing vehicles. Airport

pavements are also subjected to a number of loading cycles during their design life, ranging from about several thousand to several hundred thousand. Concrete structures supporting dynamic machines may also fail due to repeated loadings causing complex stress states. To make the problem even more complex, very often failure is the result a steady decrease in the stiffness of the structural element, rather than by the propagation of a single macroscopic crack [1]. From the design point of view, the classical methods used to assess the fatigue performance of concrete are mainly empirical and are based on the cumulative fatigue damage approach, well-established for the analysis of the fatigue

T





44

behaviour of metals [2,3]. Hence, the empirical S-N curve, relating the fatigue life or cycles to failure, N, to the applied stress range, or S, is often used both for plain [4-6] and fibre-reinforced concretes [7-12]. A schematic representation of the Wöhler’s diagram is shown in Fig. 1a, where the cyclic stress range, max min , is plotted as a function of the number of cycles to failure, N. In this diagram, we introduce the range of stress at static failure,

max min min (1 )u u uR , where u is the material tensile strength, and we define the endurance or fatigue

limit, fl , as the stress range that a sample will sustain without fracture for 71 10N cycles, which is a conventional

value that can be thought of as “infinite” life. With the advent of fracture mechanics, a more ambitious task was undertaken, i.e., to predict, or at least understand, the propagation of cracks. Plotting the crack growth rate, da/dN, as a function of the stress-intensity factor range,

max minK K K , most of the experimental data can be interpreted in terms of a power-law relationship, i.e., according to the so-called Paris’ law [13,14]. In spite of the fact that this approach requires a more elaborate testing procedure than for the S-N curves, it has also been applied to plain concrete [15-17] (see a schematic representation of the Paris’ curve in Fig. 1b). Note that the power-law representation presents deviations for very high values of K approaching

cr IC(1 )K R K [18], where ICK is the material fracture toughness, or for very low values of K approaching the

threshold stress-intensity factor range, thK [19]. Again, in close analogy with the concept of fatigue limit, the fatigue threshold

is defined in a conventional way as the value of K below which the crack grows at a rate of less than 91 10 m/cycle.

(a) (b)

Figure 1: schemes of the (a) Wöhler and (b) Paris’ curves with the related fatigue parameters.

Yet, the fatigue behaviour of quasi-brittle materials in bending or even in compression is not completely understood in terms of all the influential variables, such as the type of the loading cycle, cycling rate, structural size, crack length and, perhaps most important of all for FRC, fibres parameters. For instance, it has been experimentally shown in [15-17] that the coefficients entering the Paris’ law depend on the material composition, on the structural size, and on the crack length, potentially explaining the large scatter in the reported values available in the literature. This result has also the fundamental implication that the Paris coefficients cannot be treated as true material constants. Similarly, the S-N curves are found to be dependent on the size of the tested specimen [11], on the loading conditions [1], as well as on the material composition [8]. From the modelling point of view, an important research effort towards simulating S-N curves of plain concrete and FRC in cyclic bending was put forward in [11,20]. This required the use of nonlinear fracture mechanics theories coupled with a damage law for modelling the degradation of the concrete cohesive crack stresses and of the fibres bridging in the process zone due to cyclic loading. However, in spite of the useful predictive capabilities of these models, their complex mathematical formulation limits their applicability to design purposes and confines them to the research environment. Moreover, although the damage model can be tuned on the basis of experimental data as proposed in [10], other choices for the expression of the damage variable are also possible (see e.g. [21]) and the connection existing between the evolution of damage and the outcome of the fatigue simulation has not yet been completely clarified.




45

Following an independent line of research, the use of the concepts of dimensional analysis and incomplete self-similarity originally proposed by Barenblatt and Botvina [22,23] seems to be very promising in understanding the phenomenon of fatigue. Applying these concepts to metals, the present author has determined the relationship between the Paris’ law parameters C and m on the basis of theoretical arguments [18]. Then, the Barenblatt and Botvina’s analysis of the size-scale effects on the Paris’ law parameter m has been extended in [24] to the Paris’ law parameter C. Finally, a dimensional analysis of the cumulative fatigue damage and of the damage tolerant approaches has been proposed in [25], interpreting the observed deviations from the Paris and Wöhler regimes within a unified theoretical framework. This approach promises to bring simplicity in what has been up to now considered as an unexplained set of data, putting into evidence the role of the most influential variables on the fatigue behaviour of materials. The aim of the present study is to extend the dimensional analysis approach to quasi-brittle materials, determining the main dimensionless numbers influencing the d d/a N - K and the S-N curves. More specifically, the possibility of incomplete self-similarity in the dimensionless numbers related to the microstructural size, the crack size and the structural size is explored in details. As a result, different aspects that have always been so far treated separately are herein interpreted within a unified mathematical framework. The corresponding incomplete self-similarity exponents and their dependencies on the dimensionless numbers are also quantified on the basis of relevant experimental data taken from the literature. The new proposed correlations provide useful information for design purposes.

GENERALIZED MATHEMATICAL REPRESENTATIONS OF FATIGUE The generalized Paris law

ccording to the pioneering work by Barenblatt and Botvina [22], the following functional dependence can be put forward for the analysis of the phenomenon of fatigue crack growth:

IC thd , , ; , , , , , ;1 ,d u

aF K K K h d a E R

N (1)

where the governing variables are summarized in Tab. 1, along with their physical dimensions expressed in the Length-Force-Time class (LFT).

Variable definition Symbol Dimensions

Ultimate tensile strength u 2FL

Fracture toughness ICK 3/ 2FL

Frequency of the loading cycle 1T

Stress-intensity factor range K 3/ 2FL

Threshold stress-intensity factor range thK 3/ 2FL

Stress range 2FL

Fatigue limit fl 2FL

Structural size h L

Microstructural size (aggregate or fibre size) d L

Crack length a L

Elastic modulus E 2FL

Loading ratio R

Table 1: Governing variables of the fatigue crack growth phenomenon.

A




46

Considering a state with no explicit time dependence, it is possible to apply the Buckingham’s Theorem [26] to reduce the number of parameters involved in the problem (see also [27-29] for the application to the quasi-static case). As a result, we have:

2 2 2 2IC th

2 2 2IC IC IC IC IC

2

IC

d , , , , , ;1d

u u u

u u

iu

K Ka K Eh d a R

N K K K K K

K

(2)

where i ( 1, ,7)i are dimensionless numbers. Note that 3 corresponds to the square of the dimensionless number Z introduced by Barenblatt and Botvina [13] and to the inverse of the square of the brittleness number s introduced by Carpinteri [17-19]. The number 5 was firstly considered by Spagnoli [8] for the analysis of the crack-size

dependence of the Paris’ law parameters. Here, it has to be noted that, according to Irwin, the ratio 2IC / uK , which is

used to made dimensionless the variables h , d and a , is proportional to the plastic zone size, pr .

The number 4 is related to the microstructural length scale and has been introduced in [24] in order to interpret the effect of the grain size in metals on the coefficient C of the Paris’ law according to the experimental findings by Chan [30]. As far as plain concrete is concerned, this microstructural length scale should correspond to the average size of the aggregates. For FRC, the variable d should correspond to the fibre diameter and Eq. (2) can be rewritten in order to put into evidence the fibre volumetric content. This parameter is computed as the ratio between the total area of fibres contained in a transversal cross-section of the specimen and the cross-section area of the specimen itself,

2/ 100 /f fv A A f d h . Here, the function f represents a shape function which depends on the transversal

cross-section of the fibres and on their amount. For a square specimen cross-section of side h and for circular fibres we have 100 / 4ff n , where fn is the total number of crossed fibres. In this case, instead of considering the

dimensionless representation (2), where all the variables with a physical dimension of a length are made dimensionless using the plastic zone size, it is more convenient to use the following alternative form:

2 2 2IC th

2 2IC IC IC IC

2

IC

d , , , , , , ;1d

u u

u u

iu

K Ka K d l Eh a R

N K K K h K d

K

(3)

where the dimensionless number 4 has been replaced by *4 4 3/ / /fd h v f . Moreover, the number

6 /l d (fibre length/fibre diameter) has also been suitably introduced in Eq. (3) and it corresponds to the fibre aspect

ratio. At this point, we want to see if the number of quantities involved in the relationships (2) or (3) can be reduced further from seven (or height). This can occur either in the case of complete or incomplete self-similarities in the corresponding dimensionless numbers. In the former situation, the dependence of the mechanical response on a given dimensionless number, say i , disappears and we can say that i is non essential for the representation of the physical phenomenon. In

the latter situation, a power-law dependence on i can be proposed, which usually characterizes a physical situation intermediate between two asymptotic behaviours. Considering the dimensionless number 1 IC/K K , it has to be noticed that it rules the transition from the

asymptotic behaviours characterized by the condition of nonpropagating cracks, when thK K , to the pure Griffith-

Irwin instability, when crK K . Moreover, incomplete self-similarity in 1 would correspond to a power-law




47

dependence of the crack growth rate on the stress-intensity factor range, which is experimentally confirmed by the use of the Paris’ law for concrete [15-17]. Therefore, complete self-similarity in 1 cannot be accepted and the assumption of

incomplete self-similarity holds whenever K is far lower than ICK , i.e., for 1 1 . As far as 3 is concerned, this number is expected to be particularly important in quasi-brittle materials, since it compares the structural size-scale h with

the plastic zone size. As regards 4 (or *4 for FRC), we know that, in metals, incomplete self-similarity is attained for

p/ 1d r , i.e., for 4 1 [25,30]. A similar reasoning applies for 5 , that is incomplete self-similarity in this number is

expected when the crack length is comparable with the process zone size [17]. As regards 8 , a dependence on R is very

often observed and therefore incomplete self-similarity in 8 is expected for 80 1 .

Hence, assuming incomplete self-similarity in the dimensionless variables 1 , 3 , 4 , 5 and 8 , the following generalized representation of fatigue crack growth is derived starting from Eq.(3):

1 2 43

5

3 51 2 4

3 1 2 3 4 2 3 4

2 2 2IC

1 2 6 72 2IC IC IC

1 2 6 72 2 2 2 2(1 )

IC

d (1 ) , ,d

, ,(1 )

fu u

u

fu

vKa Kh a R

N K K f K

K h v a Rf K

(4)

where the exponent i and, consequently, the dimensionless function 1 , cannot be determined from considerations of

dimensional analysis alone. Moreover, it is important to remark that the exponents i may depend on the dimensionless

numbers i . Eq. (4) can be regarded as a generalized Paris’ law (see the classical expression d d ma N C K superimposed to Fig. 1b), in which the main functional dependencies of the parameter C have been now explicitated.

Figure 2: the effect of incomplete self-similarity in 3 , 4 , 5 and 8 on the Paris’ curves.

In this mathematical framework, it emerges that the incomplete self-similarity exponent 1 simply corresponds to the

Paris’ law parameter m that can be evaluated as the slope of the bilogarithmic d da N vs. K curve (see Fig. 2). Usually, m varies from 5 up to 30 in quasi-brittle materials [31], a value particularly high as compared to metals, where m ranges between 2 and 4. For 1K , the intercept of the straight line with the vertical axis provides the logarithm of the multiplicative parameter C (see Fig. 2). Under the assumptions of incomplete self-similarity discussed above, C is no longer a true constant, but it is rather a power-law function of h , fv and a . Regarding the exponents 2 and 3 , it is

reasonable to expect them less than zero, with a reduction of C by increasing either the size of the specimen or the amount of fibres. In fact, this would imply a shift of the vertical asymptote corresponding to max ICK K towards




48

higher values of K . This trend is consistent with the common observation that the fracture toughness of concrete is an increasing function of the size of the tested sample [32] and of the fibre volumetric content [10]. On the contrary, short cracks are expected to grow faster than long cracks [19] and therefore the exponent 4 is expected to be positive valued. The same reasoning holds for the loading ratio. It is important to note that the above scaling law has been derived by considering the stress-intensity factor range as the driving parameter for crack growth. However, in quasi-brittle materials, where the cohesive crack model is routinely applied to describe crack propagation, it is also possible to find numerical results relating the crack growth rate to the range of energy release rate, G . Power-laws of the Paris form relating the crack growth rate to G still hold [33] and Eq. (1) could be replaced by the following expression:

F thd , , ; , , , , , ;1d u

aF G G G h d a E R

N (5)

where the fracture energy, FG , has been suitably introduced instead of the fracture toughness ICK and the existence of a

threshold range of energy release rate, thG , has been postulated. Hence, Eq. (3) would become:

thF

F F F F

d , , , , , , ;1d

u u

u u

GGa G d l Eh a R

N G G G h G d

(6)

where we recognize that the dimensionless number governing the size-scale effects is no longer equal to 2 2

IC1/ /us h K as in Eq. (3), but it is now represented by the inverse of the energy brittleness number,

E F1/ /us h G . However, it is possible to demonstrate that Eq. (6) implies Eq. (3) and viceversa. Therefore, the dimensionless representation in Eq. (3) is equivalent to that in Eq. (6) and no additional dimensionless numbers have to be introduced. Recalling the Irwin’s relationship between the stress-intensity factor and the energy release rate, it is possible to recast Eq. (6) in terms of stress-intensity factors. In doing that, we note that the energy brittleness number Es

is equal to the stress brittleness number s times the inverse of the dimensionless number / uE [34], i.e.,

2E / / us s E . Therefore, Es is nothing but a linear combination of two dimensionless numbers already defined in

Eq. (3). Hence, considering plane stress conditions, Eq. (6) can be rewritten as follows:

2IC th

2 2IC IC IC IC

d , , , , , , ;1d

u u

u u

K K E Ea K d l Eh a R

N E K K K h K d

(7)

which can be further manipulated by replacing the Young’s modulus with the tensile strength, since the value of the ratio between these two variables is already taken into consideration by the dimensionless number / uE . Therefore, after such a substitution, Eq. (7) becomes identical to Eq. (3). The inverse implication is straightforward and can be gained by applying the inverse of the Irwin’s relationship. THE GENERALIZED WÖHLER REPRESENTATION

o far, the crack growth rate has been chosen as the main output parameter characterizing the phenomenon of fatigue crack growth. However, it is also possible to consider the number of cycles, N, as the parameter representative of fatigue. Following this alternative route, we postulate again the following functional dependence:

IC, , ; , , , , , ;1u flN F K h d a E R (8)

where the definition of the governing variables is provided in Tab. 1. Considering a state with no explicit time dependence, it is possible to apply the Buckingham’s Theorem [26] to reduce the number of parameters involved in the problem:

S




49

2 2

2 2IC IC

, , , , , , ;1fl u ui

u u u

d l EN h a R

K h K d

(9)

where is a dimensionless function. Note that Eq. (9) has been derived from Eq. (8) by choosing u , ICK and the plastic zone size as the main variables whose suitable combination provides a dimensionless number. At this point, we want to see if the number of quantities involved in the relationship (9) can be reduced further from height. In close analogy with the procedure carried out for the Paris’ law, we assume incomplete self-similarity in 1 ,

3 , 4 , 5 and 8 (the same conditions apply to the dimensionless numbers as in the derivation of Eq. (4)), obtaining:

1 2 43

5

531 2 4

2 3 43 1 2 3 4

2 2

2 2 6 72 2IC IC

2 2 6 72 2 2 2

IC

1 , ,

, ,1

fu u

u

f

u

vN h a R

K f K

h v a Rf K

(10)

Eq. (10) represents a generalized Wöhler relationship of fatigue and encompasses the empirical S-N curves as limit cases. For instance, the S-N curve in Fig. 1a can be approximated by the Basquin power law:

1 n n nu flN N k (11)

where u is the range of stress at static failure, fl is the fatigue limit corresponding to a conventional fatigue life of

71 10N cycles, is the stress range corresponding to a fatigue life N and k is a constant. Equating the first and the third terms in Eq. (11), we obtain he following power-law equation:

1n n nuu

n

RN

(12)

Hence, the Basquin power law is predicted as a limit case of Eq. (10) when 1 n , 2 3 4 0 and 5 n .

Figure 3: the effect of incomplete self-similarity in 3 , 4 , 5 and 8 on the S-N curves.

Therefore, in this framework, the incomplete self-similarity exponent 1 simply corresponds to Basquin parameter n

that can be evaluated as the slope of the vs. N curve in a bilogarithmic diagram (see Fig. 3). For 1N , the




50

intercept of the straight line with the vertical axis provides the logarithm of the multiplicative parameter u (see Fig. 3).

Under the assumptions of incomplete self-similarity discussed above, we expect 2 less than zero, with a reduction of

u by increasing the size of the specimen. Note that this trend is fully consistent with the well-known size-scale effects

on the tensile strength of quasi-brittle materials [32] and with the recent findings in [35]. Similarly, we expect 4 0 as

for metals [36]. As regards the fibre volumetric content and the loading ratio, an opposite trend is expected, with 3 and

5 positive valued (see Fig. 3). APPLICATION TO QUASI-BRITTLE MATERIALS

n this section, the generalized Paris and Wöhler representations of fatigue are applied to plain concrete and FRC, providing a dimensional analysis interpretation to some of the most relevant experimental trends found in the literature. More specifically, the effect of the microstructural size (fibre diameter), of the crack size and of the

structural size are carefully discussed. The effect of the microstructural size As shown in Section Generalized mathematical representations of fatigue, the use of fibres is beneficial from the fatigue standpoint, since an increase in fibre diameter (or, equivalently, in volumetric content) corresponds to a decrease in crack growth rate for a given value of K and to an increase in the cycles to failure for a given value of . The quantify such an effect, the experimental flexural fatigue data by Johnston and Zemp [8] can be profitably used. They tested square concrete beams in cyclic bending without fibres or with smooth wire fibres with different volumetric contents ( fv 0.5,

1.0 and 1.5%). All the fibres have an aspect ratio / 75l d . The original S-N results are reported in the bilogarithmic diagram of Fig. 4, along with the best-fitting power-law regression curves. According to Eq. (10), the exponent of N corresponds to 11/ and we note that it is almost

independent of the fibre volumetric content, ranging from 0.0218 to 0.0249 . For this type of fibres, the average value of 1 is then equal to 43 . On the other hand, the multiplicative coefficient of the variable N corresponds to

u and this parameter is significantly affected by the presence of fibres. Plotting the values of u vs. the volumetric

content in Fig. 5 and determining the best-fitting power-law equation, we find 0.266.37u fv . According to Eq. (10)

evaluated in correspondence of 1N , the exponent of fv is equal to 3 1/ , leading 3 10.26 11.3 .

Figure 4: The effect of the fibre volumetric content, fv , on the S-N curves as a result

of incomplete self-similarity in 4 (experimental data from [8], aspect ratio / 75l d ).

I




51

Figure 5: Determination of the incomplete-self-similarity exponent 3.

As pointed out by Johnston and Zemp [8], the fibre aspect ratio has an important effect on the fatigue behaviour of FRC, whereas the fibre type (smooth wire, surface deformed wire, melt-extract and slit sheet) is of secondary importance. To show this effect in terms of dimensional analysis arguments, let us consider the experimental results by Singh et al. [12]. They tested FRC beams under fatigue loading with different volumetric content of fibres ( 1.0, 1.5fv and 2.0% ).

However, instead of using a single fibre type as in [8], 50% of their total weight has an aspect ratio of / 20l d and the remainder has an aspect ratio of / 40l d . Again, as for the previous case, the higher the volumetric content, the higher the fatigue life for a given applied stress range (see Fig. 6). However, now the S-N curves have an average slope equal to

0.030 , corresponding to an exponent 1 33 . This value is significantly higher than 43 found in the case of a

higher aspect ratio equal to 6 / 75l d . According to dimensional analysis, this difference has to be ascribed to the

different value of the aspect ratio and to the fact that the power-law exponent 1 depends on 6 . From the engineering point of view, the use of longer fibres is clearly advantageous, since it increases the fatigue life of the specimen for a given applied stress range, reducing the exponent 1 .

Figure 6: S-N curves of beams tested in flexural fatigue with 50% of fibres with / 20l d and

the remainder with / 40l d (experimental data taken from [12]). The effect of the crack size Modified Paris’ laws taking into account the effect of the crack size have been proposed both for metals and for quasi-brittle materials. For metals, several researchers have questioned the validity of the similitude hypothesis, which states that




52

“two different sized cracks embedded into two different sized bodies subjected to the same stress-intensity factor range should grow at the same rate”. As a support to the theories against the similitude hypothesis, we mention the experimental results by Newman et al. [37], who observed that “in the threshold regime there is something missing in the (closure) model”, and those by Forth et al. [36], revealing that similitude does not hold in Region I (the near-threshold region) and also in the lower portion of Region II. To overcome this problem, Molent et al. [39] and Jones et al. [40] have recently proposed a generalized Frost and Dugdale crack growth law, assuming that the crack growth rate is proportional to the

accumulated plastic strain, averaged over a characteristic length ahead of the crack tip, ' (1 '/ 2) 'd / d m ma N C a K , where C’ and m’ are regarded as material constants. This equation states that d / da N is not only a function of the stress-intensity factor range, but also of the crack length. Such a generalized Frost and Dugdale crack growth equation was successfully used to predict the growth of near micron sized cracks in both coupon and full scale aircraft fatigue tests and interpret a large amount of experimental data that could not be modelled using the Paris’ law. For concrete, a detailed experimental examination of crack propagation in flexural fatigue [17] has shown that the crack growth rate is not a monotonic increasing function of the crack length. For cracks shorter than the crack length at peak load in quasi-static monotonic loading, a deceleration stage was found, where da/dN is a decreasing function of a. Afterwards, an acceleration stage takes place and da/dN can be well approximated according to the classical Paris’ law [17]. To model the deceleration stage, Kolloru et al. [17] proposed an empirical relationship between da/dN and the crack length, apparently independent of the Paris’ law. Actually, it can be interpreted as a particular case of our proposed

generalized Paris’ law, simply allowing a crack-size dependence of the coefficient C , i.e., setting 1 2( / 2)n nC a

4 1 2( / 2)n n , where 1n and 2n are the power-law exponents for the two regimes found in [17]. The effect of the structural size The effect of the structural size can be highlighted by considering the experimental data by Bažant and Xu [15] for normal strength concrete and by Bažant and Shell [16] for high strength concrete, subsequently re-examined by Spagnoli [41]. From their experimental results on self-similar beams tested in cyclic bending, the computed Paris’ law exponent 1m

was found to be dependent on the structural size. Plotting m vs. 3 in Fig. 7, we recognize that m is a linear decreasing

function of 3 for each type of concrete. The effect of the ultimate tensile strength is also important, since it affects the

dimensionless number 7 / uE . For normal strength concrete we have 7 9686 , whereas for high strength

concrete we have 7 4303 and the slope of the linear relationship between m and 3 turns out to be an increasing

function of 7 (see Fig. 7).

Figure 7: Size-scale effects on the Paris’ law exponent 1m for normal and high strength concrete

(experimental data from [15,16] reinterpreted in [41]).




53

The experimental data by Bažant and Xu [15] for normal strength concrete and by Bažant and Shell [16] for high strength concrete can also be used to assess the hypothesis of incomplete self-similarity in 3 and therefore the size-scale effect

on the parameter C . Plotting the Paris’ law coefficient C vs. 3 in a bilogarithmic diagram and computing the best-

fitting power-law regression curves, we find 2 4.7 for normal strength concrete and 2 1.1 for high strength concrete (see Fig. 8).

Figure 8: Size-scale effects on the Paris’ law coefficient C (evaluated using K in MPa m and da/dN in m/cycle) as a result of

incomplete self-similarity in 3 for normal and high strength concretes (experimental data from [15,16] reinterpreted in [41]).

Incomplete self-similarity in 3 is also expected to occur in the Wöhler regime, as recently put into evidence for metals in [35] on the basis of fractal concepts. For concrete specimens, the experimental data by Zhang and Stang [9] and Murdock et al. [42] show that the S-N curves corresponding to two different sizes are almost parallel to each other and translate vertically. In particular, the larger the beam size, the lower stress range at static failure (see Fig. 8). This result is in perfect agreement with the present theoretical predictions and with the numerical findings by Zhang et al. [11]. In this case, however, it is not possible to compute the incomplete self-similarity exponent 2 , since we have only two values of

( 1) uN for different beam depths.

Figure 9: The effect of the structural size on the S-N curves (experimental data taken from [9,42]).




54

CONCLUSIONS

n the present paper, a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth has been proposed in order to highlight and explain the deviations from the classical power-law equations used to characterize the fatigue behaviour of quasi-brittle materials. It has been theoretically demonstrated that both the

parameters entering the Paris’ law and the Wöhler equation are microstructural-size, crack-size and size-scale dependent. From the theoretical point of view, these anomalous dependencies are due to incomplete self-similarities in the corresponding dimensionless numbers. More specifically, as far as the Paris’ law is concerned, it has been shown that the higher the structural size or the volumetric content of fibres, the lower the crack growth rate for a given stress-intensity factor range. Conversely, the higher the initial crack size or the loading ratio, the higher the crack growth rate. Regarding the S-N curves, the higher the structural size or the initial crack length, the lower the fatigue life for a given stress range. The opposite trend is noticed for the volumetric content of fibres and the loading ratio. Finally, we have also shown that the slopes of the S-N curves are dependent on the fibre aspect ratio. Similarly, the slope of the Paris’ curve is found to be dependent on the size of the specimen and on the ratio between the elastic modulus of concrete and its tensile strength. All these information are expected to be extremely useful for the design of experiments, since the role of the different dimensionless numbers governing the phenomenon of fatigue has been elucidated. REFERENCES [1] D. A. Hordijk, Heron, 37 (1992) 1. [2] A. Wöhler, Z. Bauwesen (1860) 10. [3] O. H. Basquin, Proc. ASTM, 10 (1910) 625. [4] J. W. Murdock, C. E. Kesler, ACI J., 55 (1959) 221. [5] R. Tepfers, ACI J., 76 (1979) 919. [6] J. E. Butler, Strain, 26 (2008) 135. [7] V. Ramakrishnan, G. Oberling, P. Tatnall, SP-105-13, ACI Special Publication, ACI, Detroit (1987) 225. [8] C. D. Johnston, R. W. Zemp, ACI Mater. J., 88 (1991) 374. [9] J. Zhang, H. Stang, ACI Mat. J., 95 (1998) 58. [10] J. Zhang, H. Stang, V.C. Li, ASCE J. Mat. in Civ. Engrg., 12 (2000) 66. [11] J. Zhang, V.C. Li, H. Stang, ASCE J. Mater. Civil Engng., 13 (2001) 446. [12] S.P. Singh, Y. Mohammadi, S.K. Madan, J. Zhejiang University Science A, 7 (2006) 1329. [13] P. Paris, M. Gomez, W. Anderson, 13 (1961) 9. [14] P. Paris, F. Erdogan, J. Basic Eng. Trans. ASME, 58D (1963) 528. [15] Z.P. Bažant, K. Xu, ACI Mater. J., 88 (1991) 390. [16] Z.P. Bažant, W.F. Shell, ACI Mater. J., 90 (1993) 472. [17] S.V. Kolloru, E.F. O’Neil, J.S. Popovics, S.P. Shah, ASCE J. Engng. Mech., 126 (2000) 891. [18] Al. Carpinteri, M. Paggi, Engng. Fract. Mech., 74 (2007) 1041. [19] M. Paggi, Al. Carpinteri, Chaos, Solitons and Fractals, 40 (2009) 1136. [20] T. Matsumoto, V.C. Li, Cement & Concrete Composites, 21 (1999) 249-261. [21] K. L. Roe, T. Siegmund, Engng. Fract. Mech., 70 (2003) 209. [22] G. I. Barenblatt, L. R. Botvina, Fat. Fract. Engng. Mater. Struct., 3 (1980) 193. [23] G. I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics. Cambridge: Cambridge University Press,

(1996). [24] M. Ciavarella, M. Paggi, Al. Carpinteri, J. Mech. Phys. Solids, 56 (2008) 3416. [25] Al. Carpinteri, M. Paggi, Int. J. Fatigue, in press, doi: 10.1016/j.ijfatigue.2009.04.014 [26] E. Buckingham, ASME Trans., 37 (1915) 263. [27] Al. Carpinteri, RILEM Mat. Struct., 14 (1981) 151. [28] Al. Carpinteri, Engng. Fract. Mech., 16 (1982) 467. [29] Al. Carpinteri, RILEM Mat. Struct., 16 (1983) 85. [30] K. Chan, Scripta Metal. Mater., 32 (1995) 235. [31] N. A. Fleck, K. J. Kang, M. F. Ashby, Acta Metall. Mater., 42 (1994) 365. [32] Al. Carpinteri, Mech. Mater., 18 (1994) 89.

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[33] Y. Xu, H. Yuan, Proc. of the 12th International Conference on Fracture, Ottawa, Canada, (2009). [34] Al. Carpinteri, In: Size-Scale Effects in the Failure Mechanisms of Materials and Structures (Proceedings of a IUTAM

Symposium, Torino, Italy, 1994), Ed. A. Carpinteri, E & FN SPON, London, (1996) 3. [35] An. Carpinteri, A. Spagnoli, S. Vantadori, Int. J. Fatigue, 31 (2009) 927. [36] J.C. Newman, Jr., E.P. Phillips, M.H. Swain, Int. J. Fatigue, 21 (1999) 109. [37] J.C. Newman, Jr., A. Brot, C. Matias, Engng. Fract. Mech., 71 (2004) 2347. [38] S.C. Forth, W.M. Johnston, B.R. Seshadri, Proc. 16th European Conf. Fracture, Alexandroupolis, Greece, (2006). [39] L. Molent, R. Jones, S. Barter, S. Pitt, Int. J. Fatigue, 28 (2006) 1759. [40] R. Jones, L. Molent, S. Pitt, Int. J. Fatigue 29, (2007) 1658. [41] A. Spagnoli, Mech. Mat., 37 (2005) 519. [42] J.W. Murdock, C.E. Kesler, ACI J., 55 (1959) 221.



G. Bolzon et alii, Frattura ed Integrità Strutturale, 10 (2009) 56-63; DOI: 10.3221/IGF-ESIS.10.07

56

Mechanical characterization of metal-ceramic composites

G. Bolzon, M.Bocciarelli, E.J. Chiarullo

Politecnico di Milano , Dipartimento di Ingegneria Strutturale, [email protected]

RIASSUNTO. I compositi metallo-ceramici rappresentano una classe di materiali per uso strutturale che richiede una adeguata caratterizzazione meccanica. Le difficoltà ed i costi associati con la produzione e la lavorazione di questi compositi scoraggiano l’esecuzione di prove meccaniche tradizionali, che richiedono l’impiego di un numero statisticamente significativo di provini di dimensioni e geometria proibitivi in questo contesto. Risulta invece indicata la prova di indentazione strumentata, rapida ed economica, abbinata a tecniche di analisi inversa che combinano l’informazione sperimentale con la simulazione del comportamento del materiale durante la prova, come si mostra in questo contributo basato sull’esperienza acquisita nell’ambito del Network di Eccellenza su ‘Knowledge-based Multi-component Materials for durable and safe performance’ (KMM-NoE). ABSTRACT. Metal-ceramic composites represent a class of quasi-brittle materials for advanced structural applications that require adequate mechanical characterization. Difficulties and costs associated with material production and specimen extraction prevent the execution of a statistically meaningful number of standard laboratory tests. Parameter calibration methodologies based on instrumented indentation and inverse analysis represent fast and reliable identification procedures in the present context, as shown by the present contribution, based on some experience achieved in the framework of the European Network of Excellence on ‘Knowledge-based Multi-component Materials for durable and safe performance’ (KMM-NoE). KEYWORDS. Metal-ceramic composites; constitutive models; quasi–brittle fracture; mechanical characterization; instrumented indentation; parametric identification; inverse analysis. INTRODUCTION

etal-ceramic composites represent a class of advanced materials of growing interest for their application in several technological fields, ranging from energy production and biomechanics as witnessed, e.g., by the website www.kmm-noe.org. Significant dimensional stability, reduced thermal expansion, increased wear and

damaging resistance at high temperature, good mechanical strength (mainly in compression) make these composites interesting replacement of metals as structural components and protective coatings in high demanding applications. Main limitations concerning their usage consist of: difficult processing; high production costs; strong influence on the overall material properties of micro-structural details; brittle behaviour, though mitigated by the metal phase when compared with pure ceramics. The composition and the production process of these composites are usually designed in order to optimize their effective thermo-mechanical response, which require to carefully control the possible presence and the evolution of local damages and defects. The estimation of macroscopic properties is often based on homogenization techniques, which often neglect micro-structural details playing a significant role for brittle and quasi-brittle materials [1-8]. A proper experimental validation of the mechanical properties is therefore mandatory. On the other hand, difficulties and costs associated with specimen production prevent the execution of a statistically meaningful number of standard laboratory investigations. A practical methodology for material characterization is based on instrumented indentation, which represents a fast and economical test, which can be performed at different scales on small material portions, even directly on the structural

M




http://www.kmm-noe.org/


57

components. The combination of experimental information with the test simulation and inverse analysis [9] represents the envisaged approach to material characterization as the properties to be determined becomes numerous and less amenable to direct separate measurement, as in the present context. This approach can return the bulk and fracture properties of metal-ceramic composites in a robust and reliable manner, by exploiting experimental data relevant to both the traditional indentation curves and the mapping of the residual displacement field, as shown e.g. in [10-16] and illustrated in the next Sections with the aid of some meaningful application example. INDENTATION TOOL

ndentation test represents a practical methodology for the characterization of traditional and innovative materials. With respect to more traditional experimental investigation, indentation can be performed on small material portion, does not require to extract laboratory specimens, can be performed even in situ, directly on the structural

component, to continuous monitoring and diagnosis purposes. The outcome of instrumented indentation, namely the curves that represent the relationship between the penetration depth and the force exerted on the equipment tip, see Fig. 1, reflect constitutive and fracture properties of the sampled material, though in an indirect way. Quantitative calibration of parameters entering the selected constitutive model can then be returned by inverse analysis procedures, which combine experimental data and the simulation of the laboratory test as shown, e.g., in [9, 17, 18]. The inverse analysis problem can be formulated as the minimisation, with respect to the unknown parameters, of a norm that quantifies the overall discrepancy between the measured quantities and the corresponding values computed through a mathematical or numerical model. Indentation test can be analysed in the large deformation regime by finite element (FE) approaches, as already done in previous analyses [10-16] or by alternative numerical techniques, like in [19, 20]. Numerical simulation represents an useful tool also to the purpose of designing the experimental test, and selecting the most appropriate measurements, e.g. by sensitivity analysis [21].

Figure 1: Comparison between the experimental mean indentation curves obtained from conical (Rockwell) tests on an Al/TiB2 specimen with 50% TiB2 weight content and the corresponding curves recalculated

by a FE model of the test, supplemented by the identified material parameter set.

Let the indentation forces iF (i=1, …, N) and the corresponding penetration depth Mid represent the experimental

information collected from the mean indentation curves, visualised in Fig. 1. For any given value iF , numerical analysis

returns the corresponding penetration depth ( )Cid z as a function of the constitutive parameters contained in the selected material or fracture model (elastic moduli, yield limits, fracture energy, …), here collected by vector z. The discrepancy between quantities gathered from the experimental apparatus and from the simulation of the test can then be defined by the norm:

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2

1

( )( )N

Ci Mi

i i

d dw

z

z (1)

where iw represent weights on the displacement components Mid , assumed to be proportional to the variances which quantifies the measurement discrepancies, also visualised in Fig. 1. The optimum value of the sought parameters is represented by the entries of vector optz , which corresponds to the

minimum discrepancy. These values can be returned by a number of numerical methods implemented in widely available optimisation tools [22]. In this kind of application, the discrepancy function ( ) z is expected to be non-convex and to admit multiple minimum points, often almost equivalent from the engineering point of view, in the sense that the corresponding parameter values approximate to the same extent the available experimental data and return comparable representation of the real material behaviour, consistently with the selected constitutive model. The optimisation algorithm is hence run several times, starting from different initial parameter sets or, alternatively, evolutionary search techniques like genetic algorithms or soft computing methodologies are exploited; see, e.g., the review paper [20]. The computational efficiency of the simulation tool represents, hence, an important issue in this context. BULK MATERIAL PROPERTIES

ccording to a popular approach developed by Tamura et al. [23] for metal alloys and applied to different composite systems by a proper calibration of a characteristic ‘stress transfer’ parameter, the overall mechanical response of metal-ceramic composites is governed by the metal phase. Pressure-insensitive elastic-plastic laws,

like the classical Hencky–Huber–Mises (HHM) model, are therefore mostly considered at the macro-scale [24-26] and supplemented by constitutive parameters (elastic modulus, yield limit, hardening coefficients), evaluated on the basis of some mixture theory that depends on volume fractions. In the proposal by Bocciarelli et al. [15], volume fractions govern the transition from HHM model toward Drucker–Prager (DP) constitutive law, capable to describe the mechanical behaviour of ceramics [14]. The elastic domain defined by the traditional DP yield criterion with linear isotropic hardening is represented by:

11 ' ' 02 ij ijf I k h (2)

where: 'ij denote the components of the deviatoric stress tensor; 1I represents the first stress invariant, i.e. the trace of

the tensor collecting the stress components ij ; λ (>0) is the cumulative multiplier of the plastic deformations, which

develop as 0f and 0f (the superimposed dot denotes a rate quantity); α, k and h are constitutive parameters.

Internal friction α and initial cohesion k depend on the initial tensile and compressive yield limits, 0t and 0c , respectively, as follows:

0 0 0 0

0 0 0 0

23 ,3

c t c t

c t c t

k

(3)

while parameter h governs material hardening. HHM criterion can be recovered from DP model as the value of the internal friction coefficient α is set equal to zero.

The plastic rate components pij of the strain tensor are assumed to develop orthogonally to a potential surface ( )ijg as

follows:

11, ( ) ' '2

pij ij ij ij

ij

gg I

(4)

where β represents the dilatancy coefficient.

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Parameters governing DP constitutive law have been inferred from available experimental information gathered from conical Rockwell indentation of Al/TiB2, Al/ZrO2 and Cu/Al2O3 with about 40 to 50% weight metal content. Associative plasticity has been then introduced as reasonable hypothesis, so that α=β in relations (4). Poisson’s ratio has been a priori fixed due to the expected low influence of this parameter on indentation results; see [10, 14] and references therein. Then, the unknown overall material parameters to be returned by inverse analysis are: the elastic modulus E; the initial compression yield stress 0c ; the internal friction angle α and the hardening modulus h. Results listed in Tab. 1 concern the case of an Al/TiB2 sample with 50% TiB2 weight content produced by spark plasma sintering in a joint research project with Inasmet-Tecnalia (San Sebastian, Spain), partner of the KMM NoE [8]. The identified parameter values are given in terms of the average and of the standard deviation of several converged optimization solutions, which returned small similar values of ( )opt z starting from different initial z vectors. The

scatter on the value of the internal friction angle returned by the exploited inverse analysis procedure is relatively large, nevertheless α values were found much larger than zero in all considered situations, thus indicating that hydrostatic stress component plays a significant role in controlling the mechanical response of the considered metal-ceramic composites. The accuracy of the selected material model and of the optimal parameter set obtained from the present identification procedure can be appreciated by the comparison between the experimental information and the corresponding recalculated curves, drawn in Fig. 1.

Elastic modulus E [GPa]

Initial yield limit in compression

0c [MPa]

Internal friction angle

Arctan(α) [°]

Hardening modulus h [MPa]

Al/TiB2 57.9 (±3.7) 240.4 (±36.5) 10.7 (±4.3) 1032 (±91)

Table 1: Identified material properties after different initializations of the discrepancy minimisation algorithm for an Al/TiB2 sample with 50% TiB2 weight content.

The results of this research work showed that the overall material properties are strongly influenced by embedded defects and local damages, as earlier observed for different material systems [6]. Classical homogenization rules and mixture laws, which do not consider micro-structural details, fail in returning reliable quantitative prediction of constitutive parameters even in the elastic range.

FRACTURE PROPERTIES

he brittleness of the composite materials under investigation is mitigated by the ductility of their metal phase. In fact, during fracture processes, large plastic deformations produce metal ligaments which bridge crack surfaces as shown e.g. in Fig. 2. This image is relevant to a fracture experiment performed at the Technical University

Darmstadt (TUD), partner of the KMM Network, on a Cu/Al2O3 composite with interpenetrating network structure [1-3]. Fracture can be induced in a brittle material sample to parameter identification purposes by the sharp corners of a Vickers pyramidal tip, as shown in Fig. 3 for the case of a pure zirconia (ZrO2) specimen. Semi-empirical formulae based on linear elastic fracture mechanics (LEFM) permit to correlate the material toughness to the length of the cracks, measured from the centre of the imprint left by indentation [27, 28] which can be observed after unloading. However, experimental investigations show a systematic dependence of such estimated toughness values on the level of the applied indentation load. This result, inconsistent with LEFM assumptions, is likely originated by various error sources like the experimental difficulties in getting reliable measurement of the actual crack length and the influence of possible residual stresses, besides the existence of a crack-bridging zone at the crack tip [1, 4]. Furthermore, indentation results of brittle and quasi-brittle materials are affected by big scatters and influenced by microstructure [29, 30]. These limitations can be mitigated by an identification approach based on inverse analysis, as introduced above, comparing results gathered from the experiment and from the test simulation. Material separation during fracture propagation can be introduced in FE simulation of the indentation tests by interface elements, as shown e.g. in Fig. 4 and 5. The relationship between displacement discontinuities and the cohesive tractions transmitted across the fracture surface of metal-ceramic composites can be described by relatively simple and

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computational effective cohesive model originally proposed for mode I fracture for metals and bimetallic interfaces by Rose et al. [31], further extended in [32, 33] and widely employed in different application fields; see, e.g., [12, 14, 16, 37] and references therein.

Figure 2: Detail of a fracture experiment on a Cu/Al2O3 composite (courtesy of TUD)

Figure 3: Result of Vickers indentation on a ZrO2 specimen.

Figure 4: Simulation result of Vickers indentation on a ZrO2 specimen.




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Figure 5: Contour map of the crack opening displacements of the half-penny shape fracture surface produced by indentation (simulation results)

According to this popular formulation, the normal cohesive stress σ transferred during progressive mode I fracture processes can be described by the relationship:

( ) c c

c

w

w

ww e we

(5)

where: e is the Neper number; c is the maximum cohesive normal traction; w represents the opening displacement and

cw is a characteristic value of it, associated to the fracture energy fG of the material by the formula:

0

( )f c cG w dw e w

(6)

In mixed model fracture formulation, the opening displacement w is replaced by a scalar measure of the displacement jump vector across the interface and a further parameter is introduced to describe the shear strength of the interface [32, 33]. Preliminary studies have shown that the indentation curve, the geometry of the residual imprint and the crack length reflect to different extent fracture properties like c and fG , entering relation (5) and (6). The biggest sensitivity to a

change in these parameters is shown by the fracture length but, in turn, the actual position of the crack tip is rather difficult to establish with the required precision. For this reason, information about the geometry of the residual imprint and in-plane displacement components on the specimen surface after fracture propagation can be acquired by several kinds of microscopes nowadays available on the market, possibly endowed with suitable image correlation tools, see e.g. [35, 36]. These data can be exploited to enhance the identification of fracture properties through indentation test [12, 14, 16], including them in the discrepancy function (1), compared to the corresponding computed quantities. The application, now in progress, of reliable and robust identification procedures should also allow to verify the range of applicability of approaches based on modified mixture laws, proposed for fracture properties of metal-ceramic composites by Jin et al. [37], in analogy with [23].

CLOSING REMARKS

he envisaged increasing use of metal-ceramic composites for structural applications in the near future require adequate mechanical characterization, which can be better based on instrumented indentation and inverse analysis rather than on standard laboratory tests, due to difficulties and costs associated with material production and

specimen extraction processes. Recently developed parameter calibration methodologies, gathering experimental information from the measurement of the residual deformed configuration of the specimen after the test, besides from the indentation curves, are generally bound to enhance the identification of both bulk and fracture parameters.

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The application of these novel fast and reliable identification techniques can assist the selection of the constitutive models better suited to describe the behaviour of these composites and permits to appreciate the influence of micro-structural details on the overall mechanical characteristics. In fact, left porosity and micro defects generated by the production processes, generally neglected by material parameter prediction approaches based on classical homogenization rules, have strong influence on the properties of these quasi-brittle composites.

ACKNOWLEDGEMENT

U financial support to the project ‘Knowledge-based Multi-component Materials for durable and safe performance’ (KMM-NoE, NMP3-CT-2004-502243) is gratefully acknowledged.

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