Fibre Models. H. J. Herrmann and F. Kun

Transcription

1 Fibre Models H. J. Herrmann and F. Kun Computational Physics, IfB, HIF, E12, ETH, Hoenggerberg, 8093 Zürich, Switzerland Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, P.O.Box:5, Hungary Abstract. Fibre models have been introduced as simple models to describe failure. They are based on e probability distribution of broken fibres. The load redistribution after a fibre yields can be global or local and e first case can often be solved analytically. We will present an interpolation between ese e local and e global case and apply it to experimental situations like e compression of granular packings. Introducing viscoelastic fibres allows to describe e creep of wood. It is even possible to deal analytically wi a gradual degradation of fibres and consider damage as well as healing. In is way Basquin s law of fatigue can be reproduced and new universalities concerning e histograms of bursts and waiting times can be uncovered. INTRODUCTION Fracture and damage of composites is a very important scientific and technological problem which has attracted an intensive research over e past decades. Composite is a general term here for a broad class of materials which have a strongly heterogeneous microstructure like concrete or asphalt (also called particle composites) and for assembledge of subunits which are organized to build up superstructures, e.g. fiber reinforced composites where fiber are embedded in a matrix to improve e mechanical performance of e material. One of e first eoretical approaches to e fracture of composites was e fiber bundle model (FBM) introduced by Peires and Daniels [1, 2]. These early works initiated an intense research in bo e engineering and physics communities making fiber models one of e most important eoretical approaches to e damage and fracture of composite materials. Over e past years several extensions of e classical FBM have been worked out considering stress localization, e effect of matrix material between fibers, time dependent response, ermally activated breakdown [1, 2], damage and healing. Here we present recent advances in fiber bundle modeling of continuous breakage, of e shear failure of glued interfaces [3, 4] and of fatigue fracture of bituminous materials like asphalt [5]. We show various extensions of FBM including variable ranges and breaking in various steps. In order to account for e complex deformation states of interface elements under shear, we discretize e interface in terms elastic beams which can have stretching and bending deformation and fail due to e two deformation modes [3]. To analyze e effect of e finite load bearing capacity of failed interface regions which remain in contact, we assume at e beams/fibers can have a plastic behavior retaining a fraction of eir failure load[4]. Finally, we show at fatigue fracture of materials can be studied by means of fiber bundle models by introducing an ageing mechanism of fibers due to e accumulation of damage over e loading history of e

2 σ σ a) L F L b) E σ σ c c D( ) c) d) /2 FIGURE 1. a) Schematic FBM setup. b) Constitutive curve of a single fiber. c) Typical constitutive curve of a fiber bundle. d) Typical avalanche size distribution under GLS. system and subsequent healing over a longer time scale [5]. THE CLASSICAL FIBER BUNDLE MODEL We represent e disordered solid as a discrete set of parallel fibers of number N, organized on a regular lattice, see Fig. 1a. The fibers can solely support longitudinal deformation which allows to study only loading of e bundle parallel to fibers. When e bundle is subjected to an increasing external load F, e fibers behave linearly elastic until ey break at a failure load σ i, i = 1,...,N, as it is illustrated in Fig. 1b. The elastic behavior of fibers is characterized by e Young modulus E, which is identical for all fibers. The strengs σ i are independent identically distributed random variables wi e probability density p(σ ) and distribution function P(σ ). The randomness of breaking resholds is assumed to represent e disorder of heterogeneous materials. A widely used distribution in FBMs is e Weibull distribution P(σ ) = 1 exp [ ( σ ) m ] λ, where m and λ denote e Weibull index and scale parameter, respectively. After a fiber has failed, its load has to be shared by e remaining intact fibers. Historically, two extremal cases of load sharing are distinguished: in global load sharing (GLS), e load is equally redistributed over all intact fibers in e bundle. In e oer case of local load sharing (LLS), e entire load of e failed fiber is redistributed equally over its local neighborhood (usually nearest neighbors) in e lattice considered, leading to stress concentrations along failed regions (see Fig. 1a). Contrary to GLS models, LLS problems normally have to be solved numerically. The macroscopic constitutive behavior of e FBMs en takes e form σ() = E [1 P(E)], (1) where [1 P(E)] is e fraction of intact fibers at e deformation [6, 7]. A

3 representative example of σ() is presented in Fig. 1c for e case of Weibull distributed streng values. Wi stress controlled conditions, after each fiber breaking e load dropped by e broken fiber has to be redistributed over e surviving intact ones. The subsequent load redistribution after consecutive fiber failures can lead to an entire avalanche of breakings. For GLS, it was found at for a broad class of disorder distributions, e distribution D of avalanches of sizes follows a power law distribution wi an exponent 5/2 [8, 9] (see Fig. 1d) D( ) 5/2. (2) VARIABLE RANGE OF INTERACTION Here we introduce a one-parameter load transfer function to obtain a more realistic description of e interaction of fibers [10]. Varying its parameter, e load transfer function interpolates between e two limiting cases of load redistribution, i.e. e GLS and LLS schemes. The additional load received by an intact fiber i depends on its distance r i j from a fiber j which has just been broken. Furermore, elastic interaction is assumed between fibers such at e load received by a fiber follows a power law form. Hence, in e discrete model e stress-transfer function F(r i j,γ) takes e form F(r i j,γ) r γ i j, (3) where γ is an adjustable parameter and r i j is e distance of fiber i to e rupture point (x j,y j ). It can be seen at in e limits γ 0 and γ e load transfer function Eq. 3 recovers e two extreme cases of load redistribution of fiber bundle models, i.e. global and local load sharing, respectively. Computer simulations of e model described above wi e fibers arranged on a two dimensional square lattice of size L and Weibull distributed streng values, (m = 2, λ = 1), were performed varying e effective range of interaction γ over a broad interval. The avalanche size distribution, e cluster size distribution, and e ultimate streng of e bundle for several system sizes L were recorded. Two distinct regimes can be clearly distinguished: for small γ, e streng σ c is independent of e system size L. At a given point γ = γ c a crossover is observed, where γ c falls in e vicinity of γ = 2. For large γ all curves decrease wi N as σ c (N) α lnn. (4) This qualifies for a genuine short range behavior as found in LLS models, where e same relation was obtained for e asymptotic streng of e bundle [11]. It is wor noting at a similar crossover at γ 2 from GLS to LLS behavior is also present in e avalanche size distribution. Moreover, a detailed analysis of e distribution of cluster sizes, i.e. clusters of broken fibers preceding failure, confirms is crossover.

4 σ a) σi b) 3 σ σ1 σ2 FIGURE 2. The damage law of a single fiber of e continuous damage model for quenched (a) and annealed (b) disorder, when multiple failure is allowed. The horizontal lines indicate e damage resholds σ i. CONTINUOUS DAMAGE Here we introduce a so-called continuous damage fiber bundle model (CDFBM) as an extension of e classical FBM by generalizing e damage law of fibers. We assume at e stiffness of fibers gradually decreases in consecutive failure events [12, 2]. The continuous damage model is composed of N parallel fibers wi identical Youngmodulus E and random failure resholds σ i. The fibers are assumed to have linear elastic behavior up to breaking (brittle failure), but at e failure point e stiffness of e fiber is reduced by a factor a, where 0 a < 1, i.e. e stiffness of e fiber after failure is ae. A fiber can now fail more an once and e maximum number k max of failures allowed is a parameter of e model. Once a fiber has failed, its damage reshold σ i can eier be kept constant for e furer breakings (quenched disorder, see Fig. 2a) or new failure resholds of e same distribution can be chosen (annealed disorder, see Fig. 2b), which can model some microscopic rearrangement of e material after failure. However, e model can also be considered as e discretization of e system on leng scales larger an e size of single fibers, so at one element of e model consists of a collection of fibers wi matrix material in between. In is case e microscopic damage mechanism resulting in multiple failure of e elements is e gradual cracking of matrix and e breaking of fibers. After failure e fiber skips a certain amount of load which has to be taken by e oer fibers. For e load redistribution we assume infinite range of interaction among fibers (GLS); furermore, an equal strain condition is imposed which implies at stiffer fibers of e system carry more load. At a strain, e load of a fiber i at has failed k(i) times reads as f i () = Ea k(i), (5) where Ea k(i) is e actual stiffness of fiber i. It is important to note at, in spite of e infinite interaction range, Eq. 5 is different from e usual global load sharing, where all e intact fibers always carry e same amount of load. In e following, e initial fiber stiffness E will be set to unity.

5 t d f T F l z ζ FIGURE 3. The sheared interface is discretized in terms of elastic beams (le f t), which suffer stretching and bending deformation (middle) and fail due to two deformation modes (right). FAILURE OF INTERFACES UNDER SHEAR Shear failure of glued interfaces is treated here by extending e classical fiber bundle model to study interfacial failure. Our model represents e interface as an ensemble of parallel beams connecting e surface of two rigid blocks, see Fig. 3. The beams are assumed to have identical geometrical extensions (leng l and wid d) and linearly elastic behavior characterized by e Young modulus E. In order to capture e failure of e interface, e beams are assumed to break when eir deformation exceeds a certain reshold value. Under shear loading of e interface, beams suffer stretching and bending deformation resulting in two modes of breaking. The stretching and bending deformation of beams can be expressed in terms of a single variable, i.e. longitudinal strain = l/l, which enables us to map e interface model to e simpler fiber bundle models. The two breaking modes can be considered to be independent or combined in e form of a von Mises type breaking criterion. The streng of beams is characterized by e two reshold values of stretching 1 and bending 2 a beam can wistand. The breaking resholds are assumed to be randomly distributed variables of e joint probability distribution p( 1, 2 ). Assuming e breaking modes to be independent, a single beam breaks if eier its stretching or bending deformation exceeds e respective breaking reshold 1 or 2, i.e. failure occurs if f()/ 1 1 or g()/ 2 1, where f() and g() describe e stretching and bending breaking modes, respectively. The constitutive behavior of e interface can be obtained by integrating e load of single beams over e 2 max 1 max intact ones in e plane of breaking resholds One gets σ = d 2 d 1 p( 1, 2 ). Assuming e resholds of e two breaking modes to be independently distributed, e disorder distribution factorizes p( 1, 2 ) = p 1 ( 1 )p 2 ( 2 ) and σ() takes e simple form σ() = [1 P 1 ( f())][1 P 2 (g())]. To determine e behavior of e system for complicated disorder distributions it is necessary to work out a computer simulation technique. The presence of two breaking modes substantially reduces e critical stress σ c and strain c (σ c and c are e value and location of e maximum of e constitutive curves) wi respect to e case when failure of elements occurs solely under stretching [1, 2]. The coupling of e two breaking modes in e form of e von Mises criterion gives rise to furer reduction g() f()

6 of e streng of e interface. Simulations revealed at in spite of e complicated microscopic process of damaging, e size distribution of avalanches shows e same behavior as for simple FBMs, i.e. it has a power law form of an exponent 5/2 which is universal. PLASTIC FIBER BUNDLES When an interface gradually fails under shear, damaged regions of e interface can still transmit load contributing to e overall load bearing capacity of e interface. This can occur, for instance, when e two solids remain in contact at e failed regions and exert friction force on each oer. In many applications e glue between e two interfaces has disordered properties but its failure characteristics is not perfectly brittle, e glue under shear may also yield carrying a constant load above e yield point. In order to capture is effect in FBMs, we assume at after e breaking of a fiber at e failure reshold, it may retain a fraction 0 α 1 of its ultimate load σ i, i.e. it will continue to σ i transfer a constant load ασ i between e surfaces. Plastic behavior implies at e load carried by e broken fibers is independent of e external load, furermore, it is a random variable due to e randomness of e breaking resholds. Varying e value of α, e model interpolates between e perfectly brittle failure (α = 0) and perfectly plastic (α = 1) behavior of fibers. The load stored by e failed fibers increases e overall streng of e bundle, and it reduces e load increment redistributed over e intact fibers, which strongly affects e failure process of e interface [4]. The failure process of e bundle is dominated by e competition of fiber breaking by local stress enhancement due to load redistribution and by local weakness due to disorder. Our detailed analysis revealed at e relative importance of e two effects is controlled by e parameter α. Below e critical point α < α c high stress concentration can develop around cracks so at e failure of e bundle occurs due to localization. Above e critical point α α c e macroscopic response of e LLS bundle becomes practically identical wi e GLS constitutive behavior showing e dominance of disorder. Analyzing e evolution of e micro-structure of damage wi increasing α, e transition proved to be continuous analogous to percolation [4]. FIBER BUNDLE MODEL FOR FATIGUE AND HEALING If composites are subject to periodic external loading wi an amplitude below e tensile streng, ey often show a gradual accumulation of deformation which can even lead to macroscopic failure over a finite time. This subcritical crack grow and failure called fatigue fracture is one of e most important processes which limits e lifetime of structural components in applications. Fatigue fracture is e typical distress of asphalt in pavements and roadways due to e repeated traffic loading. Figure 4a shows at increasing e number of loading cycles N cycle e deformation accumulates and e system approaches macroscopic failure at a finite number N f of cycles wi a diverging deformation rate d/dt. Studying e lifetime N f of e specimen as a function of e external load σ 0 /σ c (Fig. 4b) ree regimes of e fatigue process can be distinguished:

7 p=e / c aa bb ÜÔ Ö Ñ ÒØ Å t f t Æ ÝÐ / c N cycle N f ÕÙ Ò Ö Ñ ¼ FIGURE 4. a) versus e number of loading cycles N cycle. The continuous lines are fits by Eq. (6). Inset: Load on single fibers in FBM as function of time at different values of σ 0 /σ c. b) The lifetime N f as function of load σ 0 /σ c. approaching e tensile streng σ 0 σ c rapid failure occurs, while at e oer extreme a so-called fatigue limit σ l /σ c can be identified below which no macroscopic failure occurs and e system has an infinite lifetime. For intermediate load values a so-called Basquin regime is found, where e lifetime has a power law dependence of e external load N f (σ 0 /σ c ) α. For e exponent α = 2.2 ± 0.1 was obtained (Fig. 4b). In spite of e large amount of experimental results gaered over e past decades, is process is still not understood. In order to capture e main ingredients of fatigue failure of asphalt we recently introduced a fiber bundle model [5]. Fibers fail due to two physical mechanisms, namely, immediate breaking occurs when e local load exceeds e streng of fibers, furermore, intact fibers undergo a damage accumulation process by e nucleation of microcracks during e loading history of e system. Fibers are assumed to have a finite damage tolerance, i.e. when e amount of accumulated damage reaches a reshold value e fiber fails. The two breaking resholds of immediate breaking and damage tolerance are independent random variables. Healing of microcracks is captured in e model by limiting e range of memory of e system over which e loading history contributes to e accumulated damage. Assuming global load sharing, under a constant external load σ 0 e evolution equation of e fiber bundle for fatigue failure can be cast in e form t σ 0 = [1 F(a 0 e (t t ) τ p(t ) γ dt )][1 G(p(t))] p(t), (6) where F and G are e cumulative distributions of e reshold values of damage tolerance and immediate breaking, respectively. Eq. (6) has to be solved for e load of single fibers p(t) as a function of time t, which is simply related to e deformation of e bundle p(t) = E(t). The exponential term in e argument of F takes into account

8 e healing of microcracks by limiting e range of memory to a finite value τ. We find at our model provides an excellent quantitative agreement wi e experimental findings as demonstrated by Figure 4a,b). CONCLUDING REMARKS In is brief review we showed at e classical Fiber Bundle Model can be improved to account for complex deformation states, plastic behavior and ageing of materials providing a quantitative insight into e failure process of a broad class of composite materials. In e limiting case of global load sharing most of e characteristic quantities can be obtained in closed analytic forms, while e realistic treatment of localized interactions requires very large computational effort. ACKNOWLEDGMENTS We ank Raul Cruz Hidalgo, Stefano Zapperi, Frank Raischel and José Soares Andrade Jr. as collaborators in some of e reported work. This work was supported by SFB381. F. Kun acknowledges financial support of NKFP-3A/043/04 and OTKA T H. J. Herrmann is grateful for e Max Planck Prize. REFERENCES 1. H. J. Herrmann and S. Roux (eds.), Statistical Models for e Fracture of Disordered Media, (Nor- Holland, Amsterdam, 1990); M. J. Alava, P. Nukala and S. Zapperi, Adv. in Phys. 55, 349 (2006). 2. F. Kun, S. Zapperi, and H. J. Herrmann, Eur. Phys. J. B 17, 269 (2000). 3. F. Raischel, F. Kun, and H. J. Herrmann, Phys. Rev. E 72, (2005). 4. F. Raischel, F. Kun, and H. J. Herrmann, Phys. Rev. E 73, (2006). 5. F. Kun, M. H. A. S. Costa, R. N. Costa Filho, J. S. Andrade Jr, J. B. Soares, S. Zapperi, and H. J. Herrmann, cond-mat/ R. da Silveira, Am. J. Phys. 67, 1177 (1999). 7. D. Sornette, J. Phys. A 22, L243 (1989). 8. M. Kloster, A. Hansen and P. C. Hemmer, Phys. Rev. E 56, 2615 (1997). 9. A. Hansen and P. C. Hemmer, Phys. Lett. A 184, 394 (1994). 10. R. C. Hidalgo, Y. Moreno, F. Kun and H. J. Herrmann, Phys. Rev. E 65, (2002). 11. S. L. Phoenix and I. J. Beyerlein, in A. Kelly C. Zweben, (Eds.): Comprehensive Composite Materials, volume 1, chapter 1.19, Pergamon-Elsevier Science (2000), p R. C. Hidalgo, F. Kun and H. J. Herrmann, Phys. Rev. E 64, (2001).