Fibre bundle models for creep rupture analysis of polymer matrix composites

10 Fibre bundle models for creep rupture analysis of polymer matrix composites F. K un, University of Debrecen, Hungary Abstract: Fibre bundle models are one of the most important theoretical approaches to the investigation of the failure of fibre reinforced composites under various loading conditions. The chapter first presents the basic concept of fibre bundle models. It then reviews extensions of the modelling approach to capture physical mechanisms responsible for the time-dependent response and creep rupture of composites. The chapter discusses the macroscopic time evolution of the fibre bundle, the microscopic process of rupture, and finally outlines methods to estimate the rupture life of composites. Key words: creep rupture, fibre bundle model, global load sharing.

10.1 Introduction Under a constant external load materials typically exhibit a time-dependent deformation and fail after a finite time. This creep rupture process also substantially affects the technological application of fibre-reinforced composites which calls for a thorough theoretical understanding. Fibre bundle models (FBM) are one of the most important theoretical approaches to the damage and fracture of fibrereinforced composite materials which have also been applied to investigate creep rupture phenomena. The basic concept of fibre bundle modelling was introduced by Peirce (1926) to understand the strength of cotton yarns. Cotton threads were represented by fibres with different load-bearing capacity organized in a bundle. In his pioneering work, Daniels (1945) provided the sound probabilistic formulation of the model and carried out a comprehensive study of bundles of threads assuming equal load sharing after subsequent failures. Already this basic setup of the model provided a surprisingly deep insight into the microscopic dynamics of failure of fibrous systems. The first attempt to capture fatigue and creep effects in FBMs was made by Coleman (1958), who proposed a time-dependent formulation of the model, assuming that the strength of loaded fibres is a decreasing function of time. Later on these early works initiated an intense research in both the engineering (Harlow and Phoenix, 1978; Harlow and Phoenix, 1991; Phoenix, et al., 1997) and physics (Alava et al., 2006; Herrmann and Roux, 1990; Chakrabarti and Benguigui, 1997; Curtin, 1998; Sornette, 1989; Andersen et al., 1997; Pradhan and Chakrabarti, 2003) communities making fibre bundle models successful not only for the study of composites but also for the understanding of the damage and fracture of the broader class of disordered materials. 327 © Woodhead Publishing Limited, 2011

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During the past few decades, the development of fibre bundle models encountered two kinds of challenges: on the one hand, it is important to work out realistic models of materials failure which have a detailed representation of the microstructure of the material, the local stress fields and their complicated interaction. Applications of materials in construction components require the development of analytical and numerical models which are able to predict the damage histories of loaded specimens in terms of the characteristic microscopic parameters of the constituents. In this context, fibre bundle models served as a starting point to develop more realistic micromechanical models of the failure of fibre-reinforced composites including also polymer matrix composites (PMC). Analytical methods and numerical techniques have been developed making possible realistic treatment of even large-scale fibrous structures (Evans and Zok, 1994; Du and McMeeking, 1995; Curtin and Scher, 1997; Curtin, 1998; Chudoba et al., 2006; Phoenix and Beyerlein, 2000; Kun and Herrmann, 2000; Hidalgo et al., 2002a; Kun et al., 2003; Kovacs et al., 2008). On the other hand, it is important to reveal universal aspects of the fracture of composites which are independent of specific material properties relevant on the micro level. Such universal quantities can help to extract the relevant information from measured data and make it possible to design monitoring techniques of the gradual degradation of composites’ strength and construct methods to forecast catastrophic failure events (Alava et al., 2006; Garcimartin et al., 1997; Vujosevic and Krajcinovic, 1997; Johansen and Sornette, 2000; Pradhan and Chakrabarti, 2003; Pradhan et al., 2005; Kun and Herrmann 2000; Kun et al., 2003a, 2003b; Nechad et al., 2005, 2006). The damage and fracture of disordered materials addresses several interesting problems also for statistical physics. It is challenging to embed the failure and breakdown of materials into the general framework of statistical physics clarifying its analogy to phase transitions and critical phenomena. For this purpose fibre bundle models provide an excellent testing ground of ideas offering also the possibility of analytic solutions (Alava et al., 2006; Baxevanis and Katsaunis, 2006; Baxevanis and Katsaunis, 2007; Baxevanis, 2008; Pradhan and Chakrabarti, 2003; Pradhan et al., 2005; Kun et al., 2000, 2003a, 2003b; Kovacs et al., 2008). Fibre-reinforced composites (FRCs), where long parallel fibres are embedded in a matrix material, provide an improved mechanical performance compared to their constituents. Subjecting the composite to an external load, the matrix material typically suffers multiple cracking or yields at load levels much below the strength of the fibres. It has the consequence that practically all the load is kept by the fibres and the matrix material mainly affects the stress transfer following fibre breakings (Jones, 1999; Evans and Zok, 1994). Motivated by these observations, fibre bundle models consist of a set of parallel fibres having statistically distributed strengths. The sample is loaded parallel to the fibres’ direction, and the fibres fail if the load on them exceeds their threshold value. In stress-controlled experiments, after each fibre failure the load carried by the broken fibre is redistributed among the intact ones. In basic FBMs the matrix material only affects the load transfer from the broken to the

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intact fibres. The behaviour of a fibre bundle under external loading strongly depends on the range of interaction, i.e. on the range of load sharing among fibres. Exact analytic results on FBM have been achieved in the framework of the mean field approach, or global load sharing, which means that after each fibre breaking the stress is equally distributed on the intact fibres, implying an infinite range of interaction and a neglect of stress enhancement in the vicinity of failed regions (Daniels, 1945; Sornette, 1989; Hemmer and Hansen, 1992; Andersen et al., 1997; Garcimartin et al., 1997; Kloster et al., 1997; Kun and Herrmann, 2000; Moreno et al., 2000; Hidalgo et al., 2002b; Kun et al., 2003a, 2003b; Kovacs et al., 2008; Pradhan and Chakrabarti, 2003). In spite of their simplicity, FBMs capture the most important aspects of material damage and they provide a deep insight into the fracture process. Based on their success, FBMs have served as a starting point for more complex models like the micromechanical models of the failure of fibrereinforced composites (Harlow and Phoenix, 1991; Phoenix et al., 1997; Curtin and Scher, 1997; Du and McMeeking, 1995; Kun and Herrmann, 2000; Hidalgo et al., 2002a, 2002b). Over the past few years several extensions of FBM have been carried out by considering stress localization (local load transfer) (Harlow and Phoenix, 1991; Phoenix et al., 1997; Curtin, 1998; Hidalgo et al., 2002b), the effect of matrix material between fibres (Harlow and Phoenix, 1991; Phoenix et al., 1997; Hidalgo et al., 2002a, 2002b), possible non-linear behaviour of fibres (Kun et al., 2000; Hidalgo et al., 2001), coupling to an elastic block (Roux et al., 1999), and thermally activated breakdown (Phoenix and Tierney, 1983; Newman and Phoenix, 2001; Curtin and Scher, 1997; Roux, 2000; Scorretti et al., 2001; Yoshioka et al., 2008). Modelling the creep rupture of fibre-reinforced composites requires a substantial extension of FBMs to capture the physical mechanisms which lead to timedependent response of the composite under a constant external load. In this chapter we first present the basic formulation of the classical fibre bundle model and briefly summarize the most important recent results obtained on the macroscopic response and microscopic damage process in the framework of FBMs. For the investigation of creep rupture of fibre composites two modelling approaches will be discussed. First we consider a bundle of visco-elastic fibres which have an instantaneous breaking, then we construct a bundle of brittle fibres which undergo a slow relaxation process after failure. Assuming global load sharing, the time evolution of the deformation and of the gradual damaging of the creeping system will be analyzed analytically, while the microscopic process of failure will be investigated by means of computer simulations. Based on fibre bundle models, reliable methods will be deduced which make it possible to estimate the rupture life or to forecast the imminent catastrophic failure event of composites.

10.2 Fibre bundle model In the framework of the fibre bundle model, the composite is represented as a discrete set of parallel fibres of number N organized on a regular lattice (square,

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triangular, . . .; see Fig. 10.1a. The fibres can solely support longitudinal deformation which make it possible to study only loading of the bundle parallel to the fibre axis. When the bundle is subjected to an increasing external load, the fibres are assumed to have perfectly brittle response, i.e. they have a linearly elastic behaviour until they break at a failure load σ ith, i = 1, . . ., N as illustrated in Fig. 10.1b. The elastic behaviour of fibres is characterized by the Young’s modulus E, which is identical for all fibres. The breaking of fibres is assumed to be instantaneous and irreversible such that the load on broken fibres drops down to zero immediately at the instant of failure (see Fig. 10.1b); furthermore, broken fibres are never restored (no healing). The strength of fibres σ ith, i = 1, . . ., N, i.e. the value of the local load at which they break, is an independent identically distributed random variable with the probability density p(σth) and distribution function . The randomness of breaking thresholds is assumed to represent the disorder of fibre strength, and hence it is practically the only component of the classical FBM where material-dependent features (e.g. amount of disorder) can be taken into account. After a fibre fails its load has to be shared by the remaining intact fibres. The range and form of interaction of fibres, also called the load sharing rule, is a crucial component of the model which has a substantial effect on the micro and macro behaviour of the bundle. Most of the studies in the literature are restricted to two extreme forms of the load sharing rule: in the case of global load sharing (GLS), also called equal load sharing (ELS), the load is equally redistributed over all intact fibres in the bundle irrespective of their distance from the failed one (Daniels, 1945; Coleman, 1958; Sornette, 1989; Curtin and Scher, 1997; Phoenix, 2000; Kloster et al., 1997; Pradhan et al., 2005). The GLS rule corresponds to the mean field approximation of FBM where the topology of the fibre bundle becomes

10.1  (a) Basic setup of the fibre bundle model. The system is represented as a bundle of parallel fibres. (b) Single fibres have a linearly elastic behaviour with an identical Young’s modulus E up to failure at a random threshold value σth or εth (brittle breaking).

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irrelevant. Such a loading condition naturally arises when parallel fibres are loaded between perfectly rigid platens, like for the wire cable of an elevator. FBM with global load sharing is a usual starting point for more complex investigations since it makes it possible to obtain the most important characteristic quantities of the bundle in closed analytic forms (Phoenix, 2000; Phoenix and Beyerlein, 2000; Kloster et al., 1997; Curtin, 1997; Hemmer and Hansen, 1992; Pradhan et al., 2005; Pradhan and Hansen, 2003; Andersen et al., 1997; Sornette, 1989; Hidalgo et al., 2001). In the other extreme of the local load sharing (LLS), the entire load of the failed fibre is redistributed equally over its local neighbourhood (usually nearest neighbours) in the lattice considered, leading to stress concentrations along failed regions. Due to the non-trivial spatial correlations, the analytic treatment of LLS bundles has serious limitations (Gomez et al., 1993; Phoenix et al., 1997; Harlow and Phoenix, 1978; Harlow and Phoenix, 1991) most of the studies here rely on large-scale computer simulations (Hansen and Hemmer, 1994; Curtin, 1998; Harlow and Phoenix, 1978; Hidalgo et al., 2002a, 2002b). Such localized load sharing occurs when a bundle of fibres is loaded between plates of finite compliance. In the following we briefly summarize the most important results of FBMs on the microscopic failure process and macroscopic response of disordered materials under quasi-static loading conditions focusing on the case of equal load sharing. Loading of a parallel bundle of fibres can be performed in two substantially different ways: when the deformation ε of the bundle is controlled externally, the load on single fibres σi, i = 1, . . ., N is always determined by the externally imposed deformation ε as σi = Eε, i.e. no load sharing occurs and consequently the fibres break one by one in the increasing order of their breaking thresholds. At a given deformation ε the fibres with breaking thresholds σ ith < Eε are broken; furthermore, all intact fibres keep the equal load Eε. Hence, the macroscopic constitutive behaviour σ (ε) of the bundle can be cast in the analytic form

[10.1]

where the term 1 – P(ε) provides the fraction of intact fibres at the deformation ε. For a broad class of threshold distributions the constitutive curve Eq. 10.1 has a linearly increasing behaviour for low deformations followed by a quadratic maximum and a softening regime. The stochastic strength of fibres is well described by the Weibull distribution

[10.2]

where the parameters ρ and λ depend on material properties. The value of λ sets the scale of fibres’ strength, while the Weibull exponent ρ controls the amount disorder in the system. Figure 10.2 presents the constitutive curve Eq. 10.1 of FBMs with Weibull distributed thresholds for different values of the Weibull exponent ρ. In stress-controlled experiments gradually increasing the external

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10.2  Constitutive curves of fibre bundles with Weibull distributed fibre strength for different values of the Weibull exponent ρ. The value of the scale parameter λ and of Young’s modulus E was fixed λ = 1, E = 1.

load σ, a more complex failure process arises: breaking events are followed by the redistribution of load over the remaining intact fibres which might induce further breakings. It follows that the breaking of a single fibre can trigger an entire avalanche of breakings which either stops and the bundle stabilizes, or the avalanche destroys the entire bundle leading to macroscopic failure. The catastrophic avalanche appears when the maximum of σ(ε) is reached, hence the position εc and value σc of the maximum define the critical strain and stress of the bundle. It can be observed in Fig. 10.2 that decreasing the amount of disorder, i.e. increasing the Weibull exponent ρ, the macroscopic failure of the bundle becomes more and more brittle. Analytic calculations revealed that the macroscopic strength of fibre bundles depends on the size of the bundle σc(N), i.e. assuming equal load sharing over the fibres σc(N) rapidly converges to a finite number with increasing N. However, when the load redistribution gets localized to a small neighbourhood of failed fibres, the bundle becomes more brittle with a lower strength than its equal load sharing counterpart, and in the limit of N → ∞ the strength of the bundle tends to zero with the logarithm of the number of fibres σc(N) ∝ 1/ln N (Kloster et al., 1997; Phoenix and Beyerlein, 2000). As the external load approaches the ultimate tensile strength σc, damaging of the bundle proceeds in larger and larger breaking bursts. Kloster et al. (1997) and Pradhan et al. (2005) pointed out that the distribution P of burst sizes ∆ in FBMs has a power law

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functional form P(∆) ∝ ∆–κ with an exponent κ = 5/2 which is universal for a broad class of fibre strength distribution.

10.3 Fibre bundle models for creep rupture In order to model creep rupture of fibre-reinforced composites the classical fibre bundle model outlined above has to be complemented with physical mechanisms which capture the time-dependent response of composite materials. In this chapter we consider two physical origins of time-dependent behaviour which are of high importance to model polymer composites. First, we assume that the fibres are visco-elastic and exhibit a time-dependent response under a constant load. Then we consider the case when the fibres are brittle without any time dependence; however, due to the plastic response of the surrounding matrix material broken fibres undergo a slow relaxation process. The slow redistribution of the load after failure events gives rise to an overall time-dependent response of the entire system. In the literature, time-dependent fibre bundles have also been introduced by assuming that brittle fibres have a finite (stochastically distributed) lifetime when subject to a constant load (Coleman, 1958; Newman and Phoenix, 2001; Roux, 2000; Scorretti et al., 2001; Yoshioka et al., 2008). This breaking mechanism may arise due to thermally activated crack nucleation and damage accumulation processes. These types of modelling approaches will not be presented here.

10.3.1  Bundle of viscoelastic fibres One of the simplest physical mechanisms which results in a time-dependent response of a system under a constant external load is the visco-elastic behaviour of the constituents. Following the general presentation of FBMs in Section 10.2, the visco-elastic fibre bundle model of creep rupture of composites consists of N parallel fibres having visco-elastic constitutive behaviour. For simplicity, the pure linearly visco-elastic fibres are modelled by a Kelvin-Voigt element which consists of a spring and a dashpot in parallel, which is illustrated in Fig. 10.3a (Du and McMeeking, 1995; Hidalgo et al., 2002a, 2002b). The constitutive equation of . fibres can be cast into the form σ0 = βε + Eε, where σ0 is the constant external load, β denotes the damping coefficient, and E is the Young’s modulus of fibres. In order to capture failure in the model a strain-controlled breaking criterion is imposed, i.e. a fibre fails during the time evolution of the system when its strain exceeds a i , i = 1, . . ., N drawn from a probability distribution P(ε). For breaking threshold εth the stress transfer between fibres following local failure events, equal load sharing is assumed, i.e. the excess load is equally shared by all the remaining intact fibres, which provides a satisfactory description of load redistribution in unidirectional long fibre reinforced composites (Phoenix et al., 1997; Curtin, 1997; Jones, 1999). The construction of the model is illustrated in Figure 10.3a. In the framework of global load sharing, most of the quantities describing the behaviour of the fibre

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10.3  (a) The viscoelastic fibre bundle model. Intact fibres are modelled by Kelvin-Voigt elements. (b) Deformation–time diagram ε(t) of the model obtained analytically at different external loads σ below and above σc.

bundle can be obtained analytically. In this case the time evolution of the system under a steady external load σ0 is described by the first-order differential equation

[10.3]

where the visco-elastic behaviour is coupled to the failure of fibres (Hidalgo et al., 2002a; Kun et al., 2003). Note that the above equation of motion has a similar structure to the constitutive equation of simple fibre bundles Eq. 10.1; the viscoelasticity just introduces a delay term in the equation. To describe the creeping process of the fibre bundle, Eq. 10.3 has to be solved for the deformation ε(t) at a constant external load σ0. For the behaviour of the solutions ε(t), two distinct regimes can be distinguished depending on the load level σ0: when σ0 falls below a critical value σc, Eq. 10.3 has a stationary solution εs, which can be obtained by . setting ε = 0 so that

[10.4]

It means that as long as this equation can be solved for εs at a given external load σ0, the solution ε(t) of Eq. 10.3 converges to the stationary value ε(t) → εs when t → ∞, and the system suffers only a partial failure. However, when σ0 exceeds . the critical value σc no stationary solution exists; furthermore, the derivative ε remains always positive, which implies that for σ0 > σc the strain of the system ε(t) monotonically increases until the system fails globally at a finite time tf (Hidalgo et al., 2002b; Kun et al., 2003a, 2003b). The behaviour of ε(t) is presented in Figure 10.3b for several values of σ0 below and above σc for Weibull

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distributed breaking thresholds with the parameters λ = 1, ρ = 2. It follows from the above argument of stationary solutions that the critical value of the load σc is the ultimate tensile strength (UTS) of the bundle. The creep rupture of the viscoelastic bundle can be interpreted so that for σo ≤ σc the bundle is partially damaged implying an infinite lifetime tf = ∞ and the emergence of a stationary macroscopic state, while above the critical load σ0 > σc global failure occurs at a finite time tf, but in the vicinity of σc the global failure is preceded by a long-lived steady state. The nature of the transition occurring at σc can be characterized by analyzing how the creeping system behaves when approaching the critical load both from below and above. For σ0 ≤ σc the bundle relaxes to the stationary deformation εs through a gradually decreasing breaking rate of fibres. It can be shown analytically that ε(t) has an exponential relaxation to εs with a characteristic time scale τ that depends on the external load σ0 as τ ∝ (σc – σ0)–1/2 for σ0 < σc, i.e. when approaching the critical point from below the characteristic time of the relaxation to the stationary state diverges according to a universal power law with an exponent –1/2 independent of the form of the disorder distribution P. Above the critical point the lifetime tf defines the characteristic time scale of the bundle which can be cast in the analytic form tf ∝ (σ0 – σc)–1/2 for σ0 > σc, so that tf also has a power law divergence at σc with a universal exponent –1/2 like τ below the critical point. Hence, for global load sharing the system exhibits scaling behaviour on both sides of the critical point indicating a continuous transition at the critical load σc. It has been discussed in the previous section that even in the case of equal load sharing, the macroscopic strength of fibre bundles exhibits a size effect, which gets more pronounced when the load sharing is localized. We have shown analytically that the lifetime of creeping fibre bundles shows a similar size effect. Fixing the external load above the critical point σ0 > σc, the lifetime tf (N) of a finite bundle of N fibres can be cast into the analytic form

[10.5]

which means that tf (N) exhibits a universal scaling tf (N) – tf (∞) ∝ 1 / N with respect to the number N of fibres of the bundle (Hidalgo et al., 2002a; Kun et al., 2003). Here tf (∞) denotes the lifetime of the infinite system N → ∞. It has to be emphasized that the precise form of the threshold distribution affects only the multiplication factor of the N dependence.

10.3.2  Bundle of slowly relaxing fibres Another important microscopic mechanism which can lead to macroscopic creep is the slow relaxation of fibres after breaking. In this case, fibres of the composite are linearly elastic until they break; however, after breaking they undergo a slow relaxation process, which can be caused, for instance, by the sliding of broken fibres with respect to the matrix material or by the creeping matrix. To take into

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account this effect, our approach is based on the model introduced by Du and McMeeking (1995), Fabeny and Curtin (1996), Hidalgo et al. (2002b), and Kun et al. (2003), where the responses of a visco-elastic-plastic matrix reinforced with elastic and also visco-elastic fibres have been studied. The model consists of N parallel fibres, which break in a stress-controlled way, i.e. by subjecting a bundle to a constant external load, fibres break during the time evolution of the system when the local load on them exceeds a stochastically distributed breaking i , i = 1, . . ., N. Intact fibres are assumed to be linearly elastic i.e. threshold σ th σ = Ef εf holds until they break, and hence for the deformation rate it applies

[10.6]

Here εf denotes the strain and Ef the Young’s modulus of intact fibres, respectively. The main assumption of the model is that when a fibre breaks its load does not drop down to zero instantaneously; instead it undergoes a slow relaxation process introducing a time scale into the system. In order to capture this effect, the broken fibres with the surrounding matrix material are modelled by Maxwell elements as illustrated in Figure 10.4a, i.e. they are conceived as serial couplings of a spring and a dashpot which result in a non-linear response

[10.7]

Here σb and εb denote the time dependent load and deformation of a broken fibre, respectively. The relaxation process of broken fibres is characterized by three

10.4  (a) Bundle of brittle fibres which undergo a slow relaxation process after failure. Broken fibres are modelled by Maxwell elements which are serial couplings of a spring and a dashpot. (b) Deformation–time diagrams of the model obtained at different load levels below and above the critical load σc.

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parameters Eb, B and m, where Eb is the effective stiffness of a broken fibre, and the exponent m characterizes the strength of non-linearity of the element. We study the behaviour of the system for the region m ≥ 1. Assuming global load sharing for the load redistribution, the constitutive equation describing the macroscopic elastic behaviour of the composite reads as

[10.8]

The second term on the right-hand side of the above equation takes into account that broken fibres carry also a certain amount of load σb(t), furthermore, P(σ (t)) and 1 – P(σ (t)) denote the fraction of broken and intact fibres at time t, respectively (Du and McMeeking, 1995; Fabeny and Curtin, 1996; Kun, 2003; Kovacs, 2008). It can be seen from Eq. 10.8 that under a constant external load σ0, the load of intact fibres σ will also be time dependent due to the slow relaxation of the broken ones. Due to the boundary condition illustrated in Fig. 10.4a, the two time derivatives . . have to be always equal: εf = εb. The differential equation governing the time evolution of the system can be obtained by expressing σb in terms of σ using the boundary condition and Eq. 10.7, and substituting it finally into Eq. 10.8 [10.9] In order to determine the initial condition for the integration of Eq. 10.9 we note that upon subjecting the undamaged specimen to an external stress σ0 all the fibres attain this stress value immediately due to the linear elastic response. Hence the time evolution of the system can be obtained by integrating Eq. 10.9 with the initial condition σ (t = 0) = σ0. Since intact fibres are linearly elastic, the deformation-time history ε(t) of the model can be deduced as ε (t) = σ (t) / Ef , which has an initial jump to ε0 = σ0 / Ef . It follows that those fibres which have breaking thresholds σ thi smaller than the externally imposed σ0 immediately break. To characterize the macroscopic behaviour of the composite the solutions σ(t) of Eq. 10.9 have to be analyzed at different values of the external load σ 0. Similarly to the previous model, two different regimes of σ(t) can be distinguished depending on the value of σ 0: if the external load falls below the critical load σ c a stationary . solution σs of the governing equation exists which can be obtained by setting σ = 0 in Eq. 10.9

[10.10]

This means that until the above equation can be solved for σs the solution σ(t) of Eq. 10.9 converges asymptotically to σs resulting in an infinite lifetime tf of the composite. Note that Eq. 10.10 also provides the asymptotic constitutive behaviour of the model which can be measured by quasi-static loading. If the . . external load falls above the critical value the deformation rate ε = σ / Ef always

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remains positive resulting in a macroscopic rupture in a finite time tf (Kun et al., 2003; Kovacs et al., 2008). Representative examples of the solution ε(t) = σ(t) / Ef can be seen in Figure 10.4b for Weibull distributed breaking thresholds with the parameters λ = 1, ρ = 2. The behaviour of the system shows again universal aspects in the vicinity of the critical load σc. Similarly to the previous model, it can also be shown that the lifetime tf of the bundle has a power law divergence when the external load approaches the critical point from above

[10.11]

for σ0 > σc. It is important to emphasize that the exponent is universal in the sense that it is independent of the disorder distribution of the breaking thresholds; however, it depends on the stress exponent m of broken fibres (Kun et al., 2003; Kovacs et al., 2008). In order to numerically justify the analytically deduced behaviour of the time to failure tf as a function of the distance from the critical point, computer simulations were performed for several different values of the creep exponent m. In Fig. 10.5a the results are presented for m = 1.5 and m = 2.5. The slope of the fitted straight lines agrees very well with the analytic predictions of Eq. 10.11. The size scaling of the time to failure tf was analyzed by simulating the creep rupture of bundles of size N = 5 × 102–107 setting a Weibull distribution with λ = 1, ρ = 3 for the breaking thresholds. We found that tf (N) converges to the lifetime of the infinite system tf (∞) according to the universal law Eq. 10.5 independently of the value of the exponent m. In Fig. 10.5b the best fit was obtained for both curves with slope 1 ± 0.05 for both m values in excellent agreement with the analytic predictions.

10.5  (a) Lifetime of the fibre bundle as a function of the distance from the critical load σ0 – σc. Results of computer simulations (symbols) are compared to the analytic results (continuous lines) for two different values of the stress exponent m. (b) The lifetime of finite bundles exhibits a universal scaling which is independent even of the value of m.

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10.3.3  Microscopic process of failure One of the main advantages of fibre bundle studies of creep rupture of composites is that FBMs also provide a detailed physical picture of the microscopic process of rupture (Du and McMeeking, 1995; Fabeny and Curtin, 1996; Kun et al., 2003; Kovacs et al., 2008). In the framework of FBMs, assuming equal load sharing after fibre breakings the macroscopic time evolution of a creeping system can be analyzed in details by analytical means; however, the microscopic process of creep rupture is accessible only by means of computer simulations. On the micro level, during the creep process the fibres break in a single avalanche which either stops and the bundle stabilizes after the breaking of a finite fraction of fibres (σ0 < σc), or the avalanche continues until macroscopic failure occurs (σ0 > σc). Inside this avalanche, due to the disordered breaking thresholds, fibres may break in faster or slower sequences leading to fluctuations of the breaking rate. The process of fibre breaking on the micro level can easily be monitored experimentally by means of acoustic emission techniques. Except for the primary creep regime, where a large amount of fibres break in a relatively short time, the time of individual fibre failures can be recorded with high precision. The microscopic evolution of the rupture process can be characterized by the waiting times ∆t between consecutive fibre breakings and their distribution providing information also on the cascading nature of breakings. By analyzing waiting times based on the acoustic emission techniques in experiments, valuable diagnostic tools can be designed. Representative examples of waiting times ∆t obtained by computer simulations of the slowly relaxing fibre bundle are shown in Fig. 10.6 for loads below (Fig. 10.6a) and above σc (Fig. 10.6b). It can be seen that in both cases at the beginning of the creep process a large number of fibres break, which results in short waiting times, i.e. all the ∆ts are small at the beginning. Below the critical load σ0 < σc, at the macro level a stationary state is attained after a finite fraction of fibres break. Approaching the stationary state ∆t becomes larger (Fig. 10.6a) and reaches a maximum value on the plateau of ε(t) (compare to Fig. 10.4b). Above the critical load σ0 > σc, however, the slow plateau regime with long waiting times is followed by a strain acceleration (Fig. 10.4b) accompanied by a large number of breakings resulting again in small ∆t values (Fig. 10.6b). Varying the stress exponent m the qualitative behaviour of ∆t in Fig. 10.6 does not change. Besides the overall tendencies described above, the waiting times ∆t show quite an irregular local pattern with large fluctuations and have a non-trivial distribution. We determined the distribution function f (∆t) on both sides of the critical point σc varying the stress exponent m within a broad range. Examples of f (∆t) are presented in Fig. 10.7 for a bundle of 106 fibres with the stress exponent m = 2.0, where a power law form of f (∆t) can be observed on both sides of the critical load σc. For σ0 < σc no cutoff function can be identified, the power law prevails over six to seven orders of magnitude in ∆t up to the largest

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10.6  Waiting times ∆t between consecutive fibre breakings as a function of time t for a bundle of N = 105 fibres at a load level σ0 below (a) and above (b) the critical load σc.

values. For σ0 > σc the power law regime is followed by an exponential cutoff which shifts to higher ∆t values when approaching the critical load from above. Hence, the distribution function f(∆t) can be cast into the functional form

[10.12]

where the cutoff waiting time ∆t0 is a decreasing function of the external load σ0 for σ0 > σc. The value of the exponent α is different on the two sides of σc but inside one regime it is independent of the actual value of σ0. It is interesting that computer simulations revealed a strong dependence of α on the stress exponent m. The inset of Fig. 10.7 demonstrates that with increasing m the value of α decreases both in the under-critical and over-critical cases and tends to the same limit value α → 1 at large m. It was shown analytically (Kun et al., 2003; Kovacs et al., 2008)

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10.7  Distribution of waiting times f (∆t) for several different external loads below and above the critical value. The inset presents the exponent α of the distribution as a function of the creep exponent m.

that approaching the critical load from above, ∆t0 rapidly increases and has a power law divergence as a function of σ0 – σc

[10.13]

This behaviour implies that in the limit σ0 → σc the exponential cutoff disappears and f(∆t) becomes a pure power law. The cutoff exponent β can be calculated analytically as a function of the stress exponent m of the Maxwell elements: since .  the cutoff ∆t0 is proportional to the inverse of the minimum strain rate ε –1m, it follows that β = m. Below the critical point σc the distribution of waiting times does not have a cutoff function; furthermore, changing the load in this regime σ0 < σc not only reduces the statistics of the results (at lower loads less fibres break) but the functional form of f(∆t) does not change (see Fig. 10.7). The behaviour of ∆t0 as a function of σ0 in Eq. 10.13 shows that for σ0 < σc the system is always in the state of ∆t0 → ∞ so that in Eq. 10.12 a pure power law remains. We note that a similar power law functional form of the waiting time distribution has recently been found by Baxevanis and Katsaunis (2007) and by Baxevanis (2008) in FBMs using a different type of rheological element to describe the response of fibres and of the surrounding matrix material. These studies showed that the power law form of the waiting time distribution is universal; however, the value of the exponent depends on the precise rheology of elements.

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10.3.4  Predicting the lifetime of fibre bundles One of the most challenging problems for theoretical studies on creep rupture of composites is to derive observables and scaling laws which allow for an accurate prediction of the lifetime of samples from short-term measurements (Monkman and Grant, 1956; Vujosevic and Krajcinovic, 1997; Johansen and Sornette, 2000; Guedes, 2006; Brinson et al., 1981; Binienda et al., 2003). Computer simulations of the fibre bundle model of slowly relaxing fibres showed that the rupture life tf of finite bundles has large sample-to-sample fluctuations due to the quenched disorder of fibre strength. In order to characterize these fluctuations, the probability distribution of lifetime p(tf) was determined numerically, which proved to have a log-normal form, i.e., the logarithm of tf has a normal distribution with the form

[10.14]

where 〈lntf〉 and s(lntf) denote the mean and standard deviation of the logarithmic lifetime lntf (Kovacs et al., 2008). In order to demonstrate the validity of Eq. 10.14, in Fig. 10.8a the standardized distribution is presented, namely, s(lntf)p(lntf) is plotted as a function of (lntf – 〈lntf〉) / s(lntf) together with the standard Gaussian . An excellent agreement can be seen between the numerical results and the standard Gaussian for all the stress exponents m considered. The inset of Fig. 10.8a demonstrates that both the mean 〈lntf〉 and standard deviation s(lntf) of the rupture life increase exponentially with the stress exponent of the material (Kovacs et al., 2008). The result implies that in the case of high m values relevant for experiments, large fluctuations of tf arise. Consequently, the lifetime estimation for finite samples requires the development of methods which provide reliable results for single samples without averaging (Binienda et al., 2003; Brinson et al., 1981; Guedes, 2006; Vujosevic and Krajcinovic, 1997). . Based on the evolution of the rate of deformation ε(t), the creep rupture process . can be divided into three regimes. In the primary creep regime, ε(t) rapidly decreases with time. The secondary creep is characterized by a slowly varying, almost steady deformation rate, which is then followed by strain acceleration in . the tertiary regime for high-enough external loads. Therefore, the strain rate ε(t) . attains a minimum with a value εm at the so-called transition time tm between the secondary and tertiary creep regimes (see Fig. 10.8b for representative examples obtained by numerical solution of the bundle of slowly relaxing fibres). In laboratory experiments, the failure time tf of the specimen is usually estimated . from the variation of the deformation rate ε(t) based on the Monkman-Grant (MG) relationship (Monkman and Grant, 1956). The Monkman-Grant relation is a semiempirical formula which states that the time-to-failure of the system tf is uniquely . related to the minimum creep rate εm in the form of a power law , where the MG exponent ς depends on material properties. The advantage of the MG relation

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10.8  (a) The probability distribution of the lifetime has a universal lognormal functional form. The inset shows that both the average lifetime and its standard deviation increase exponentially with increasing stress exponent. (b) Strain rate of the bundle of slowly relaxing fibres for three different load values above the critical load. The inset demonstrates the validity of the Monkman-Grant relation for the model for the case of m = 2.

is that once the relation is established from short-term tests, the rupture life tf can be . determined just following the system until the minimum of ε(t) is reached (Brinson et al., 1981; Binienda et al., 2003; Guedes, 2006; Vujosevic and Krajcinovic, 1997). For the bundle of slowly relaxing fibres one can obtain analytically the MonkmanGrant relation

[10.15]

which shows that the MG exponent of the model depends on the stress exponent of relaxing fibres ς = 1 – 1/2m. The inset of Fig. 10.8b presents results of computer simulations which justify the validity of the Monkman-Grant relation and the m dependence of the MG exponent for the model (Kovacs et al., 2008). It has recently been pointed out that besides the MG relation, the lifetime tf can directly be related also to the transition time tm between the secondary and tertiary creep regimes (Nechad et al., 2005, 2006). Experiments on different types of composites revealed that tf = 3/2tm holds from which tf can be obtained from a measurement of tm with a significantly shorter duration (Nechad et al., 2005, 2006). For the model of slowly relaxing fibres, Fig. 10.9a presents tf obtained by computer simulations as a function of tm for a uniform distribution between 0 and 1 and for a Weibull distribution of breaking thresholds varying the stress exponent m. We emphasize that each symbol in Fig. 10.9a stands for a single sample with different realizations of the disorder and different values of the external load above the corresponding critical point σ0. It can be observed that all the points fall on the same straight line with relatively small deviations implying a linear relationship tf = atm, where the parameter a has a universal value a = 2.05

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10.9  (a) Lifetime tf of the bundle as a function of the transition time tm for two different values of the ratio Eb / Ef of the Young’s modulus of broken and intact fibres. The linear relation can be nicely observed. (b) The inset shows that in the tertiary creep regime a power law acceleration of the strain rate is obtained with an exponent γ, which depends on m. The main panel of the figure presents the m dependence of the exponent γ.

independent of the type of disorder and of the stress exponent m. The value of a ≈ 2 implies a symmetry of the time evolution of the system with respect to the transition time, which might not be valid for certain type of materials as it is indicated also by the experiments of Nechad et al. (2005, 2006). It can be observed in Fig. 10.9b that during damage-enhanced creep processes in the tertiary creep regime the macroscopic failure of the specimen is approached by an acceleration of the strain rate and breaking rate of fibres. An analogous effect has been observed in rupture experiments on disordered materials with increasing external load, where the acoustic emission rate has a power law singularity at catastrophic failure which also allowed for the possibility of predicting imminent failure events (Andersen et al., 1997; Alava et al., 2006; Johansen and Sornette, 2000; Garcimartin et al., 1997). In the framework of FBM with slowly relaxing fibres, computer simulations were carried out analyzing the functional form of . strain rate ε(t) in the vicinity of the failure time tf . The numerical calculations . revealed a power law divergence of ε as a function of the distance from tf

[10.16]

which is illustrated in Fig. 10.9b Extensive simulations showed that the value of the exponent γ does not depend on the external load σ0 and on the disorder distribution, but is a decreasing function of the stress exponent m governing the relaxation of broken fibres (see the inset of Fig. 10.9b). Similar power law acceleration was revealed analytically by the model of Nechad et al. (2005, 2006), which was also confirmed experimentally. Since Eq. 10.16 contains the rupture

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life of the system, it makes it possible to forecast the imminent failure event of the composite under a steady external load.

10.4 Summary and outlook Under high steady stresses fibre composites may exhibit time-dependent failure called creep rupture, which limits their lifetime and consequently has a high impact on their applicability for construction components. Fibre bundle models, as one of the most important modelling approaches to the failure of fibre-reinforced composites, provide a rich theoretical framework in which creep rupture phenomena of FRCs can also be investigated. In this chapter we have given an overview of the basic concepts of fibre bundle modelling and presented extensions of FBMs which capture physical mechanisms relevant for the time-dependent response and creep rupture of polymer matrix composites. We presented a fibre bundle model of viscoelastic fibres, where the rheological behaviour of single fibres was modelled by Maxwell elements. Assuming equal load sharing over intact fibres, we determined analytically the macroscopic time evolution of the bundle characterized by the deformation-time diagram at different load levels. It was discussed how the time evolution of the accumulating damage changes when the external load approaches the ultimate tensile strength of the system both from below and above. As a consequence of long-range load sharing, a universal power law behaviour of the characteristic time scale of the bundle was revealed where the exponent was independent of any materials details. A more detailed investigation was devoted to the bundle of slowly relaxing fibres where intact fibres exhibit brittle breaking; however, they undergo a slow stress relaxation after failure. It was demonstrated that on the macro level the model provides realistic deformation-time diagrams and in the vicinity of the ultimate tensile strength of the system the lifetime shows again a universal power law behaviour. The exponent of the power law is only influenced by the non-linearity exponent of stress relaxation. The chapter presented a detailed analysis of the microscopic damage accumulation process focusing on the breaking sequence of fibres. The disordered strength of fibres results in large fluctuations of the waiting times elapsed between consecutive fibre breakings so that the damage accumulation process can only be characterized by the probability distribution of waiting times. The waiting time distribution proved to have a power law decay with an exponent which has different values below and above the critical load but only depends on the stress exponent of the relaxation process. Another important consequence of the disordered fibre strength is that the rupture life of the system has a lognormal distribution with a relatively large standard deviation. Special emphasis was put on methods which can be used to predict the rupture life of composites based on fibre bundles. Besides the Monkman-Grant relation it was demonstrated that a unique relation can be established between the lifetime of

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creeping fibre bundles and the transition time between the secondary and tertiary creep regimes which can also be exploited for lifetime estimates based on shortterm measurements. The acceleration of damage accumulation has been found in heterogeneous materials to show universal behaviour as the critical time of failure is approached. A similar universal power law divergence of the creep rate was obtained, which provides possibilities to forecast the imminent catastrophic failure event of structural components subject to a steady load. When analysing the creep rupture of fibre-reinforced composites a number of challenges have to be overcome, among which the correct representation of the load transfer and redistribution among fibres, as well as between fibres and the embedding matrix, taking into account the multiple fibre cracks and debonding mechanisms of the fibre-matrix interface, are of outmost importance. Theoretical approaches designed at the mesoscopic structural level of composites, such as the shear load, fibre bundle, micromechanical, and continuum damage mechanics models, proved to be very successful during the past decade in modelling various aspects of the damage and fracture of composites under different loading conditions. These modelling approaches provide important alternatives to continuum mechanics based numerical models. Modelling approaches which also require large-scale computer simulations can benefit from the rapid development of computational power in recent years. Powerful computers make it possible to work out more detailed models, allowing for a more realistic description of the materials’ mesostructures and of the relevant physical mechanisms underlying the degradation process. A major outcome of this development is the possibility of introducing hybrid approaches which blend the advantageous features of mesoscopic and continuum approaches. Along this line, fibre bundle models of the creep rupture of polymer matrix composite materials are expected to become more realistic in the future, providing a more detailed description of the creep rupture process and also making possible the analysis of large-scale structures. The predictive power of such advanced models also makes it possible to complement the experimentally available information on the failure process by reliable numerical results of computer simulations. Over the last decade, applications of fibre-reinforced composites using polymer matrices have seen tremendous growth. In spite of the complexity of their behaviour and the unconventional nature of fabrication, the usage of such composites, even for primary load-bearing structures in military fighters and transport aircraft, as well as satellites and space vehicles, has been beneficially realised. Most of such usage constituted structural applications where service temperatures are not expected to be beyond 120°C. Research efforts are now focused on expanding the usage of such composites to other areas where temperatures could be considerably higher. The intended applications are structural and non-structural parts around the aero-engines and airframe components of aircraft. The development of reliable FRPs for these novel applications requires adequate modelling approaches where advanced fibre bundle models in combination with hybrid modelling strategies can play an important role in the future.

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10.5 Acknowledgement The author is grateful for the financial support of the Bolyai Janos research fellowship of the Hungarian Academy of Sciences.

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