Statistical physics of fracture Purusattam Ray 1 The Institute of Mathematical Sciences, Taramani, Chennai 600 113, India
ABSTRACT We discuss the physics of brittle fracture in terms of the failure of an elastic medium with quenched disorder. The disorder acts as nucleation centers for fracture. We consider two situations: (1) low disorder concentration where the failure is determined by the extreme value statistics of the most vulnerable defect (2) high disorder concentration where percolation theory is applicable. We discuss the approach to breakdown through avalanches of large number of tiny fractures with universal statistical features. Finally, we discuss briefly the roughness of the fracture surfaces which also show universal behavior.
1
Introduction
Catasprophic failure, mark of brittle materials, is known to occur as a result of the presence of quenched defects in the material. The defects distort the stress field around them. The field gets concentrated at the sharp edges of the defects and this sets off the nucleation of fracture. A first principle analysis of fracture is, however, viable only in very specific cases, like that of an isolated single defect in the form of a microcrack of suitable geometrical shape [1]. Most engineering as well as many natural materials like rocks, wood, glass (cellular), composite materials (fibre-glass, plaster..) are 1
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1
examples of heterogeneous systems where defects of various kinds, shapes and vulnerabilities (from the point of nucleation of fracture) are present all over the system. Fracture phenomena, then, becomes extremely complex due to the cooperative role played by the interacting defects over wide range of length and time scales. Since fracture, at any stage, develops from the most vulnerable defect (weakest link of a chain), a theory based on a continuous coarse-grained description is untenable and a realistic computer simulation is almost unfeasible. Here, we begin with Griffith’s law which is the starting point regarding the role of defects in fracture nucleation. We extend the idea to the the heterogeneous systems having many defects: (1) for low defect density we discuss how extreme statistics gives rise to Weibull or Gumbell distribution and the system size dependence of the fracture strength, (2) for high defect density we discuss how the scaling pictures near the percolation transition explain the system size dependence of the fracture strength. In heterogeneous brittle materials, the final rupture is preceded by large number of microfractures. The strengths and the time intervals of occurrences of these microfractures show universal power law behaviors suggesting that the fracturing of a material under stress is a strongly cooperative phenomena. The approach to failure with increasing stress is discussed. We disuss fibre bundle model which captures some crucial aspects of the dynamics in a simple way. We then extend the calcalulation to the mean-field theory of a disordered elastic network. Finally, we discuss briefly the roughness of the fracture surfaces in brittle materials. The fracture surfaces turn out to be rough with a roughness exponent which is rather high compared to what one finds in most other physical systems. The roughness also shows universal behavior: the exponent seems to be independent of the material and the mode of fracturing.
2
Griffith’s law
Griffith worked out the energetics of the crack growth in a stressed elastic medium. Explicit calculation is feasible only for simple geometry of the crack: for example a thin one-dimensional crack of elliptical geometry with major axis 2l much larger than the minor axis [1]. The crack is in a two-dimensional sheet of a material of elastic modulus E which is subjected to a tensile stress 2
σ
US
U
2l
UE σ
(a)
(b)
Figure 1: (a) A crack of lenth 2l in an elastic medium subjected to a stress σ. (b) The surface energy Us , the elastic energy UE and the total energy U is plotted against l. σ (see fig. 1a). The elastic energy released due to the presence of the crack is well approximated by UE = πl2 σ 2 /2E. The energy required to produce the crack is Us = 4lγ, where γ is the surface enery of the material per unit length. The total energy U = −UE + Us as well as energies UE and Us as functions of l are shown in fig. 1b. U has a maxima and hence a point of instability at lc = γE/σ 2 : a crack of length l ≤ lc does not propagate whereas a crack of lengthql > lc grows to reduce U. Conversely, the strength of the material is σc = γE/l (Griffith’s law) at which the crack starts propagating and the fracture nucleates. The nucleation of fracture in this picture is similar to the nucleation theory in first order phase transitions and fracture appears to be a first order transition from a phase where microcracks remain as they were to a phase where cracks propagate and the system breaks apart [2].
3
Strength of heterogeneous materials
In heterogeneous systems, defects of all sizes and shapes are present and Griffiths’s energy balance calculation cannot be extended as such. We consider the situations where the defect density is (1) low and (2) high.
3
3.1
Low defect density
When the defect density is low, the defects are statistically far apart and act independently towards the nucleation of fracture. Fracture nucleates from the weakest of these defects (the statistics is goverened by the extreme events) and propagates without affecting other defects or being affecetd by them leading to the global failure of the system (weakest link of a chain concept). The failure probability F (σ) of a sample of volume V subjected to a stress σ is then: F (σ) = 1 −
n Y
(1 − fi (σ)) ∼ = 1 − exp[−
i=1
X i
fi (σ)] ∼ = 1 − exp[−V g(σ)],
where fi (σ) is the failure probability of the i-th (i = 1....n) defect under stress σ and g(σ) is the density of defects that will fail at or above σ. Two forms of F (σ) are mostly talked about in the literature. These correspond to two forms of g(σ): (1) a power law form of g(σ) ∼ σ m gives rise to Weibull strength distribution F (σ) = 1 − exp(−cV σ m ) which gives 1 the average strength < σ >∼ V − m [3]. (2) An exponential form of g(σ) ∼ exp(−A/σ α ) gives rise to Gumbel distribution F (σ) = 1−exp(−cV exp(−Aσ α )) 1 and the average strength < σ >∼ lnV − α [4]. In a lattice where defects appear in the form of absence of a bond with probability p, the probability of a defect of size l (i.e. the probability of having l successive bonds absent) is pl . Weibull distribution results at the limit p → 0 [5].
3.2
High defect density
When the defect density is high, one cannot consider the defects to be independent of each other. Fracture, at this limit, becomes a highly cooperative process involving interactions between fracture zones around defects which are even large distance apart. The critical theory and the scaling ideas of percolation can be used, at this limit, to predict the distribution of strength of a material. As before, the system is taken to be a lattice and the disorder is introduced by removing bonds randomly with a concentration p. In the percolating regime and close to the percolation threshold pc ( i.e. for p ≤ pc ), the lattice is connected ’typically’ over the pair-connectedness correlation length ξ. The system can be visualised as a lattice with lattice constant ∼ ξ. This gives rise to the node-link-blob picture (see fig. 2a) of the percolating network where the nodes are typically ξ distance apart and are connected 4
-12
Node
-12.5
ln σ f
-13
Blob
-13.5
ξ
-14
Link -14.5
-15 2.8
3
3.2
3.4
3.6
3.8
4
ln L
(a)
(b)
Figure 2: (a) A typical cell of the node-link-blob picture of the percolating network. The nodes are typically ξ distance apart and are connected by singly-connected bonds (links) and multiply-connected bonds (blobs). (b) logσf vs. logL in the simulation of a 2d bond bending network (data taken from [9]).The straight line is a guide to the eye and has slope -1.82. by singly-connected bonds (links) and multiply-connected bonds (blobs) [6]. Cutting of one link destroys the connection between the nodes in question but cutting a bond in the blob does not. In the node-link-blob picture, the stress σ is shared by 1/ξ (d−1) number of parallel channels and the strain σ/Y (Y being the Young’s modulus) is shared by 1/ξ number of channels. The elastic energy per channel is ∼ σ 2 ξ d/Y ∼ σ 2 ξ (d+Te /ν) . Here ν is the correlation length exponent defined as ξ ∼ |p−pc |−ν and Te is the exponent that describes how the elasticity goes to zero as one approaches pc : Y ∼ |p − pc |Te ∼ ξ −Te /ν [7]. Over a distance of the order ξ, the number of singly-connected bonds varies 1 statistically as ξ ν and all the bonds taken together varies as ξ db [6]. db is the fractal dimension of the percolating lattice. Assuming all the bonds to be singly-connected we get the lower bound of the strain energy per bond as σ 2 ξ (d+Te /ν−1/ν) . Assuming all the bonds to share the strain energy equally, we get the upper bound of the energy per bond as σ 2 ξ (d+Te /ν−db ) . The above argument gives a bound for the exponent Tb which describes how the strength of a percolating system goes to zero as p → pc , σc ∼ |p − pc |Tb [8]:
5
Te + dν − 1 Te + (d − db )ν ≥ Tb ≥ . (1) 2 2 At dimensions of the sytem d ≥ 6, two far away points in the system is predominantly connected by singly-connected bonds [6]. The bounds mentioned above for Tb become equal and the equalities become exact [8]. For systems at d ≥ 6 with bond-stretching and bond-bending forces, ν = 1/2, db = 2 and Te = 4 which gives Tb = 3. For these systems 2.8 ≥ Tb ≥ 2.3 in 2d and 2.9 ≥ Tb ≥ 2.5 in 3d. Simulation results of elastic networks validate the size dependence of the strength σc ∼ L−Tb /ν as predicted from the percolation theory (see fig. 22). The values of the exponents Te and Tb for common percolating elastic networks and the experimental results are given in table.1. The observed Tb -values are always in good agreement with the inequalities (1) given above. System
Dimension
Te
Tb
Central-force system
2
1.4 ± 0.02 [10]
∼ 1 [11]
Bond-bending force system
2 Mean field
3.96 ± 0.04 [12] 4 [7]
≥ 2.26 [8] 3 [8]
Experimental results
2
3.7 ± 0.2 [13]
2.3 ± 0.5 [13]
Table 1: The rigidity exponent Te and the strength exponent Tb are given for various elastic networks and experimental realizations.
4
Approach to breakdown
Recent experiments have shown that the final fracture in brittle materials is preceded by numerous microfractures inside the materials. These tiny ruptures are commonly imperceptible except that the acoustic emissions from these ruptures can be magnified and recorded with the help of ultrasonic detectors. For large class of brittle materials the energies associated with these emissions scales as E ∼ (Fc −F )−γ , where F is the applied stress and Fc is the stress at which macroscopic fractures occur breaking the system apart. Histogam P (ǫ) of the energies ǫ behaves as: P (ǫ) ∼ ǫ−β . Likewise, histogram 6
P (δt) of the time intervals δt between the bursts of emissions behaves as: P (δt) ∼ (δt)−α . Experimental results of the exponents for different brittle materials are listed in table 2. Material Chipboard wood, fiber glass [14] Synthetic plaster [15] Composite materials [16] Cellular glass [17] Plexiglass [18]
α β γ 1.9 ± 0.1 1,51 ± 0.05 0.27 ± 0.05 1.3 ± 0.1 0.4 ± 0.1 – 1.9 ± 0.1 1.55 ± 0.05 0.28 ± 0.05 1.5 ± 0.1 1.27 ± 0.01 – ∼ 1.8 – –
Table 2: Experimental results of the exponents α, β and γ associated with the acoustic emissions of the microfractures for different brittle materials. The scale-invariance manifested in the power-law form of P (ǫ) and P (δt) tells us that the development of fracture in heterogeneous systems is a phenomenon correlated over large length and time scales. The crucial points are that a defect grows only when its stress intensity exceeds (threshold mechanism) the static fatigue limit of the material. Secondly, at any stress level, only the most vulnerable defects grow (extremal mechanism). Fracture in heterogeneous systems then corresponds to the dynamical response of a threshold activated extremal dynamical system to an external driving (stress or strain). The system under stress has a large number of microscopic metastable states differing in internal stress distribution and crack structure and the dynamics takes the system from one metastable state to another by nucleating a microcrack and emitting energy thereby.
5
Extremal dynamics in random threshold systems
A simplified model of heterogeneous systems taking into accont the extremal dynamics and random thresholds is as follows: consider a network where the bonds are Hookean springs (of identical spring constant) and mimic the heterogeneity by assigning a random breaking threshold τ drawn from a distribution P (τ ) to each of the springs. Under stress a spring can be stretched till the threshold value beyond which it ruptures irreversibly. Susequent to rupture, the released load is redistributed over the remaining intact springs 7
P(f) Bars
2R
Fibres
0
(a)
f
1
(b)
Figure 3: (a) A schematic diagram of a fibre bundle model with fibres between two rigid bars. (b) A uniform distribution of P (f ) vs. f in the range [0, 1]. of the network. A breaking up of a spring mimics the onset of fracture. It can lead to further breaking up of the springs and the breaking process continues or the breaking event may stop whereat the stress level on the network is to be increased to induce further breaking. We discuss the breakdown properties of random spring network model below.
5.1
Fibre bundle model
Fibre bundle model is a simple model to capture the essentials of the threshold activated extremal dynamical process. The model consists of identical fibres (Hookean spring for example) between two bars which are subjected to tensile forces (fig. 3a). The fibres can withstand stress upto a threshold value f beyond which they break irreversibly. The threshold f is different for different fibres and has a distribution P (f ) (see fig. 3b). An important point in fibre bundle models is the redistribution of the load when a fibre breaks. The extra load is shared equally by all the existing fibres in equal load sharing (ELS) model whereas by only the neighboring fibres in the local load sharing (LLS) model. The dynamics in ELS is as follows: the R f /ρ fraction ρ of the intact fibres at load F is given by ρ = Nn = [1− 0 P (x)dx], where n is the number of intact fibres of total number N and f = F/N is the stress on each fibre initially. For uniform √ distribution P (x) in [0 : 1], ρ = 1 − f /ρ. This has solution: ρ = (1 + 1 − 4f )/2. We see that as f 8
increases, ρ decreases upto fc = 1/4 which denotes a critical stress value or the breakdown point below which a meaningful real solution does not exist. 1 1 At fc , ρ = ρc = 1/2 and at any f , ρ = ρc + 41 (fc − f ) 2 and dρ ∼ (fc − f )− 2 . df We see that the number of intact fibres and their rate of decreament, both exhibit power law. The breaking of fibres take place in avalanches whose size s for stress f follows the distribution: D(s, f ) ∼ s−3/2 exp(−s/s0 ) where s0 ∼ (fc − f )−1 [19]. The above results are valid for any sufficiently regular distribution P (x) [20]. One can likewise find solution for the dynamics of the fibre bundles under ELS rule. If ρ(t, f ) is the fraction of fibres at time t under load f , then ρ(t, f ) at successive time steps follow the recurrence relation ρ(t+ 1, f ) = 1 −f /ρ(t, f ). This has a solution: ρ(t, f ) − ρ(0, f ) ∼ exp(−t/τ ) where the time scale τ ∼ (fc − f )−1/2 [21]. For LLS scheme, closed form solution is difficult to get even in 1d. It was shown that the avalanche size distribution has a narrow power law regime: D(s) ∼ s−4.5 exp(−s/s0 ) and size dependence appear in the average bundle strength: fc ∼ 1/logN [22] (see [23] for a review).
5.2
Random threshold network
Random threshold networks have received lots of attention in the context of breakdown in elastic or conducting systems. Typical such network is a lattice, whose bonds are either elastic springs (elastic network) or conductors (conductng network), subjected to a stress (or a potential difference). Each bond i has a threshold Di , if the bond is strained or the current ovver the bond exceeds the threshold the bond snaps irreversibly. Once a bond breaks, the electric or elastic fields are recalculated over the network by solving the relevant Laplace’s equation. This may trigger further bond snapping. Otherwise the external field is raised to initiate further bond failure. The question asked here is how the bond failure process proceeds as the external field is slowly raised till the entire system fails. In the mean-field theory, the fluctuations in the elastic response at different parts of the network arising due to the breaking of the bonds are ignored and replaced by an average response which depends on the concentration of the intact bonds [24]. Consider the conductivity of a bond i to be σi = 1 till the current through the bond Ii < Di and zero otherwise. The total P dissipated energy is: E(σ) = 21 i σi [(∆V )2i − Di2 ], where (∆V )i is the voltage difference across the bond i. In terms of the fixed current I through P the network, E(I, σ) = 21 (I 2 /G(σ) − i σi Di2 ). In the realm of mean-field 9
theory, we write the total conductivity of the lattice as G(σ) = 2φ − 1, P where φ = i σi /L2 . In terms of φ, the expression for energy becomes: P E(I, σ) = 12 i σi [I 2 /L2 φ(2φ − 1) − Di2 ]. The solution for φ can be found Ra self-consistently from: φ = 1 − q 0 P (D)dD, where P (D) is the distribution of the threshold D and a = I/[L φ(2φ − 1)]. For any analytic and normalizable distribution ρ(D), φ has a solution. The defines the critical values φc anf f c. Close divergence of the susceptibility dφ df ∼ (fc −f )−1/2 , where to fc , φ behaves as: (φ−φc ) ∼ (fc −f )1/2 and < s >= dφ df < s > is the average size of the avalanches of broken bonds with increasing I or equivalently f . This defines the exponent γ = 1/2. The mean field avalanche size distribution comes out to be: P (s) ∼ s−τ F (s(fc − f )κ ), with τ = 3/2 and κ = 1. These exponents satisfy the scaling relation: k(2−τ ) = γ. Extensive computer simulation studies [24, 25, 26] of random resistor and spring networks in two and three dimensions show that both the two models of breakdown exhibit the mean-field scaling behaviors and the exponents on approaching the rupture point. The reason may be the presence of long range interacion in these models. At the breakdown point, the macroscopic quantities like elasticity or conductivity are discontinuous and go to zero sharply. The characteristic crack size stays always finite at the breakdown point even when the system size is large. The statistics of the global quantities (like the total number of broken bonds) display scaling in analogy to a first order phase transition. The behaviors observed in these models suggest that breakdown is a first order transition and the nucleation of fracture is very similar to spinodal nucleation in thermally driven homogeneous systems. The outcome of the fibre bundle model and the mean-field theory is qualitatively similar to what one observes in the experiment. The integrated distribution of the the burst energies was found to follow a power law with an exponent roughly equal to -2 [16]. If we assume that s here is proportional to the energy E in [16], then this result should be compared to −(τ +κ) = −5/2. The discrepancy may be due to the statistics but most likely there is appreciable local load sharing which causes the results to deviate from that of mean-field theory.
6
Roughness of fracture surfaces
Lastly, we discuss the universal roughness behavior of the surfaces produced in brittle fractures. Experimental studies indicate that the fracture surfaces 10
a)
0
10
0
= 0.75
1
log (δ/δ )
δ/δo = 5 δ/δo = 9 δ/δ = 18 o δ/δ = 33 o δ/δ = 63 o δ/δ = 119 o δ/δ = 223 o δ/δ = 421 o δ/δ = 793
10
10
3d
ζ
log (σ)
−1
2 1/2
−2
P(∆’h(δ) ) (2σ )
800 µm b)
10
−1 10
3
o
o
−3
10
2
y = e−x
−4
10
−5
10 100 µm
−6
10
−8
−6
−4
−2
0
2
4
6
8
2 1/2
∆’h(δ) / (2σ )
(a)
(b)
Figure 4: (a) Zoom of the crack front images (i) in paper sheet (ii) in PMMA, (b) Statistical distribution of the height fluctuations P (∆′ h) for fracture surface in a 3d granite block (taken from [28]). are self-affine and scaling properties of the roughness are same (to within the experimental errors) for most brittle and weakly ductile materials. The typical deviations ∆h of the surface as a function of the distance δ along the fracture surface scale as ∆h ∝ δ ζ [27]. ζ is called the roughness exponent. Fig. 4(a) shows the fracture front images in (i) paper sheet and (ii) PMMA (taken from [28]). Fig. 4(b) shows the results of the height fluctuations in 3d granite block along the direction perpendicular to the fracture propagation. The statistical distribution of the height fluctuations P (∆′ h) is obtained at different length scales δ, where ∆′ h = ∆h− < ∆h >, < ∆h > being the average height √ of the surface over√the length δ. Fig. 4(b) shows the semilog ′ plot of P (∆ h) 2σ 2 versus ∆′ h/ 2σ 2 and at large length scales δ, the plot shows the parabolic shape of a Gaussian distribution. The inset shows the varuation of the standard deviation of the distribtion with δ: σ ∝ δ ζ with ζ ≈ 0.75. The results indicate that the height fluctuations P [∆h(δ)] over a length δ in fracture surfaces follow the scaling form: P [∆h(δ)] ∼ δ −ζ G[∆h(δ)/δ ζ ], where the rescaling function G is Gaussian and there exists a unique scaling exponent ζ. This indicates that fracture surface is rough and has a monoaffine scaling. In 2d, ζ ≈ 0.6 and in 3d, ζ ≈ 0.8 [28]. These values of the roughness exponent is much higher than what is predicted from the existing theories[29]. 11
The reason for this high value ζ is not clear yet. It has been suggested that the universal roughness of brittle fracture and the high value of the roughness exponent may be explained if one considers fracture propagation as a percolation phenomenon in damage coalescence processes [30]. References 1. J. F. Knott, Fundamentals of Fracture Mechanics, (Butterworths, 1973). 2. B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids, Cambridge University Press, Cambridge (1975). 3. A de S Jayatilaka in Fracture of Engineering brittle Materials, (Applid Science Publishers Ltd., London, 1979). 4. P. M. Duxbury, P. L. Leath and P. D. Beale, Phys. Rev. B 36, 367 (1987). 5. P. Ray and B. K. Chakrabarti, Sol. St. Commun. 53, 477 (1985). 6. A. Aharony and D. Stauffer, Introduction to Percolation theory, Taylor and Francis, London (1994). 7. Y. Kantor and I. Webman, Phys. Rev. Lett. 52, 1891 (1984). 8. P. Ray and B. K. Chakrabarti, Phys. Rev. B 38, 715 (1988). 9. M. Sahimi and S. Arbabi, Phys. Rev. B 47, 713 (1993). 10. A. M. Lemieux, P. Breton and A. M. S. Tremblay, J. Phys. (Paris) Lett. 46, L1 (1985). 11. P. Ray and B. K. Chakrabarti, J. Phys. C 18, L185 (1985). 12. D. J. Bergman, Phys. Rev. b 31, 1696 (1985); J. G. Zabolitzky, D. J. Bergman and D. Stauffer, J. Stat. Phys. 44, 211 (1986). 13. L. Benguigui, Phys.Rev. Lett. 53, 2028 (1984); L. Benguigui, Phys. Rev. Lett 54, 1464 (1985); K. Sieradzki and R. Li, Phys. Rev. Lett. 56, 2509 (1986); L. Benguigui, P. Ron and D. J. Bergman, J. de. Physique 48, 1547 (1987). 12
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