10/21/2019 Si String Section Crack
Where G 0( t, f) = K 0( t, f) 2/ E is the reference energy release rate which, as K 0( t, f), would result from the same loading with a straight crack front within the mean plane y = 0 at the mean position x = f( t). Henceforth, the operator a indicates averaging of the variable a( z) over the spatial coordinate z.
Rear side of an Arco Solar module with Tedlar cracking. Percentage of failed. Figure 7: String efficiency of Casaccia PV Plant norma. The power loss of module PWX1 (Section 3.2.1), in particular, is traceable to fill factor loss. Loss of fill.
The two quantities G 0 and K 0 coincide with the ones that would have been defined in the conventional LEFM approach when ignoring front distortions. They only depend on the specimen geometry and imposed load (of which both evolve with t) and can be determined using continuum mechanics (e.g., finite element analysis).
Slow cracking (as considered herein) implies that the solid is loaded by imposing external displacements rather than external stresses (the opposite would yield dynamic fracture with crack at speed on the order of the sound speed). This induces additional constraints: G 0 should decrease with f( t) (specimen compliance always increases with crack length) and increase with t (to drive a crack, external displacement can only increase with time). Without loss of generality, we set t = 0 at a time when the crack has just stopped and x = 0 is the crack tip's position at this time. G 0( t = 0, f = 0) = Γ where Γ = 〈Γ( x, y, z)〉 is the mean value of fracture energy.
Considering the subsequent variation f( z, t) to be small with respect to the crack length at t = 0, we can write. Where γ( x, y, z) = Γ( x, y, z) − Γ is the fluctuating part of the fracture energy.
The solution of this equation provides the space-time dynamics of the in-plane projection of the crack front. Subsequently, it gives the time evolution of the fracture velocity at the continuum-level scale, v( t) = d f/d t. This is the relevant observable characteristics of the crack dynamics in standard experiments of fracture mechanics. Principle of Local Symmetry and Equation of Path We now derive an equation of path by making use of the principle of local symmetry. To do so, we take inspiration from the work of Katzav et al. to model crack path in a model 2D situation and extend it for three-dimensional solids. The idea is to decompose the front propagation along x-axis into infinitesimal straight kinks of length ℓ, identified with the microstructural length-scale characterizing the spatial distribution of Γ( x, y, z) (see Figure:right).
Consider now the kink occurring at the front location z between time t and t+d t. Leblond et al's formulas relate the stress intensity factors after kinking K′ i( z) ( i = I, II, III) to the stress intensity factors before kinking K i( z) and to the kink angle δθ = θ( z, t + d t) − θ( z, t) = ℓ ∂ 2 h/∂ x 2( x = f( z, t), z) via. To introduce the effect of microstructural disorder, a spatially distributed noise term K 0ϑ ( x, y, z) is added to the right-handed side of the equation. In the situation of inter-granular crack growth in a material made of sintered grains (e.g. Sandstone), for example, this additional term would translate the difference between the kink angle predicted by the principle of local symmetry and that truly selected so that the crack propagates between the cemented grains. Finally, combining the resulting equation with the asymptotic formula for stress intensity factors (Equation 1), one gets.
Here, the form of the functions f( u) and g( u) is expected to be universal. The exponents 1/σ and 1/Δ are also predicted to be universal: 1/σ = 0.69 ± 0.010 and 1/Δ = 0.385 ± 0.010 ,. Conversely, the effect of a finite c is not uncovered yet. By yielding some overlap between the avalanches , it can significantly alter the dynamics. In particular, a recent work has evidenced a transition line between a crackling-like dynamics made of irregular power-law distributed jumps and a continuum-like dynamics ruled by the conventional LEFM theory. Figure presents typical times profiles of the continuum-level scale velocity v( t) for different values of c and k in the crackling regime. Here and thereafter, the system size N and the disorder strength η ˜ are set constant, N = 1024 and η ˜ = 1, returning at the end of this section to a brief discussion on their effect.
Note the irregular jumps characteristics of the underlying avalanching dynamics. Note also the qualitative changes in the signal appearance as k and c are modulated: Pulses become shorter as k increases, and the pulse density increases as c increases. Note finally that, due to the finite value of c, v( t) does not vanish between the pulse, but becomes equal to a small value proportional to c (prefactor dependent of the Runge Kunta scheme). The avalanches are then identified with the bursts where v( t) is above a prescribed reference level v c = 10 −3. Their duration D is defined as the interval between the two successive intersections of v( t) with v c, and their size S is defined as the integral of v( t) − v c between the same points. Top: Distribution of the avalanche size measured (A) for various values of k at constant c = 10 −4 and (B) for various values of c at constant k = 10 −2. In both cases, N = 1024 and η ˜ = 1 and the axes are logarithmic.
The power-law exponent τ is found to be independent of the parameters and compatible to the universal value τ = 1.28 (black dashed line) predicted for c → 0 and k → 0. The lower cutoff is found to be independent of the parameters: S min ≈ 10 −3 (vertical dash line). The upper-cutoff is found to decrease with k and to increase with c (resp. To be independent of c) when c is large enough (resp.
When c is small enough). Bottom: Distribution of the avalanche duration measured (A′) for various values of k at constant c = 10 −4 and (B′) for various values of c at constant k = 10 −2.
In both cases, N = 1024 and η ˜ = 1 and the axes are logarithmic. The power-law exponent α is found to be independent of the parameters and compatible to the universal value α = 1.50 (black dashed line) predicted for c → 0 and k → 0.
The lower cutoff is found to be independent of the parameters: D min ≈ 3 (vertical dash line). The upper-cutoff is found to decrease with k and to increase with c (resp.
To be independent of c) when c is large enough (resp. When c is small enough). The power-law distributions observed in Figure also exhibit upper cutoffs S 0 and D 0. These cutoffs decrease with k in all cases. The effects of c is of two types:. For large k/small c the cutoff does not depend on c (only on k).
At small k/large c, the cutoff increases with c and the distribution also displays a bump at large sizes and durations. Direct computation of the cutoff is quite imprecise.
Hence, the selection of the typical length and time scales is studied via variations of the mean values 〈 S〉( c, k) = ∫ ∞ S min S × P( S c, k)d S (Figure ) and 〈 D〉 = ∫ ∞ D min D × P( D c, k)d D (Figure ). At large enough k, both 〈 S〉 and 〈 D〉 are independent of c. This large k regime is attributed to a regime of pseudo-isolated (pi) avalanches. The distributions are then expected to take forms similar to that of Equation 10. As a result, 〈 S〉 pi is expected to take the form 〈 S〉 pi ≈ S τ − 1 min S 2 − τ 0 ∝ k −(2 − τ)/σ, 〈 D〉 pi ≈ D α − 1 min D 2 − α 0 ∝ k −(2 − α)/Δ. These two scaling are compatible with the observations at large k (dash line in Figures ). As a synthesis, the mean avalanche size and duration are found to take the following form at large k.
(A) Evolution of the mean avalanche size 〈 S〉 as a function of the unloading factor k for different loading rates c. The curves collapse for large k onto a c-independent power-law curve. Black thick dash line depicts a fit over the collapsed region using Equation 11a: 〈 S〉 pi ≈ 10 −2 k −(2 − τ)/σ with τ = 1.28 and 1/σ = 0.69. (B) Evolution of the mean avalanche duration 〈 D〉 as a function of the unloading factor k for different value of the loading rate c.
At large k, the curves collapse onto a c-independent power-law curve. Black thick dash line is a fit over the collapsed region using Equation 11b: 〈 D〉 pi ≈ 11 k −(2 − α)/Δ with α = 1.50 and 1/Δ = 0.385. In both (A,B) N = 1024 and η ˜ = 1 and the axes are logarithmic. The different symbols correspond to different values of c, given in the legends. Let us try now to characterize the effects of the avalanche overlap when k becomes small or c becomes large. Previous work evidenced a transition between the crackling dynamics studied here and a continuum-like dynamics when c becomes large enough or k small enough.
This transition is believed to coincide with the point where the avalanche overlap percolates throughout the entire system. At constant η ˜ and N, this transition was shown to be fully driven by the ratio c/ k.
We hence plotted, in Figure, 〈 S〉/〈 S〉 pi and 〈 D〉/〈 D〉 pi (first order estimation of the number of individual avalanches having merged together to form the bursts detected from the signal v( t)), as a function of the control parameter c/ k. A coarse collapse is observed. As expected, the master curves diverge at the transition value between crackling and continuum-like dynamics (materialized by the vertical dash lines in the main panels of Figures ).
(A) Main panel: Variation of 〈 S〉/〈 S〉 pi as a function of c/ k for different k. (A) Inset: 〈 S〉 as a function of c for different k. (B) Main panel: Variation of 〈 D〉/〈 D〉 pi as a function of c/ k for different k. (B) Inset: 〈 D〉 as a function of c for different k.
In both (A,B) N = 1024 and η ˜ = 1 and the axes are logarithmic. The different symbols correspond to different values of k, given in the legend between graphs (A,B). In the main panels of both graphs, the vertical dash lines indicates the transition value c/ k between crackling and continuum-like dynamics as determined in Nukala et al. for N = 1024 and η ˜ = 1. Where the two functions ( u) and ( u) exhibit a plateau at small u, and both diverge at the same value u c.
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The value of the exponents τ, α, 1/σ and 1/Δ are well known : They can e.g., be related to the so-called roughness exponent ζ and dynamic exponent κ classically defined in the realm of critical depinning transition: τ = 2 − 1/(1 + ζ), α = 1 + ζ/κ, 1/σ = (1 + ζ)/2 and 1/Δ = κ/(1 + ζ). Conversely, the precise origin of the exponents b SN, b S η ˜, b DN, b D η ˜, c N, and c η ˜ and their link with ζ and κ remain to be uncovered. Effect of the system size N and disorder strength η ˜ on the avalanche statistics. Main panel, top: Distribution of the avalanche size measured (A) for different N at constant c = 2 × 10 −5, k = 10 −2 and η ˜ = 1, and (B) for different η ˜ at constant c = 2 × 10 −5, k = 10 −2 and N = 1024.
The axes are logarithmic. Main panel, bottom: Distribution of the avalanche duration measured (A′) for different N at constant c = 2 × 10 −5, k = 10 −2 and η ˜ = 1, and (B′) for different η ˜ at constant c = 2 × 10 −5, k = 10 −2 and N = 1024. All axes are logarithmic. In all panels, the power-law exponents τ and α are compatible with the universal values τ = 1.28 and α = 1.50 (inclined dashed line) predicted for c → 0 and k → 0. Regarding the size, the lower cutoff S min is observed to be independent of η ˜ and to decrease with N.
Regarding the duration, the lower cutoff D min decreases with both N and η ˜. D min) is quantitatively defined as the intersection of the power-law regime with exponent τ (resp. Α) and the saturation value for P observed at small S (resp. The red and blue dash lines in ( A,A′,B′) present illustrations for N = 128 and N = 1024, respectively.
The variation of S min with N is shown in the inset of a. The variations of D min with N and η ˜ are shown in the insets of (A′,B′). The different symbols correspond to different values for c (from 2 × 10 −6 to 10 −4). The red line are fits: S min ∝ N − a SN with a SN = 1.7 ± 0.1, D min ∝ N − a DN with a DN = 0.6 ± 0.1, and D min ∝ η ˜ − a D η ˜ with a D η ˜ = 1.2 ± 0.1. Effect of the system size N and disorder strength η ˜ on the mean size and duration of avalanches.
Top, inset: Mean avalanche size 〈 S〉 as a function of c for constant k = 10 −2 and (A) different N at constant η ˜ = 1 and (B) different η ˜ at constant N = 1024. Bottom, inset: Mean avalanche duration 〈 D〉 as a function of c and constant k = 10 −2 and (A′) different N and constant η ˜ = 1 and (B′) different η ˜ and constant N = 1024.
In all cases, the axes are logarithmic. Main panels: Curve collapse obtained by plotting 〈 S〉/ N − b SN vs c/N −c N (A), 〈 S〉/ η ˜ − b S η ˜ vs c/N − c η ˜ (B), 〈 D〉/ N −b DN vs c/N −c N (A′), and 〈 D〉/ η ˜ − b D η ˜ vs c/N −c η ˜. The fitted exponents are found to be b SN = 1.3 ± 0.1, b S η ˜ = 0.7 ± 0.1, b DN = 0.45 ± 0.1, b D η ˜ = 0.65 ± 0.1, c N = 0.65 ± 0.1, c η ˜ = 1.05 ± 0.1. Where the mean avalanche size 〈 S〉 and duration 〈 D〉 are expressed with real length and time units, and Ġ, G′, W, Γ, μ, ℓ and γ ˜ are recalled to be the loading rate, the unloading factor, the specimen width, the fracture energy, the mobility, the microstructure length-scale, and the contrast in local fracture energy. The value of the exponents are recalled to be τ = 1.280 ± 0.01 (predicted), α = 1.500 ± 0.01 (predicted), 1/σ = 0.69 ± 0.01 (predicted), 1/Δ = 0.385 ± 0.01 (predicted), b SN = 1.3 ± 0.1 (fitted), b S η ˜ = 0.7 ± 0.1 (fitted), b DN = 0.45 ± 0.1 (fitted), b D η ˜ = 0.65 (fitted), c N = 0.65 ± 0.1 (fitted), and c η ˜ = 1.05 ± 0.1 (fitted).
Roughness of Fracture Surfaces We turn now to the topography h( x, z) of the post-mortem fracture surfaces as predicted by equation 9b. Figure reports typical topographies for different values of the two external parameters A and θ ˜. When A is close to 1, the surface seems to be statistically isotropic while as A gets smaller, the surface appears more elongated in the direction of z. Conversely, the parameter θ ˜ only affects the range swept by the roughness.
Note that in almost all elastic solids, Poisson ratio ν lies between 0 and 0.5, which impose a finite interval for A = (2 − 3ν)/(2 − ν): 1/3 ≤ A ≤ 1. Herein, only A within this interval are considered. To characterize quantitatively the spatial distribution of fracture roughness, we adopted the classical procedure and computed the structure function S(Δ r →) = 〈( h( r → + Δ r →) − h( r →)) 2〉. Here, the operator 〈〉 denotes averaging over all positions r → = ( x, z). First, we computed the structure function S z(Δ z) along 1D profiles taken parallel to z (mean direction of the crack front). The procedure is the following: (i) an initially straight front was first propagated over a distance equal to 10 N to obtain a statistically stationary regime; (ii) 10000 subsequent profiles h( x i, z) separated by a distance x i+1 − x i = 1 were recorded; (iii) the structure function S i z(Δ z) was computed for each of these profiles; and (iv) finally, these 10000 individual structure functions were averaged to get S z(Δ z).
Figure depicts typical examples of S z(Δ z) curves for different values of N, θ and A. S z goes as S z = p zlog(Δ z/λ z) up to an upper cutoff set by the system size N. This logarithmic scaling is anticipated to extend over the whole range of length-scales as N → ∞. Note that logarithmically rough crack surfaces were also predicted in earlier theoretical works analyzing crack propagation through a three-dimensional heterogeneous solid. As a plus, the present model allows relating the prefactor p z and characteristic length-scale λ z with the fracture parameters: p z is found to scale as θ ˜ 2/ A (Figure ), while λ z is independent of both A and θ ˜ (Figure ). Finally, the structure function along z is.
(A) Slope p z associated with the curve S z vs. Δ z as a function of A at θ ˜ = 1 (main), and as a function of θ ˜ at A = 1 (inset).
In the inset, the axes are logarithmic. In both graphs, the red lines correspond to fits p z = C/A (main) and p z = C θ ˜ 2 where C = 0.32 ± 0.02. (B) Characteristic length-scale λ z associated with the curve S z vs. Δ z as a function of A at constant θ ˜ = 1 (main) and as a function of θ ˜ at constant A = 1 (inset). In both graphs, the red lines correspond to fits λ z = 0.24 ± 0.03. Here, ± indicates a 95% confident interval. (A) Slope p x associated with the curve S x vs.
Δ x as a function of A at constant θ ˜ = 1 (main) and θ ˜ at constant A = 1 (inset). In the inset, the axes are logarithmic. In both graphs, the red lines are fits p 1 = C x/ A (main) and p 1 = C θ ˜ 2where the fitted parameter is found to be C = 0.31 ± 0.02 (95% confident interval). (B) Characteristic length-scale λ x associated with the curve S x vs. Δ x as a function of A at constant θ ˜ = 1 (main) and θ ˜ at constant A = 1 (inset). In both graphs, the red lines are fits λ x = 0.21 ± 0.02/ A (main) and λ x = 0.21 ± 0.02 (inset).
Here, ± indicates a 95% confident interval. Where S x, S z, Δ x and Δ z are expressed with real length units, and ν, ℓ and θ ˜ are recalled to be the Poisson ratio, the microstructure scale, and disorder contrast. Concluding Discussion Stress enhancement at crack tips makes the macroscale failure behavior observed extremely sensitive to the presence of disorder at the microstructure scale. This translates into crackling dynamics and rough fracture surfaces, which, by essence, cannot be addressed within the conventional LEFM framework. In this paper, we have used the RT-CM approach to obtain quantitative relations between some statistical observables characteristic of these two aspects and the fracture parameters: Loading rate (time derivative of the energy release rate), specimen geometry (specimen thickness and unloading factor), conventional mechanical constants (fracture energy, Poisson ratio), and microstructural disorder (microstructure scale and disorder strength). Over a certain range of the fracture parameters, this RT-CM approach predicts crackling dynamics : The crack growth splits up into discrete jumps, which are power-law distributed in size and duration. The characteristic exponents associated to these power-laws are universal.
Conversely, the scales covered by these scale-free features are non-universal and, in particular, the mean size and duration of the crack jumps are found to depend on the fracture parameters according to scaling laws that are uncovered. These scaling laws can be understood over a certain range of the fracture parameters, in the regime of pseudo-isolated avalanches addressable via standard functional renormalization theory , –. Conversely, the effect of the avalanche overlapping is not understood. On-going work aims at analyzing the distribution of the local avalanches as detected in the space-time diagrams of the front dynamics, in order to understand the coalescence process. Also, this RT-CM approach predicts rough fracture surfaces.
The fracture roughness can be characterized by computing the structure function, which exhibits logarithmic scaling. The associated prefactor and characteristic length-scale are found to depend on the Poisson ratio, microstructure length-scale, and disorder strength according to laws that were uncovered. This may have interesting applications: It allows one to infer the microstructure parameters (the access of which could be made difficult otherwise, due to the smallness of the length-scales involved) from the analysis of post-mortem fracture surfaces at larger scale. Work in progress aims at testing the scaling predicted here for the structure functions against fractography experiments achieved in oxide glasses. Note finally that the RT-CM model studied here call upon a variety of assumptions (see Section 2). An interesting perspective would be to measure to which extend these assumptions can be released. Work in this direction is currently under progress.
The model is also limited to nominally brittle fracture, with a single macroscopic crack propagating in an otherwise intact material. Promising alternative approaches have emerged from statistical and non-linear physics and may permit to address quasi-brittle fracture, with many microcracking events interacting with each-others. A central challenge in the field would be to bridge the gap between these approaches and engineering damage mechanics.
Conflict of Interest Statement The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Acknowledgments The authors would like to thank Alberto Rosso for enlightening discussions and the “Agence Nationale de la Recherche” (ANR) for financial support of the project MEPHYSTAR (ANR-09-SYSC-006-01).
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