Table 1

From: Universal Quake Statistics: From Compressed Nanocrystals to Earthquakes

(a)

Statistical Distributions

Scaling forms predicted by the model5,6,7

CCDF, C(S), of avalanche size, S

C(S) ~ S−(κ−1) GS(S/Smax)

CCDF, C(V), of stress-drop rate, V ~ S/T

C(V) ~ V−(ψ−1) GV(V/Vmax)

CCDF, C(T), of avalanche durations, T

C(T) ~ T−(α−1)GT(T/Tmax)

Power spectrum, P(ω), at frequency, ω

P(ω) ~ ω−φDω(ω/ωmin)

(b)

Fixed-stress loading conditions: slowly increasing stress, F, up to the failure stress, F c

Fixed-strain-rate loading conditions: moving the boundary at a slow strain rate, Ω

Smax ~ (Fc − F)−1/σ

Smax ~ Ω−φλ

Vmax ~ (Fc − F)−(1+ρ)/(σφλ)

Vmax ~ Ω−(1+ρ)

Tmax ~ (Fc − F)−1/(σφ)

Tmax ~ Ω−λ

ωmin ~ (Fc − F)1/(σφ)

ωmin ~ Ωλ

(c)

Exponents

Sample Sizes

κ

κ + σ

σ

φ

α

ψ

Model Predictions

Mean Field Theory (MFT)7

 

1.5

2

0.5

2

2

1

Experimental Verifications*

Nanocrystals (Molybdenum (Mo), Compression, see1,29 and Figs 2, 3, 4)

10−8 m

1.5

2

0.5

2

  

Microcrystals (Nickel (Ni), Compression32,33)

10−6 m

1.5

  

2

  

Bulk Metallic Glass (BMG) (Cu47Zr47.5Al518, Zr45Hf12Nb5Cu15.4Ni12.6Al108 and Zr64.13Cu15.75Ni10.12Al109, atomic percent) Compression.

10−3 m

1.5

2

0.5

2

2

 

Lab-scale rocks (Sidobre granite, Compression19,20,34)

10−2 m

~1.5

1.66–2.2

    

Lab-scale rocks (Westerly granite, Frictional sliding24)

10−2 m

~1.5

     

Jammed granular materials (Photo-elastic disks in Couette cells and other geometries7)

1 m

~1.5

  

1.8–2.5

~2

 

Earthquakes4,14,35

105 m

~1.5

  

2

  
  1. Table 1. (a) Model predictions for scaling forms of various distributions.
  2. Here GS(S), GV(V), GT(T) and Dω(ω) are universal scaling functions, κ, ψ, α, φ, σ, λ and ρ are universal power-law exponents7 and Smax, Vmax, Tmax and ωmin are the cutoffs of the power-law regimes of the corresponding distributions6,7.
  3. (b) Predicted scaling forms of the cutoffs for two loading-conditions, near failure6,7.
  4. (c) Comparison of model (Mean Field Theory (MFT)) exponents with different experiments, showing strong agreement. Open entries indicate predictions for future experiments. MFT predicts that λ = 1 and ρ = 17. Our experiments reveal an exponent of κ = 3/2 for nanocrystals down to 75 nm in size. Additional predictions are given in the SI.
  5. *Exponents from experiments and observations quoted throughout this paper have a 10% error range due to statistical fluctuations. As shown in Figure 3A–C for compressed nanocrystals, bulk metallic glasses (BMGs) and rocks, power-law fits for small stresses (where the cutoff is small) would yield wrong exponent values, because those are skewed by the small exponential cutoff, as predicted by our model. Instead, scaling collapses like those of Figure 3G–L yield the correctly extrapolated exponents, which agree with our model predictions. Exponents from previous experiments were obtained from19,20 at the largest stresses, using that the Gutenberg Richter exponent, b, in19 is related to our exponents via b = 3(κ − 1)/2 (see5,34). For the relationship between the slip-size and the acoustic-emission signal see34, the Supplementary Information of1,36 and references therein.
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