Post on 17-Dec-2015
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Universality Classes of Constrained Crack Growth
Name, title of the presentation
Alex Hansen
Talk given at the Workshop FRACMEET,Institute for Mathematical Sciences Chennai, January 21, 2013
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Quasi-brittle materials:
Materials that respond non-linearlydue to heterogeneities.
Concrete.
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cracksstress field
The struggle between force and disorder.
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Contents:
• When the disorder dominates: the fiber bundle• When the disorder dominates: the fuse model• Scale-invariant disorder: the fuse model• Localization: Soft clamp fiber bundle model • Constrained crack growth: roughness• Intermezzo: gradient percolation• Soft clamp model in a gradient.• Dynamics of constrained crack growth
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• Peirce (1926)• Daniels (1945)
Stiff clamps
Each fiber has same elasticconstant, but different maximum load at which it fails.
Stiff clamps: Equal Load Sharing FB Model
Aka: Democratic Fiber Bundle Model
The fiber bundle model
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Fk = (N-k+1) xk
x
Average behavior fromorder statistics:
P(xk) = k/N
Fk/N = [1-P-1(xk)] xk
F/N = [1-P-1(x)]x
Flat distribution on the unit interval:P(x) = x
F/N = [1-x] x
F reaches its peak value at x=xc xc=1/2.
Signifies value at which k’th fiber fails.
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Fluctuations vs. averages
Daniels and Skyrme (1989):
(xc)=N1/3 f[N1/3(xc-<xc>)]
Sample to sample distribution of maximum elongation xc.
xc = <(xc-<xc>)2>1/2 ~ N-1/3
Fluctuations in maximum elongation
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Definition of Burst fibers fail before the force Fneeds to be increased to continue.
x
Burst of size .
9Analytical Expression for the Burst Distribution
Hemmer and Hansen (1992)
D(,xs)=- f((xc-xs))
f(y) approaches a constant for small y, and is proportional to exp(-y2) for large y.
Universal scaling exponents
Reminiscent of second order phasetransition.
Process is stopped at x = xs.
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Uniform distribution
Weibull distribution
m = 5
xs = xc
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Burst Distribution as a Signal of Imminent FailurePradhan, Hansen, Hemmer (2005)
Start recording bursts at x0 0.
Change in exponent when x0 is close to xc.
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Uniform
Weibull
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A single fiber bundle with N = 107, x0 = 0.9 xc:
Earthquakes
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The Fuse Model
Thresholddistribution
Fuse burns out if voltagedifference across it exceedsthreshold value t.
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•Strain Electrical potential•Stress Currents
Statistical distribution in thresholds, t.
Disorder:
Cracks:
Burned-out fuses
Other similar models:•Laplace: fuse model•Lamé: central-force model•Cosserat: beam model
•Disorder: Repulsionbetween cracks.
•Current distribution:Attraction between cracks.
Competition betweenDisorder and currentDistribution.
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Fuse Model in Infinite-Disorder Limit.
Fuses blow in order of weakest, next weakest, … as long as they are not screened.
Screened percolation process
Remark: Homogenization: approach material from zero-disorder limit.Statistical physics: approach material from infinite-disorder limit.
(Roux et al., J. Stat. Phys. 1988, Moreira et al., PRL, 2012)
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What is needed to reach the infinite-disorder limit?
Random number
Threshold value
Disorder parameter:
How big must be for thedisorder to dominate?
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Cumulative distribution:
Order statistics:
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Must compare threshold ratio to largest current ratio in network ~2 :
100X100 lattice:
-value for the disorder to dominate completely.
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= 0.01
= 1
= 100
(Moreira et al. 2012)
32X32: > 700
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Scaling in the infinite-disorder limit:
This value shows up in many connections…
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Strong and weak disorder in the fuse model:
* ~ L0.9
Mf: mass of final crack
Mb: mass of backbone
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Scale-invariant disorder
(Hansen et al. 1991)
Current distribution is scale free:
Histogram
Growing correlation length
i ~ (L/)
N ~ (L/)2 f
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Intensive (scale free variables):
Intensive time:
Intensive histogram:
Intensive currents:
f- formalism(multifractals)
No L dependence: Scale invariance
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Threshold distribution in intensive variables
Threshold distribution
Threshold values
Threshold distribution
Independent of L
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No spatial correlations in threshold distribution:
As L , the distribution takes on the form
This corresponds to two power law tails
= 0 for t 0 = -for t
Only the powerlaw tails survive as L
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A phase diagram for the fuse model
Diffuse loc.
Diffuse damage
Disorderless
Strong dis.
Scr. perc.
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Localization: Soft-clamp fiber bundle model
Order in which bonds fail:
• Lighter: earlier• Darker: later(Batrouni et al. 2002)
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e = E/L = 32
e = 0.0781
e = 2-6
e = 2-17
L= 128
Failure point, N pc
n= Np
(Stormo et al. 2012)
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r1 r4
r-1/4r-4
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xc = <(xc-<xc>)2>1/2 ~ N-1/3
(Daniels and Skyrme, 1989)
Wc ~L-2/3
Not an inversecorrelation lengthexponent!
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There is no phase transition
Slope remainsfinite: crossover-Not a phasetransition
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Scenario: Equal load sharing fiber bundle model until localization sets in.
System is never brittleJust percolation untillocalization sets in.
Critical pc notrelated to percolation threshold.
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Constrained crack growth: roughness.
(Santucci et al. 2010)
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From Tallakstad et al. 2011
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Two roughness exponents
Santucci et al. 2010
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Intermezzo: gradient percolation
(Hansen et al. 2007)
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Wavelet analysis of percolation front
Roughness exponent = 2/3
gradient
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Removing overhangsk 0
Roughness exponent
Gradient percolation: = 2/3
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Soft clamp model: two in one(Gjerden et al., 2012)
Stiff Soft
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Scale invariant elastic constant:
e = Ea/L
Small E is equivalent to large L.
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Soft system:roughness exponent = 0.39.
Large scales
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Stiff system:roughness exponent = 2/3.
Small scales
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High precision: Hull of Front Fractal Dimension
10/7
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Two roughness regimes:
Small scale: = 0.67 – percolation!Large scale: = 0.39 – fluctuating line.
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Family-Vicsek Scaling
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Soft system:
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Stiff system:
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From Måløy and Schmittbuhl, 2001
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Velocity distribution
52From Tallakstad et al. 2011
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Resumé:
• When the disorder dominates: the fiber bundle• When the disorder dominates: the fuse model• Scale-invariant disorder: the fuse model• Localization: Soft clamp fiber bundle model • Constrained crack growth: roughness• Intermezzo: gradient percolation• Soft clamp model in a gradient.• Dynamics of constrained crack growth