1 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

2 Quasi-brittle materials: Materials that respond non-linearly due to heterogeneities. Concrete.

3 cracksstress field The struggle between force and disorder.

4 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

5 Peirce (1926) Daniels (1945) Stiff clamps Each fiber has same elastic constant, but different maximum load at which it fails. Stiff clamps: Equal Load Sharing FB Model Aka: Democratic Fiber Bundle Model The fiber bundle model

6 F k = (N-k+1) x k x Average behavior from order statistics: P(x k ) = k/N  F k /N = [1-P -1 (x k )] x k 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=x c x c =1/2. Signifies value at which k’th fiber fails.

7 Fluctuations vs. averages Daniels and Skyrme (1989):  (x c )=N 1/3 f[N 1/3 (x c - )] Sample to sample distribution of maximum elongation x c.  x c = ) 2 > 1/2 ~ N -1/3 Fluctuations in maximum elongation

8 Definition of Burst  fibers fail before the force F needs to be increased to continue. x Burst of size .

9 Analytical Expression for the Burst Distribution Hemmer and Hansen (1992) D( ,x s )=  -  f(   (x c -x s ))   f(y) approaches a constant for small y, and is proportional to exp(-y 2 ) for large y. Universal scaling exponents Reminiscent of second order phase transition. Process is stopped at x = x s.

10 Uniform distribution Weibull distribution m = 5 x s = x c

11 Burst Distribution as a Signal of Imminent Failure Pradhan, Hansen, Hemmer (2005) Start recording bursts at x 0  0. Change in exponent when x 0 is close to x c.

12 Uniform Weibull

13 A single fiber bundle with N = 10 7, x 0 = 0.9 x c : Earthquakes

14 The Fuse Model Threshold distribution Fuse burns out if voltage difference across it exceeds threshold value t.

15 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: Repulsion between cracks. Current distribution: Attraction between cracks. Competition between Disorder and current Distribution.

16 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)

17 What is needed to reach the infinite-disorder limit? Random number Threshold value Disorder parameter:  How big must  be for the disorder to dominate?

18 Cumulative distribution: Order statistics:

19 Must compare threshold ratio to largest current ratio in network ~2 : 100X100 lattice:  -value for the disorder to dominate completely.

20  = 0.01  = 1  = 100 (Moreira et al. 2012) 32X32:  > 700

21 Scaling in the infinite-disorder limit: This value shows up in many connections…

22 Strong and weak disorder in the fuse model:  * ~ L 0.9 M f : mass of final crack M b : mass of backbone

23 Scale-invariant disorder (Hansen et al. 1991) Current distribution is scale free: Histogram Growing correlation length i ~ (L/  )   N ~ (L/  ) 2  f

24 Intensive (scale free variables): Intensive time: Intensive histogram: Intensive currents: f-  formalism (multifractals) No L dependence: Scale invariance

25 Threshold distribution in intensive variables Threshold distribution Threshold values Threshold distribution Independent of L

26 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 power law tails survive as L 

27 A phase diagram for the fuse model Diffuse loc. Diffuse damage Disorderless Strong dis. Scr. perc.

28 Localization: Soft-clamp fiber bundle model Order in which bonds fail: Lighter: earlier Darker: later (Batrouni et al. 2002)

29 e = E/L = 32 e = e = 2 -6 e = L= 128 Failure point, N p c n= Np (Stormo et al. 2012)

30 r1r1 r4r4 r -1/4 r -4

31

32  x c = ) 2 > 1/2 ~ N -1/3 (Daniels and Skyrme, 1989) W c ~L -2/3 Not an inverse correlation length exponent!

33 There is no phase transition Slope remains finite: crossover -Not a phase transition

34 Scenario: Equal load sharing fiber bundle model until localization sets in. System is never brittle Just percolation until localization sets in. Critical p c not related to percolation threshold.

35 Constrained crack growth: roughness. (Santucci et al. 2010)

36 From Tallakstad et al. 2011

37 Two roughness exponents Santucci et al. 2010

38 Intermezzo: gradient percolation (Hansen et al. 2007)

39 Wavelet analysis of percolation front Roughness exponent  = 2/3 gradient

40 Removing overhangs k  0 Roughness exponent Gradient percolation:  = 2/3

41 Soft clamp model: two in one (Gjerden et al., 2012) StiffSoft

42 Scale invariant elastic constant: e = Ea/L Small E is equivalent to large L.

43 Soft system: roughness exponent  = Large scales

44 Stiff system: roughness exponent  = 2/3. Small scales

45 High precision: Hull of Front Fractal Dimension 10/7

46 Two roughness regimes: Small scale:  = 0.67 – percolation! Large scale:  = 0.39 – fluctuating line.

47 Family-Vicsek Scaling

48 Soft system:

49 Stiff system:

50 From Måløy and Schmittbuhl, 2001

51 Velocity distribution

52 From Tallakstad et al. 2011

53 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