Stability and roughness of crack

paths in 2D heterogeneous

brittle materials

Eytan Katzav

Disordered Systems Group

King’s College London

eytan.katzav@kcl.ac.uk

In collaboration withM. Adda-Bedia & B. Derrida (LPS-ENS)

Open Statistical Physics, 7 March 2012

Cracks – moving singularities

Linear elasticity + free boundary conditions on the crack faces

yields a singular behavior of the stress field in the vicinity of the tip

ij jn 0

I

ij ij

I I

I

Kr, ~

2 r

A crack in two dimensions

Crack tip

Stress tensor

r

What can cracks do?

Apparently, much more than one can imagine…

(Andersson, 1969) (Ravi-Chandar, 2003)

Micro branching instability(Sharon&Fineberg, 1996)

Micro branching instability(Sharon&Fineberg, 1996)

They can bifurcate (Katzav et al, IJF 07)

Thermal crack(Ronsin et al., 1995)

A moving cutting tip(Roman et al., 2004)

Fast cracks(Livne et al., 06)

They can oscillate…

(Corson et al, preprint)

… and when there are many of them they can produce complex structures

Glaze of a ceramic plate(Bohn et al, 2005)

Dry mud(river bed in Costa Rica)

T-junctions

2D crack interaction

Rough surfaces

• The work of Mandelbrot, Passoja and Paullay (84) –

A first systematic study of the fractal nature of fracture surfaces.

• Bouchaud et al. (90), Måløy et al. (92) and many more…. used concepts like fractals and self-affine surfaces to describe properties of rough cracks (from nano to macro scales)

Mourot et al., 2005

L

L

Schmittbuhl & Maloy 1997

Roughness – Self-Affinity I

Statistical self-affine shapes, i.e. random walks

12h x 1 h x p1

x

h(x)

under anisotropic rescaling has the same statistical propertiesx bx

h b h

x y x xx y x x x

2

y

h xh y x h y RM

max h y min h y mi

S

n max

y y + x

x

h x x

( = 1/2)

= ½ results from uncorrelated steps

> ½ implies positive correlations while < ½ implies negative ones. log x

log h(x)

Slope =

Roughness – Self-Affinity II

Milman, Stelmashenko and Blumenfeld (PMS 94)Schmittbuhl, Vilotte and Roux (PRE 95)

… in Fourier space

x

h(x)

iq x d

q d 2

1h h x e d x

2

Fourier components

Static correlation function q q q Sh h ~ q

average

d

2

For small q!!!roughness exp.

log q

log q

An anisotropic scaling is found with two scaling exponents:

~ 0.6 along the direction of the propagation and

~ 0.8 along the front (too large!!!)

Back to the fracture surfaces … 3D

After 20+ years measurements, the full 3D problem is still debated: How to analyze? Universality? Anisotropy? No solid equations of motion.

(Ponson & al. 06)

Mortar: Mourot, Morel, Bouchaud & Valentin 05

Fracture of 2D materials – i.e. paper, concrete

2D problems:

out of plane crack path roughness

Simpler: a 2D problem with one well-defined exponent; easier experiments

Still very complicated due to dependence on the whole history

= ~ 0.6

Paper: Santucci et al. 2004, Bouchbinder et al. 2006.

Concrete: Balankin et al. 2005

= ~ 0.75

Questions:

1. Stability – under which conditions is the crack stable

2. Roughness – what determines ? why > ½ ? is it universal?

(Directed Polymer problem – Barabasi & Stanley 95)

Crash course on

Linear Elastic Fracture Mechanics

Mode I: Pure opening

Mode II: In-plane shear / sliding

Mode III: Out-of-plane shear / tearing

Mode I Mode II Mode III

The three fracture modes:

LEFM – stress field singularity

III

ij ii x jxI

jr, ~ O r2 r

TK

Stress Intensity Factor (SIF)(external loading + geometry)

universal functions T-stress(in the direction of the crack)

Crack tip

Stress tensor

r

1. In general, the SIF’s and the T-stress depend on:the geometry of the medium (infinite, strip, etc…),the shape of the crack h(x), on the loading (not easy to determine)

ij 2. The structure of the expansion and the functions are universal

Principles of crack propagation1. The Griffith criterion – an energy balance G = (Griffith, 1920) where is the fracture energy = energy invested in creating new surfaces… (equivalent to K=Kc – where Kc is the material toughness - Irwin)

IIK 02. The Principle of Local Symmetry (PLS) – at each time the crack chooses a direction such that it will propagate locally in a pure opening mode (Goldstein & Salganik, 1970).

Crack path is mostly selected by PLS,

while Griffith determines rates… we need to know KI and KII

Stability à la Cotterell & Rice, 80’ Cotterell & Rice considered a semi-infinite straight crack, that encounters a

single shear perturbation at the origin, forms a kink and continues …

T>0: crack path is unstable and grows exponentially

T<0: crack path is stable

Criticisms:

1. Infinite strip – finite strip with width H

2. Just one encounter with heterogeneity

h x ~ x h x ~ exp x

1 2

xh x1 2

II I2 x xK h x h x k T dx

1IH h x k

Based on a perturbative dependence of the SIF’s on the shape

III

i ij ix jxI

jr T, ~ O r2 r

K

The model

… many kinking events, with undisturbed propagation between events.

From which follows the basic equation: thi 1 i i kinkh h h x x x

Applying the Principle of Local Symmetry right after kinking gives

1 2

I

xh x dxII II 1 1 2 T

II I2 2H k x xI I I

K K 2x 2 2 k h x h x k

K K K

Identifying two noisy quantities, , rescaling ( )

and defining .

1 2

xh x dx

2 1 x xˆh x x x h x h x T

I

L 2L TH k

ˆ, T

I ,II

I

k11,2 K x

x Lx

h Lh

0 x 1

II

I

Kkink K2

1

The equation

- local toughness fluctuations

1 21

xh x dx

x2 xˆh x h x h 0 1x x xx T

1 x m mx x D x x 2 x - local shear fluctuations

5

1 2

10 , 20

D 0.04, D 6.25

T̂ 0

1

length scale T̂ - proportional to the T-stress

An example:

Conclusion: for ≤ 0 we get stable paths,

while for > 0 the path becomes unstable,

and we generalize the T-criterion to heterogeneous materials,

while fixing the problem of (Cotterell & Rice 80)

Stability – the T criterion

We begin by studying the -dependence

4

1 2

10 , 500

D 0.004, D 0.008

h x ~ x

T̂ 4

T̂ 0

T̂ 4

Beyond stability – the T=0 limit

We can put aside the -term since:

As long as the growth is stable, a scaling argument (strengthened by

numerical results) shows that it is not important in the large + small scales.

Actually, in physical systems we expect T to be small (less than 1)

As a consequence the model become exactly solvable in that limit.

And we can get the x-dependent Fokker-Planck equation

dP Ph , h ,x h C h h P

dx h h

where we have defined 21C

This equation has no 2nd derivatives, and is just a Liouville equation for a

deterministic evolution

Deterministic evolution…

By writing a Fokker-Planck equation for it turns out that the evolution of its PDF

become deterministic, and controlled by and

1 2h x h x h xx x

Averaging over realizations of the local toughness fluctuations, amounts to replacing the noise term by a negative constant that is proportional to the density of the heterogeneities!

21C 0

We can easily solve this equation in our configuration h 0 0, h 0 0

12

1

1x2C x y

12

0 2

sinh C C 4 x yh x e dy

Cy

C 4

2 xCh x h x h x

21C

Average path

Many shear perturbations

1 2

q q 2 21,2 2 2 2

1 C q 2C 2h h D

q C C q

Averaging over ten realizationsNO FITTING PARAMETERS!!!

4

1 2

ˆ10 , 1,T 0

D 0.04, D 4.2

Average power spectrum

Self-affinity?

What does the analytical result teach us?

1 2

22 2 2

1 C q 2C 2

q q 2 2q C C q

const small q'sh h D ~

q large q's

NO self-affinity:

•Flat on large scales

•Random-walk like on small scales

How does this compare with the measured roughness ~ 0.6-0.8?

We suspect that it is an artifact due to curve fitting by power-laws

and due to a systematic bias in real-space self-affine extraction

algorithms.

→ Flat

→ = 1/2

Self-affinity? Anything goes!

4

1 2

ˆ10 , 1, T 0

D 0.04, D 4.2

4

1 2

ˆ10 , 100, T 0

D 0.04, D 0.04

Reliability of self-affine measurements

A systematic bias in real-space self-affine measurements!

Not mentioned in: Milman, Stelmashenko and Blumenfeld (PMS 94) Schmittbuhl, Vilotte and Roux (PRE 95)

Summary and Conclusions

1. We derive an equation that describes crack paths in heterogeneous 2D

brittle media. The model becomes exact in the limit T=0.

2. Stability: The model extends the validity of the T-criterion, and fixes the

crack path prediction for stable paths – path decays into a flat

configuration.

3. Roughness: The model predicts non self-affine behavior, with different

scaling for large/small scales. Paths are globally flat as observed.

4. Bad news – No Universality

(anyway, different from the one discussed so far in the literature)

5. Good news – No Universality: description beyond roughness –

information about the bulk from measurements on the crack …

Outlook1. Revision of the experimental results along the lines presented here.

1 2

q q 2 22 2 2

1 C q 2C 2h h D

q C C q

L

L

Sintered Plexiglassunder pure Mode I

2. Can we say something about the full 3D problem?

Is it related to the simplified 2D problems?

Coupling between in-plane and out-of-plane roughness?

Outlook3. Self-affine measurements based on real-space methods are very

sensitive to oscillations/decays, and therefore not recommended for crack paths’ analysis. Better to use Fourier methods or analysis of the whole PDF.

Plouraboue & Roux 96

0.88

2

q q 2 4

A 1 Bqh h

1 Cq Dq

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

log10

qlo

g10

<h-q

h q>

4. Spectrum which is a rational function is maybe more common than what is currently believed: Crumpled paper

Fracture of 3D materials but confined onto a 2D plane

2D problems II:

inplane crack front roughness

Simpler as it is both a 2D problem and history independent (almost…)

Significantly more complicated experimentally

= 0.5 - 0.65

Schmittbuhl & Måløy 97: = 0.54

It’s a different kind of experiment:

The dynamics of the fracture front itself

is directly measured “in vivo”.L

L

Sintered Plexiglassunder pure Mode I

Wetting of an amorphous solid by a liquid has a similar mathematical

formulation.

Inplane fracture – a word on wetting

Difference: wetting is intrinsically a 2D problem while fracture is a 3D…

Is it in the same universality class? YES

(Prevost et al. 1999,

Moulinet et al. 2002)

= 0.5

(Katzav et al., 07)

Equation of motion for a moving in-plane crack (Katzav et al., 2006)

Equation of motion

* 2

0 0 0

h z h z h zh dz dz dz 3 4ˆK K K K h z, t

t 2 z z 2 2 z z z z 8 3

The nonlinear corrections are “relevant” in the RG sense

• For high velocities, = 0, z=1 (as in the linear order – Rice 85, Ramanathan & Fisher 97-98).

• At low velocities, occurs a dynamical phase transition into a phase with = 0.5, z = 1.

• Combined with the neglected irreversibility we conclude that > 0.5.

• Note that in the wetting problem where locally the irreversibility is much weaker ~> 0.5.

I cg v K h z , t K Griffith / Irwin

Applying Renormalization Group techniques (the Self-Consistent

Expansion) we find:

Outlook

L

L

2. Can we say something about the full 3D problem?

Is it related to the simplified 2D problems?

Coupling between in-plane and out-of-plane roughness?

Out-of the planeroughness 3D roughnessIn-plane roughness

+?=

T as a function of in a strip

The T-stress in a strip geometry

2d sc c

0 0

III

ij ii x jxI

jr, ~ O r2 r

TK

I 0

2 1K

The plot of T as a function of kappa in a strip