Stability and roughness of crack paths in 2D heterogeneous brittle materials

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Stability and roughness of crack paths in 2D heterogeneous brittle materials

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Stability and roughness of crack paths in 2D heterogeneous brittle materials

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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 with

M. Adda-Bedia & B. Derrida (LPS-ENS)

Open Statistical Physics, 7 March 2012

Cracks – moving singularities

A crack in two dimensions

Crack tip

Linear elasticity + free boundary conditions on the crack faces

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

Stress tensor

What can cracks do?

Apparently, much more than one can imagine…

They can bifurcate

(Katzav et al, IJF 07)

(Ravi-Chandar, 2003)

(Andersson, 1969)

Micro branching instability

(Sharon&Fineberg, 1996)

They can oscillate…

A moving cutting tip

(Roman et al., 2004)

Thermal crack

(Ronsin et al., 1995)

(Corson et al, preprint)

Fast cracks

(Livne et al., 06)

2D crack interaction

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

Dry mud

(river bed in Costa Rica)

Glaze of a ceramic plate

(Bohn et al, 2005)

T-junctions

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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

Schmittbuhl & Maloy 1997

logDh(Dx)

Slope = z

log Dx

Roughness – Self-Affinity I

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

(z = 1/2)

h(x)

Dx

x

y

y + Dx

under anisotropic rescaling has the same statistical properties

z= ½ results from uncorrelated steps

z > ½ implies positive correlations while z < ½ implies negative ones.

Roughness – Self-Affinity II

… in Fourier space

log fq

h(x)

log q

x

Fourier components

average

Static correlation function

roughness exp.

For small q!!!

Milman, Stelmashenko and Blumenfeld (PMS 94)

Schmittbuhl, Vilotte and Roux (PRE 95)

Back to the fracture surfaces … 3D

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!!!)

(Ponson & al. 06)

Mortar: Mourot, Morel, Bouchaud & Valentin 05

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

2D problems:

out of plane crack path roughness

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

z = ~ 0.6

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

z = ~ 0.75

Concrete: Balankin et al. 2005

Questions:

1. Stability – under which conditions is the crack stable

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

(Directed Polymer problem – Barabasi & Stanley 95)

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

Still very complicated due to dependence on the whole history

Crash course onLinear Elastic Fracture Mechanics

The three fracture modes:

Mode I

Mode II

Mode III

Mode I: Pure opening

Mode II: In-plane shear / sliding

Mode III: Out-of-plane shear / tearing

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

LEFM – stress field singularity

Crack tip

Stress tensor

Stress Intensity Factor (SIF)

(external loading + geometry)

universal functions

T-stress

(in the direction of the crack)

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)

Principles of crack propagation

1. The Griffith criterion – an energy balance G = G(Griffith, 1920)

where G is the fracture energy = energy invested in creating new surfaces…

(equivalent to K=Kc – where Kc is the material toughness - Irwin)

2. 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

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

- Criticisms:
- Infinite strip – finite strip with width H
- Just one encounter with heterogeneity

The model

… many kinking events, with undisturbed propagation between events.

From which follows the basic equation:

Applying the Principle of Local Symmetry right after kinking gives

Identifying two noisy quantities, , rescaling ( )

and defining .

The equation

- local toughness fluctuations

- local shear fluctuations

- proportional to the T-stress

An example:

Stability – the T criterion

We begin by studying the T-dependence

Conclusion: for T ≤ 0 we get stable paths,

while for T > 0 the path becomes unstable,

and we generalize the T-criterion to heterogeneous materials,

while fixing the problem of (Cotterell & Rice 80)

Beyond stability – the T=0 limit

We can put aside the T-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

where we have defined

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 a and

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!

We can easily solve this equation in our configuration

Average path

Many shear perturbations

Average power spectrum

Averaging over ten realizations

NO FITTING PARAMETERS!!!

Self-affinity?

What does the analytical result teach us?

→ Flat

→z = 1/2

- NO self-affinity:
- Flat on large scales
- Random-walk like on small scales

How does this compare with the measured roughness z ~ 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.

Self-affinity? Anything goes!

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 …

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Outlook

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

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?

Sintered Plexiglass

under pure Mode I

Outlook

3. 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.

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

Plouraboue & Roux 96

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2D problems II:

inplane crack front roughness

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

(z = 0.5 - 0.65)

It’s a different kind of experiment:The dynamics of the fracture front itself is directly measured “in vivo”.

Sintered Plexiglass

under pure Mode I

Schmittbuhl & Måløy 97: z = 0.54

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

Significantly more complicated experimentally

Inplane fracture – a word on wetting

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

formulation.

z = 0.5

(Prevost et al. 1999,

Moulinet et al. 2002)

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

Is it in the same universality class? YES

(Katzav et al., 07)

Equation of motion

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

The nonlinear corrections are “relevant” in the RG sense

Applying Renormalization Group techniques (the Self-Consistent

Expansion) we find:

- For high velocities, z = 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 z = 0.5, z = 1.
- Combined with the neglected irreversibility we conclude that z > 0.5.
- Note that in the wetting problem where locally the irreversibility is much weaker z ~> 0.5.

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Outlook

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 plane

roughness

In-plane roughness

3D roughness

The T-stress in a strip geometry

T as a function of k in a strip

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