Stability and roughness of crack paths in 2D heterogeneous brittle materials. Eytan Katzav Disordered Systems Group King’s College London [email protected] In collaboration with M. Adda-Bedia & B. Derrida (LPS-ENS) Open Statistical Physics, 7 March 2012. Cracks – moving singularities.
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Stability and roughness of crack paths in 2D heterogeneous brittle materials
Eytan Katzav
Disordered Systems Group
King’s College London
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
L
Lz
Rough surfaces
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
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
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:
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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