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
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Stability and roughness of crack paths in 2D heterogeneous brittle materials
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. - PowerPoint PPT Presentation
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… 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)
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