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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 eytan.katzav@kcl.ac.uk 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

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  1. 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

  2. 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

  3. 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)

  4. 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)

  5. 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

  6. L Lz 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

  7. 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.

  8. 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)

  9. 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.

  10. 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

  11. 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

  12. 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)

  13. 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

  14. 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

  15. 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 .

  16. The equation - local toughness fluctuations - local shear fluctuations - proportional to the T-stress An example:

  17. 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)

  18. 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

  19. 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

  20. Many shear perturbations Average power spectrum Averaging over ten realizations NO FITTING PARAMETERS!!!

  21. 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.

  22. Self-affinity? Anything goes!

  23. 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)

  24. 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 …

  25. L Lz 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

  26. 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

  27. L Lz 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

  28. 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)

  29. 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.

  30. L Lz 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

  31. The T-stress in a strip geometry T as a function of k in a strip

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

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