Tristan Cambonie (Postdoc ANR Mephystar and GeoSMEC), Harold Auradou (DR CNRS), Véronique Lazarus (MdC UPMC)

Tristan Cambonie (Postdoc ANR Mephystar and GeoSMEC), Harold Auradou (DR CNRS), Véronique Lazarus (MdC UPMC)

2 Do upper-crust fault segments result from the bottom-up propagation of mode 3 loaded cracks? From Klinger (2010): S plate thickness III III III a. 3D general view of the initial crack a b. after propagation S S b. 2D view from the right of previous figure From Karma, Leblond and Lazarus (2011) S a

3 Linear Elastic Fracture Mechanics Aim: Condition of crack propagation? u p Ω u E, ν F s e 1 e 3 (s) e 2 (s) T p Ω t Solution of the static elasticity problem: strains ε, stresses σ.

4 Definition of K 1 (s), K 2 (s), K 3 (s) and G(s) 1 3 2 e 1 r e 2 M process zone lim r 0 σ ij (r, θ) = 1 K p (s) f p r ij (θ), f p ij (θ) are universal functions. Williams (1952); Leblond and Torlai (1992) Stress Intensity Factors (SIFs): K 1 (s), K 2 (s), K 3 (s) Energy release rate G: G(s) de elast ds = 1 ν2 E (K 1(s) 2 + K 2 (s) 2 ) + 1 + ν E K 3(s) 2 Irwin s formula

5 Propagation threshold F Initial slit Griffith (1920) propagation threshold: G = G c In mode 1, it is equivalent to Irwin (1958) s threshold: K 1 = K c Here: K 1 (F c, E, ν) = K c F c = f (E, ν, K c )

6 What is the crack path? Mode I plane Initial slit K 2 = K 3 = 0 the path is quasi-straight

What is the crack path? Initial slit K 2 0 the crack kinks.

8 How to predict the new direction? (Leblond, 2003) II I II I l ϕ s s Max G: Maximum energy release rate Erdogan and Sih (1963) max G(ϕ) ϕ PLS Principal of local symmetry Goldstein and Salganik (1974) K 2 (ϕ) = 0 Different values of ϕ but small difference < 1.5 (Amestoy and Leblond, 1992) Both give similar results and can be used equally.

9 Application to the kink angle prediction...in agreement with experiments. Erdogan and Sih (1963); Buchholz et al. (2004)

10 Application to more complex geometries 3 Point Bending experiments. Mode I plane Initial slit β x 3 β

11 Application to more complex geometries... Application of the criterions: opposite kink angles along the front. path computed point by point and step by step by finite elements. the obtained twisted path corresponds to the experimental one. FE Buchholz et al. (2004); Lazarus et al. (2008)

Conclusion G = G c and K 2 = 0 or max G gives in practice similar and accurate predictions; extensively used. but in-plane fluctuations of the front are inevitable. Will these perturbations increase (instable) or decrease (stable) in time?

13 Straight front instability y P a x P z K 1 = 2 π Pa1/2 ; K 2 = K 3 = 0 Hence, coplanar propagation with straight front is solution of K 1 (z) = K c and K 2 = 0.

In-plane fluctuations of the front are inevitable Will these perturbations increase (instable) or decrease (stable) in time? y δ(z) a x z To answer, use perturbation approaches: Here (Rice, 1985): δk 1 (z) K 1 (a) = δ(z) 2a + 1 2π PV δ(z ) δ(z) (z z) 2 dz for δ a Other simple geometries in an infinite media: Lazarus (2011)

15 Straight front instability Rice (1985); Leblond et al. (1996); Lazarus (2011) Stability λ < λ c K (A) < K (B) Bifurcation λ = λ c = λ ca K (A) = K (B) Instability λ > λ c K (A) > K (B) a B λ A where λ c = π α a B λ A

16 Straight front stability in experiments y P P a z x Peeling instability (λ h) Adda-Bedia and Mahadevan (2006) Model example Several solutions to K 1 = K c and K 2 = 0 Not so relevant in practice: P(a), difficult to control the velocity Large wavelength instability, width has to be considered Telephone Cord Buckles Faulhaber et al. (2006); Faou et al. (2012)

2D out-of-plane perturbation of a straight crack εf(x) I x For ε 1: δk 2 = ε [ 1 π 0 2 f (0)K 1 2 f (0)A 1 Existence of f (x) 0 such as δk 2 = 0? 2 πx ] f (x)σ xx (x, 0 + ) dx x Movchan et al. (1998) Possible depending on the applied load... Obrezanova et al. (2002)

18 Instability during contraction loading Yuse et Sano, 1993, 1997; Ronsin et al., 1995; Yang and Ravi-Chandar, 2001 From Yuse and Sano, 1993, 1997

19 Instability during contraction loading Simulations: phase-field simulations constructed empirically to have K 2 = 0 (Hakim and Karma, 2009) Corson et al. (2009) variational approach to fracture based on the maximization of G (Francfort and Marigo, 1998) Video on Blaise Bourdin webpage. Bourdin et al. (2008)

20 Mode 1+3 loaded crack K 1 (s) = K 1 K 2 = 0 K 3 (s) = K 3 Path such as: PLS : K 2 = 0 Threshold : G = G c?

21 Mode 1+3 shape stability K 1 (s) = K 1 K 2 = 0 K 3 (s) = K 3 Coplanar propagation with straight crack front is solution. But, uniqueness, stability?

22 Mode 1+3 shape stability. Linear stability analysis Superposition of an exponential growing (a>0)/decreasing (a<0) oscillating, out-of-plane perturbation: ɛφ y (x, z) = ɛa y e x/a sin(kz). in-plane perturbation: Elliptical helix shape crack front ɛφ x (x, z) = ɛa x e x/a cos(kz). Leblond et al. (2011): For K 3 /K 1 > (K 3 /K 1 ) c, a is difficult to interpret experimentally. existence of growing solutions such as K 2 (z) = 0 and G(z) = G c

Mode 1+3 shape stability. Less rigourous linear stability analysis (unpublished) III III III a. 3D general view of the initial crack a b. after propagation S S b. 2D view from the right of previous figure 6 5 4 S/2 a 3 2 nu=0.2 1 nu=0.3 nu=0.4 0 0 2 4 6 8 10 K3/K1 23 For K 3 /K 1 > (K 3 /K 1 ) c (ν), S a, weak influence of K 3 K 1 and ν.

24 Instability observed in twisted samples Sommer (1969)

3PB experiments with inclined slit Mode I plane Initial slit β x 3 β 25 Global smooth twist satisfy to K 2 = 0.

26 3PB experiments with inclined slit At smaller scales: Discontinuous propagation Not uniform crack length Out-of-plane perturbations In qualitative agreement with the linear stability analysis...

27 3PB experiments with inclined slit...but linear analysis overestimates the critical threshold (K 3 /K 1 ) c can not predict the further progressive coalescence of the facets.

28 Instability predicted by phase-field simulations Pons and Karma (2010) Non-linear effects: Progressive coalescence of the facets Subcritical instability Karma et Pons (Nature 2010) Lazarus et al. (IJF 2008)

29 Further developments about mode 3 instability Analytical developments: JB Leblond (IJLRDA, UPMC) Phase-fields simulations: A. Karma and A. Pons (Northeastern University) Experiments: Tristan Cambonie, ANR founded, postdoct.

30 Experiments of ANR Mephystar. Postdoc of Tristan Cambonie (03/2013-07/2013) Wedge splitting (use of Instron testing machine at FAST) Study of the propagation path. Influence of: Mode 2 Material heterogeneities on the propagation path. Material: 1. plexiglas, 2. home-made model rock (sintered beads of polystyrene)

31 Experiments of ANR GeoSMEC Postdoc of Tristan Cambonie (08/2013-07/2014) 3PB fatigue tests (use of hydraulic testing systems at LMS, X) Wedge splitting (use of Instron testing machine at FAST) Mode I plane Initial slit β x 3 β Mode 3 instability. Plexiglas and home-made model rock (sintered beads of polystyrene)

32 Variational approach to fracture The crack state Γ minimizes E(Γ) = E elastique (Γ) + E fracture (Γ) Francfort and Marigo (1998) Equivalent to the max G criterion for preexisting crack. Applicable also to crack initiation and multi-cracks. How to perform minimization among all possible configurations? Numerical regularization: 1. Replace Γ by a continuous "damage" field α = α(m) 2. Replace E(Γ) by E l (α) chosen to have lim l 0 E l (α) = E(Γ) Bourdin et al. (2008)

33 Variational approach to fracture is able to predict quantitatively the crack shape starting from the sound solid. Drying of colloidal suspensions ( 10 nm) Gauthier et al., 2007, 2010 Maurini et al. (In press)

34 Application to segmentation? S Prediction of the rotation angle, but not of S.

35 Summary III III III a. 3D general view of the initial crack a b. after propagation S S b. 2D view from the right of previous figure Theoretical and phase-field studies of crack propagation under mode 3 is understudy. Complex and new studies (>2010). Experiments have to be performed. Aim of Tristan Cambonie s postdoc. Extension: Use of the variational approach to obtain the segments rotation angles? Application to strike-slip faults?

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