Initiation de fissure dans les milieux fragiles - Prise en compte des contraintes résiduelles

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1 Initiation de fissure dans les milieux fragiles - Prise en compte des contraintes résiduelles D. Leguillon Institut Jean le Rond d Alembert CNRS/UPMC Paris, France

2 Parvizi, Garrett and Bailey experiments (978) 2.5 Is it a Weibull effect? Controversy: Kelly (988), L. et al. (25) Applied strain (%) Inner ply thickness (mm)

3 Parvizi, Garrett and Bailey experiments (978) 2.5 Energy Parvizi et al. (978) Applied strain (%) Inner ply thickness (mm)

4 Parvizi, Garrett and Bailey experiments (978) 2 Applied strain (%).5.5 Energy Parvizi et al., (978) + Stress L. (22) Martin and L. (24) Inner ply thickness (mm)

5 3-point bending of V-notched specimens Stress criterion onset whatever the applied load for ω < π. Energy criterion (Griffith) no onset whatever the applied load if ω >. These conclusions are contradictory and do not match with the experiments, obviously onset occurs but for a finite applied load. How to state a criterion for the prediction of the crack onset? The starting point is a very simple energy balance δ + + = P K W δw Ga c And the observation that in quasi-static tests K δw

6 The coupled criterion The crack nucleation occurs abruptly. The crack jumps from to a length a without any equilibrium state in between (except if it occurs at the tip of a pre-existing crack) Two conditions must be fulfilled simultaneously (L., 22): there is enough available energy to create a crack of length a the tensile stress is greater than the tensile strength all along the presupposed crack path (i.e. between and a) δ P W ( a) Ga c σ( z) σ for z a c Gc is the material toughness σc is the tensile strength δ W P ( a) : change in potential energy, z: distance to the notch root σ ( z) is a decreasing function of z then inc G a σ a ( ) ( ) and with G σ c c inc P G ( a) δw ( a)/ a = incremental energy release rate a is unknown ( incremental vs. differential considering the limit a )

7 The coupled criterion P P P P inc W () W ( a) W ( a) W ( a+ δ a) G ( a) = ; Ga ( ) (Martin and L., 24) a δ a G inc : incremental energy release rate G: energy release rate (Griffith, 92) Crack nucleation inc G a σ a ( ) ( ) and G σ c c Crack growth Ga ( ) G c

8 Under some assumptions, the previous expressions can be explicitly derived using matched asymptotic expansions (L., 989) inc 2 2λ G a = Ak a + ( )... G a σ 2 2λ ( ) 2 Ak a... = λ + λ ( a) ka... = + The coupled criterion matched asymptotic expansions A is a scaling coefficient k is the generalized stress intensity factor of the singularity at the corner λ is the exponent of the singularity Recalling that the solution of a linear problem of elasticity can develop in the vicinity of the corner as follows (Williams, 958) U( r, θ) = C+ kr λ u( θ) +... r and θ are polar coordinates with origin at the V-notch root.

9 Then the coupled criterion takes the special form (Irwin-like form) (L., 22) a = G Aσ c 2 c λ Gc k kc = A The coupled criterion matched asymptotic expansions σ 2λ c Remind Crack nucleation inc G a σ a ( ) ( ) and G σ c c Crack growth Ga ( ) G c

10 Case : monotonic case (energy) 3-point bending on V-notched homogeneous specimen (L., 22), 4-point bending on V-notched laminated ceramics (Bermejo et al., 26)

11 Case : monotonic case (energy) F=.8 Frupt a

12 Case : monotonic case (energy) F=.97 Frupt a

13 Case : monotonic case (energy) F= Frupt a

14 Case : monotonic case (energy) F= Frupt a

15 Case 2: energy governed non monotonic case Double cleavage drilled specimen (L. et al., 25), Fiber interface debonding in composite material (Martin et al., 28) High Gc, low σc

16 Case 2: energy governed non monotonic case F=.9 Frupt a

17 Case 2: energy governed non monotonic case F= Frupt a

18 Case 2: energy governed non monotonic case F= Frupt a

19 Case 2: energy governed non monotonic case F=.2 Frupt a

20 Case 3: stress governed non monotonic case Double cleavage drilled specimen (L. et al., 25), interface debonding in composite material (Martin et al., 28), edge cracking (Chen et al., 2; L. et al., 25) Low Gc, high σc

21 Case 3: stress governed non monotonic case F=.9 Frupt a

22 Case 3: stress governed non monotonic case F=.95 Frupt a

23 Case 3: stress governed non monotonic case F= Frupt a

24 Case 3: stress governed non monotonic case F= Frupt a

25 Initiation and arrest lengths F= Frupt a initiation length lower bound for the arrest length upper bound for the arrest length

26 Mode mixity The first generalization is to consider a mode mix loading: both the critical load and the crack direction must be predicted. 4 F/F 3 β=3 2 (Yosibash et al., 26 ; L. et al., 29) ω α ω=3 β 3 6 9

27 Increasing complexity Mode mixity Symmetric loading, opening mode Data: GIc, σc (Leguillon, 22) unknown (critical load) Complex loading, opening mode Data: GIc, σc (Yosibash et al., 26) 2 unknowns (critical load + crack direction) Complex loading, mixed mode (guided crack) Data: GIc, GIIc, σc, τc (Tran et al., 22) unknown (critical load)

28 Residual stresses in an adhesive layer (epoxy). Matched asymptotic expansions λ Thermal residual stresses in an adhesive layer m t Gc k = k + k kc = ( σc Θσ) A (Henninger and Leguillon, 28) Comparison with experiments (Kian and Akisanya, 998) 2 Θ = 4 deg. (i) ( σ c, Gc) = ( 45 MPa, 46 Jm ) neglecting the residual stresses; with 2 residual stresses; (ii) ( σ c, Gc) ( 45 MPa, 9 Jm ) = with residual stresses.

29 Thermal residual stresses in a brazed assembly A similar result can be obtained in brazed assemblies of SiC structures (Nguyen et al., 22) Comparison with experiments carried out in CEA Grenoble: load at failure vs. solder thickness. The transition between the energy driven criterion and the stress driven one is visible.

30 Thermal residual stresses in layered ceramics The crack initiates at the notch root and then grows unstably and penetrates the next layer and stop as k I = not ki = kic (i.e. the cracks is closed due to the compressive residual stresses). Then after reloading, it kinks (Leguillon et al., 25). Only pure FE calculations are allowed because the layers thickness is similar to the crack length.

31 Laminated ceramics: before kinking F=.9 Frupt a

32 Laminated ceramics: kinking F= Frupt a

33 Laminated ceramics: kinking F= Frupt a

34 Laminated ceramics: stable growth F=.2 Frupt a

35 Laminated ceramics: instability F=.4 Frupt a

36 F (N) The pop-in occurs at F=52 N in the experiments and is predicted at F=46 N in the model Laminated ceramics: a complete scenario Crack kinking followed by a stable growth of the deflected crack 2 3 Instability h (mm)

37 Residual stresses in a polymer due to oxydation

38 Residual stresses in a polymer due to oxydation The challenge here is that in the oxydized layer, the Young modulus, the tensile strength and the toughness vary. Again, only pure FE calculations are allowed. A comparison with experiments (Pannier et al., 24; Leguillon et al., 26) shows a good agreement, both in terms of applied load at initiation of crack(s) and in terms of cracks density.

39 Residual stresses in zirconia due to phase change Grinding a notch in zirconia produces big stresses at the notch which induces a phase change and generates high compressive residual stresses (around 8 MPa) in a thin layer of a few micrometers (< µm). They are released by a heat treatment KK Ic (MPa m) 5 5 rr ( µm ) TL as notched TL heat treated DL as notched DL heat treated

40 . Thermal residual stresses in sub-micron films Coating intrinsic strain (%) Experiments by Andersons et al. (28) Film thickness (µm)

41 . Thermal residual stresses in sub-micron films Coating intrinsic strain (%) Hashin (energy based criterion) Film thickness (µm)

42 . Thermal residual stresses in sub-micron films Coating intrinsic strain (%) CC Film thickness (µm)

43 6 5 Scaled crack density µm.67 µm.9 µm.2 µm.32 µm Theory Scaled applied strain (µm /2 )