Measure what is measurable, and make measurable what is not so. - Galileo GALILEI. Dynamic fracture. Krishnaswamy Ravi-Chandar

Transcription

1 Measure what is measurable, and make measurable what is not so. - Galileo GALILEI Dynamic fracture Krishnaswamy Ravi-Chandar Lecture 4 presented at the University of Pierre and Marie Curie May, 04 Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics

2 Outline Crack surface roughness development Crack front waves Crack branching Tape peeling /46

3 Dynamic crack evolution 3/46

4 Surface roughening Mirror Mist Hackle Branch Ravi-Chandar and Knauss, Int J Fract, 984 4/46

5 Dynamic crack front evolution Mirror Mist Hackle Branch Ravi-Chandar and Knauss, Int J Fract, 984 5/46

6 Microcracking model for dynamic fracture Ravi-Chandar and Knauss, Int J Fract, 984 Homalite /46

7 Surface roughening and limiting speed Microcracking, instability R-C and Knauss, IJF, 984, R-C and Yang, JMPS, 996 Dynamic instability and microbranching Sharon and Fineberg, PRL, 996 Crack twisting mode III perturbations Hull, J Mat Sci, 997 Microcracking, roughening Bonamy and coworkers, 00, L p 7/46

8 Surface roughening - PMMA Ravi-Chandar and Yang, J Mech Phys Solids, 997 PMMA 8/46

9 Formation of conic marks a 00 mm 5 mm b Smekal, 953 Ravi-Chandar and Yang, J Mech Phys Solids, 997 ; Ravi-Chandar, Int J Fract, 998 PMMA 9/46

10 Plane of the microcracks - microbranches Ravi-Chandar and Yang, J Mech Phys Solids, 997 PMMA 0/46

11 Nucleus of the conic Ravi-Chandar, Int J Fract, 998 PMMA /46

12 Statistics of conic marks Density of Conic Markings - mm Region a Region b Region c 0 Focal Length - microns Position - mm 3 Focal Length - microns 0 Ravi-Chandar and Yang, J Mech Phys Solids, 997 PMMA /46 Focal Length - microns

13 Statistics of conic marks - Scheibert, Guerra, Celarie, Dalmas and Bonamy, PRL, 00 3/46

14 Speed of microcracks? Assume that microcracks grow with the same speed If microcrack speeds are of equal, then the shape of the conic is independent of the speed (R-C and Yang, 997) Dalmas et al (IJF, 03) show this by examining the detailed shapes of conics How fast do the microcracks grow? It is not really possible to examine this experimentally. To be determined by other means! This is still an open issue Can we simulate this? just the kinematics! 4/46

15 Nucleation and growth model: Flaw Nuclei Nucleation is by cavity growth Spacing and density are governed by the stress level Measured densities are in the range of 500 to 500 nuclei per mm Ravi-Chandar and Yang, J Mech Phys Solids, 997 5/46

16 Nucleation and growth model: Nucleation When the stress at a flaw nucleus reaches a critical value, the nucleus becomes an active microcrack; calculation of this requires detailed knowledge of the flaw dimensions and the detailed stress field in its vicinity. Here we assume that when an active crack front approaches a nucleus to within a critical nucleation distance, d n the flaw is nucleated into a microcrack. active crack front nucleus d n, nucleation distance Ravi-Chandar and Yang, J Mech Phys Solids, 997 s, spacing between nuclei 6/46

17 Nucleation and growth model: Growth Constant microcrack speed all active microcracks grow with the same speed regardless of the density of microcracks Constant energy flux microcrack speed depends on the number of active microcracks and the available energy Ravi-Chandar and Yang, J Mech Phys Solids, 997 7/46

18 Constant microcrack speed Ravi-Chandar and Yang, J Mech Phys Solids, 997 8/46

19 Microcrack speed Macrocrack speed Constant power input Position Position 0.5 9/ Position Ravi-Chandar and Yang, J Mech Phys Solids, 997

20 Comments Differences in density of conic marks between different types of PMMA need to be reconciled Bonamy et al have one to two orders of magnitude lower conic marks per unit area Quantitative modeling of the deformation within the fracture process zone cavitation, crack initiation, growth, coalescence, Continuum modeling based on heterogeneity? Line models and quantitative calibration/comparison Crack front waves 0/46

21 Crack front waves a quick summary Gk ˆ (, ) P ˆ( k, ) Aˆ ( k, ) G0 ˆ( Pk, ) obtained from perturbation solution of Willis and Movchan; has a simple zero corresponding to a propagating mode with speed C f ; this is the crack front wave. A( z, t) a( z, t) v t 0 Cf C R Morrissey and Rice, JMPS, 000 /46

22 Wallner lines Source Field, J Contemporary Physics, 97 /46

23 Photographs: Wallner Jill lines Glass, Sandia National Laboratories 3/46

24 Mode III perturbations Pulse amplitude - V Glass Bonamy and Ravi-Chandar, 003 4/46 z y x.7 mm E E E-06 Distance traveled 7.50E-06 - m 8.00E E Wavelength and duration for glass f MHz l mm t ns

25 Specimen P v= 890 m/s f = 5 MHz 0 mm 5/46 5

26 .7 mm 6/46 Specimen AC v= 440 m/s f = 5 MHz 6

27 Specimen AI v= 440 m/s f = 0 MHz 0 mm 7/46 7

28 Crack-ultrasonic pulse interaction Crack front v Shear wave q l z C s x vl/c s 8/46 z y x Attenuation is similar to that of bulk glass v m/s f MHz l x l v/c s mm

29 Results of Fineberg et al 9/46

30 Summary on crack fronts Theory predicts that they exist Important in roughness generation; bulk waves can also deliver energy to the cracks and cause roughness, but these CF waves are more effective. Difficult to distinguish from Wallner lines Small amplitude plane wave perturbation Perfectly linear response to mode III perturbations Responds to wavelength of input, speed of crack no inherent characteristic length! No persistent interaction between different pulses No persistence when perturbation is removed Need larger amplitude perturbations? 30/46

31 Dynamic crack branching 3/46

32 3/46

33 Crack branching Crack position, a - mm Dynamic Stress Intensity Factor, K I - MPa m / K I a Time - ms 33/46

34 Microcracking model for dynamic fracture Ravi-Chandar and Knauss, Int J Fract, 984 Homalite /46

35 Control of surface roughening and branching Ravi-Chandar and Knauss, IJF, /46

36 Fracture mechanics Outer Problem: Knowledge of constitutive laws and balance equations is adequate to calculate the energy release rate, G L p Inner Problem: Are the details of the failure mechanisms and fracture processes important only in determining the fracture energy, G? 36/46

37 Crack branching analysis Stress field based Yoffe (95): v~0.65 C R Energy-based Eshelby (970) Freund (97): v~0.53 C R Adda-Bedia et al. (005, 007) G E A v K I( ) I dynamic energy release rate Hoop Stress - f qq Angle (Radians) v/cd= /46

38 Energy balance calculations of Adda-Bedia et al K t v k v K t p I II III 0 p (, ) p ( ) p (,0),,, 0 Kp ( t, v) kl ( v) H pl l, v ', v Kl ( t,0) l Not all the H pl are available, but H 33 was calculated by Adda-Bedia et al, corresponding to the mode III problem. 0 l K ( t, v) k ( v) H, v ', v K ( t,0) III III 33 III G g( v) H, v K ( t,0) m 0 33 l III ; G G G v G v v 0.39 C ; l 0.(39.6 ); v 0 s Adda-Bedia et al. 005, 007, 38/46

39 In-plane modes G g v H l v K t g v H l v K t m 0 0 ( ), (,0) ( ), (,0) I I II II ; G G G v G v H and H are not available, but Adda-Bedia argues that in the limit of zero crack speed, these should behave similarly to H 33, corresponding to the mode III problem. v 0.58 C ; l 0.3(3.4 ); v 0 G v R influences the critical speed, but not the branching angle Adda-Bedia et al. 005, 007, 39/46

40 Dynamic lifting of a tape Inextensible, flexible tape with tension T, mass per unit length r w( x, t) w( x, t) w( x, t) transverse deflection transverse velocity slope (assumed small) T w( x, t) w w, w t L rw Tw l ct w w c 0 x t c T r speed of transverse waves ds Expect piecewise linear solutions: s dt w w s w w 0 f f s f s t x w x c In the absence of adhesion in the string, you can show that If w is prescribed, then w w / c s c Burridge and Keller, 978, Freund, 990, Duomochel et al, 008, 40/46

41 Dynamic peeling of a tape with adhesion Now let us consider adhesion of the tape to the substrate with an adhesive energy G Consider pulling one end of a semiinfinite tape at a constant speed, w. As a result, a peel front moves with a speed s Expect piecewise linear solutions: w t w s w w x But another equation is needed to determine 0 T w( x, t) l w st w, w t This is derived from energy balance: dynamic energy release rate T G L s L s w w c / G T w s c s w x s 4/46

42 Dynamic peeling of a tape with adhesion Normalized Slope Impose Griffith condition: G = G G T w G s c T w( x, t) w w, w t w x w s w qs c 5 4 l st s w qs G / T 3 s w c c G/ T w c Normalized Velocity 4/46

43 Peeling front encountering a weak patch G 0 x l G G G l x Let us peel quasi-statically: d w qs w qs w G / T ; w 0 qs d l G / T G G l qs l a At this point, the peel front encounters a weak patch; let us fix d w w G / T qs qs Quasi-statically, crack will extend until w G / T qs qs l a G l G equilibrium crack length However, the crack will propagate dynamically 43/46

44 Dynamic peeling front encountering a weak patch G G G l x But, the crack will advance dynamically, at a speed A kink wave will propagate towards the fixed end in the peeled portion of the tape t 44/46 c G 0 x l c l s s c x h c w qs l ct w, w crack tip: sw w 0 w w cw w kink: 0 dynamic energy release rate cw w w w / w s G / T w G G w G G qs s c G G s / G / G

45 Dynamic peeling front encountering a weak patch Next, we consider reflection from the fixed end of the kink wave w w cw w kink: 0 r This will arrive at the location at t l l s a / r x l a d w r 0, w r c w qs w, w t t c c l s c l a x l s G w w w w r qs qs c G The peel front arrests and everything becomes quiescent! G qs la l la G s 45/46

46 Extensions to this one-d problem Continue loading at x = 0; must consider multiple reflections Introduce speed dependence to the energy G(v) Consider flexibility straightforward extension to the bending problem Consider extensible strings more complex due to the presence of longitudinal and transverse waves 46/46