Fracture Mechanics of soft Materials: crack tip field of a material that exhibits finite extensibility

Transcription

1 Fracture Mechanics of soft Materials: crack tip field of a aterial that exhibits finite extensibility Chung-Yuen Hui Mechanical and Aerospace Engineering, Cornell University Acknowledgeent: Forer Ph.D student: Rong Long (Prof. at U. Alberta). International Workshop on Flow, Fracture, Interfaces in Soft Heterogeneous Materials. The Michelin Material Science Chair at ESPCI Paris Tech. Dec. 9 th 10 th, 013 Partial support by NSF, Materials & Surface Engineering progra, CMMI

2 What is in this talk? Brief overview of LEFM (linear elastic fracture echanics) and soe results in large deforation elastic fracture echanics Finite Extensibility: Crack tip fields of a Mode III crack in a Gent aterial. Singularity is alost invariably a clue The Boscobe valley Mystery, by Sir Arthur Conan Doyle

3 Fracture of Stiff and Soft Materials Stiff aterials: Soft aterials Sun et al., Nature, 01,489,

4 Soft aterial: Large deforation Stiff aterials: (etal, ceraics, polyer glass ) linear elasticity, plasticity T 11 / tension neo-hookean Linear elastic l 1 =L/L 0 Hyperelastic aterial: neo-hookean odel: S = W / F W 3 ( I ) = - 3 / Soft aterials (rubber, gel, biological tissue ) nonlinear elasticity, viscoelasticity A o Stress Measures Noinal stress: S True stress: T T = SF T (incopressible) P P A Stress: T 11 = P/A S 11 = P/A o I = l + l + l l 1 =L/L Stretch ratio:

5 Large deforation theory: kineatics x = (x 1, x ) = reference coordinates (undefored configuration) X = position of a aterial point x after deforation x y u(x) displaceent of a aterial pt at x Crack face before deforation x Crack face after deforation u O X y O x 1 y 1 X = x + u(x) y = (y 1, y ) = X u(0) = coordinates of a aterial pt in the defored configuration which is at x before deforation Direction of load application

6 LEFM: Crack Tip fields r KI KII KIII Tij = fij + fij + fi p r p r p r ( q ) ( q ) ( q ) I II III crack 0 q K I, K II, K III = Mode I, II, III stress intensity factors K I K II K III G = + + E E G = energy release rate Key Results In-plane stresses identical for plane stress or plane strain All stress coponents have the sae square root singularity and specified by three loading paraeters (K I,K II, K III ). Mode I and Mode II separable (pure Mode II exist) Separable solution for each Mode High Triaxial stress state directly ahead of crack tip. Stress state is hydrostatic for incopressible solids like rubbers (Mode I Plane strain).

7 Large deforation: Plane strain, Neohookean Mode I T T Linear theory KI = T = p R 11 1 = 0, T = True (Cauchy) stress tensor - / 3 / 11 T AR T = B T = T = Large deforation, f = 0 Crack profile after deforation y R crack f f R y 1 undefored crack T >> T1 T11 Uniaxial stress state R. Stephenson, J. Elasticity, 198,1,65-99 V. Krishnan, C.Y. Hui, EJP, (009)

8 Mode I Plane stress Crack tip fields (neo-hookean) Crack Crack face face before before deforation deforation x Crack face after deforation T 11 O y r q O T T1 T x 1 T 11 R f Wong & Shield, ZAMP, (1969) Geubelle, & Knauss, J. Elasticity 1994 C1C T = C / 4 r, T11 = C1, T1 = - sin q / r True stresses in current configuration: C1C C T =, T11 = C1, 4R 3/ C1 C T1 = f1( R, f) R ( RC1 sin f + C cosf ) f=p/, T ~ 1/R Krishnan, et al. Languir, 4, 008 ( ) Crack face before deforation O Crack face after deforation 8 O f=0, T ~ 1/R 1

9 Penny-shaped Crack in an infinite Hyperelastic Solid Linear theory predicts Energy Release Rate independent of Triaxiality. T Y.Y. Lin and C. Y. Hui, Intl. J. Frac. 16, 05, 004 R. Long and C.Y. Hui, Soft Material, 6, 138, 010 Triaxiality factor : S/T r 0 S T ro T G = f, E E S T E Cristiano et al, 009, J. polyer Science, part B

10 Finite strain versus sall strain theory Noralized energy release rate * G( T, S) G ( S / T ) = G( T, S = 0) Noralized energy release rate G* T=0.3E T=0.01E T S Neo-Hookean solid Hydrostatic Uni-axial S/T Linear elasticity predicts energy release rate independent of lateral tension.

11 Brief Suary: Mode I, in-plane stresses doinated by uniaxial tension in both plane stress and strain. Crack tip fields (true stresses) in plane stress not the sae as plane strain Absence of Pure Mode II Two paraeters needed to characterize crack tip field in Mode I True stress fields in defored coordinates in general not separable. Energy release rate sensitive to triaxiality 11

12 Effect of Strain Hardening Entropy controlled Strain hardening neo-hookean odel Generalized neo-hookean (GNH): 00 True stress: s n b W 1 ( I 3) 1 = b n Exponentially hardening s/ Stretch ratio l 1 Exp: J=3.5 GNH (n=,b=1) W J I - 3 = exp -1 J 50 neo-hookean Linear elastic l 1

13 Effect of hardening on energy release rate of a penny-shaped crack R. Long and C.Y. Hui, Soft Material, 6, 138, 010 neo-hookean Exponential hardening solid J M =5.09

14 Paradox: Plane stress crack, GNH Geubelle. P.H & W. G. Knauss, J. Elasticity, 35, 61-98,(1994) y c 1-1/ n 1 = Br q( q, n) y = Ar f ( q, n) c, q( q, n) deterined by solving a pt. bvp. A, B undeterined constants works for n < 1.46 What happens for larger n (ore strain hardening)? R. Long, V. Kristnan & C.Y. Hui, JMPS,59, (011) / 4n * = A ( q; ), > n 1.46 Only one unknown A y r g n n n -1 T / = A r Tˆ ( q, n) 1-1+ n-3/ True stresses: T ˆ 1 / = A r nt 1( q, n) 3-1+ n-3 T11 / = A r nt 11( q, n)

15 Exponentially hardening aterial Region I Undefored crack Region II Region III Region I & III Region II T = r f -1 ( ( q ), -3/ ( )) ( q ) ( -3/4 ln ( / )) ( q ) -1 ln ( / ) q, T = r -J ln r / r f, T = r -J r r f ( ) ( ) ( -5/ ( )) ( q ) ( -7/ 4 ( )) ( q ) T = r -J r r f -1 0 T = r -J ln r / r f, T = r -J ln r / r f, I I /r q = 3p/4 0.1 Defored Crack 10 4 q=3p/4 crack x 0 II Region II T / -1/rln(r) III III 10 q = 0 crack q= x r/a

16 What about a aterial with finite extensibility? (stress approaches infinity at finite extension) Arruda & Boyce (1993) JMPS Gent (1996) Rubber Che. Technology W J g = - ln1- J

17 Finite extensibility (stress approaches infinity at finite extension) Boyce & Arruda, Rubber Cheistry and Technology, 000

18 Finite extensibility: Gent s odel Mode III Analysis: Long & Hui, Proc. R. Soc. (011) x Crack x 1 Solution valid for a rigid lower aterial Gent s strain energy density for Mode III: W g = w, + w, J g = - ln1- J 1 g 1 18 x 3 Knowles 1977 Intl. J. Fracture, 13, 611

19 Governing equations & boundary conditions Knowles 1977 Intl. J. Fracture, 13, 611 W dw / dg dw / dg w,, w,, g g J g = - ln1- J = 0 g = w, + w, 1 ( ) w, x 0,x = 0 = 0 1 x Crack faces traction free x 1

20 Nonlinear PDE to linear PDE: hodograph transfor Rice 1967, JAM Idea: Map physical plane (x 1, x ) to strain plane (g 1, g ) by x = / g x g f g x 1 f f = p / = 0 g 1 J J - g = + g g g g g f 0 linear PDE

21 Exact solution in Strain Coordinates Introduce crack tip coordinate h = 1- g J + = c 1 k h + a k h sin( k + 1) f k= 0 arbitrary = 1 known Doinant crack tip behavior h -> 0 h c1k sin( k + 1) f k= 0 Infinite nuber of paraeters is needed to specify the crack tip field

22 Sall Scale Yielding K r w r sin / III ( ) = ( q ) crack r q Exact solution in strain-plane 1 = - - r lnr + r sinf r g r = J c 10 = -, c 1k = 0

23 Relation between physical and strain plane (SSY) r lnr 1 1- r sin cotq = cot f +, r = g / J f - - ( r 1) 4( r 1) r f 4( r ) r = ln cos + ln r q f = q - p/ h = 1- r 0 r = 1 at crack tip r f r = 1 g / J f = q + p/ g1 / J

24 Exact solution: Sall Scale Yielding Sall scale yielding: approxiate crack tip fields Region II / = 0, = 1 KIII p r Region I, III K cos K cos sin r J r III III 1 =, = J Crack tip field not separable, highest stress behind the crack

25 Asyptotic field near crack tip Region I Region III q r Region II I,III: II: = 1 = J J r q r ( q ) q ( 0) ( q ) cos q, sinq J q =, >> 1 r Contains unknown function q Asyptotic field in Linear theory : -KIII q KIII q 1 = sin, = cos p r p r

26 Results Strain hardening allows stress state near crack tip to be ore triaxial. Crack tip fields of highly strain hardened aterial can change behavior rapidly across a boundary layer In Mode I cracks, highest stress does not occur directly ahead of the crack tip For Gent s aterial, crack tip field can not be characterized by a finite nuber of paraeters (unknown function).