Small crack energy release rate from configurational force balance in hyperelastic materials

Transcription

1 Small crack energy release rate from configurational force balance in hyperelastic materials M. Aït-Bachir, E. Verron, W. V. Mars, and P. Castaing Institut de Recherche en Génie Civil et Mécanique, Ecole Centrale Nantes BP 92101, Nantes cedex 3, France 1

2 Industrial motivation Rubber anti-vibration mount end of fatigue life = onset of macroscopic crack ( 1mm) 2

3 Experimental observation macroscopic crack = propagation of microscopic defects crack plane is specifically oriented w. r. t. loading conditions cycles Crack initiation and growth under proportional tension-torsion [Mars, 2006] 3

4 Experimental observation macroscopic crack = propagation of microscopic defects crack plane is specifically oriented w. r. t. loading conditions cycles Crack initiation and growth under proportional tension-torsion [Mars, 2006] 4

5 Experimental observation macroscopic crack = propagation of microscopic defects crack plane is specifically oriented w. r. t. loading conditions cycles Crack initiation and growth under proportional tension-torsion [Mars, 2006] 5

6 Experimental observation macroscopic crack = propagation of microscopic defects crack plane is specifically oriented w. r. t. loading conditions cycles Crack initiation and growth under proportional tension-torsion [Mars, 2006] 6

7 Our previous works: a continuum approach Continuum Idealized microstructure Model Defect = material plane = oriented surface dsn in the undeformed configuration - Andriyana's thesis (2006) - Aït-Bachir's thesis (2010) - Verron, E. et Andriyana, A. Definition of a new predictor for multiaxial fatigue crack nucleation in rubber. J. Mech. Phys. of Solids, 56, (2008). 7

8 Our previous results A criterion: the minimum principal configurational stress ( Σ, 0) Σ = min T P I =1, 3 with Σ =WI F Σ = W σ max the maximum true stress A critical plane: the material plane subjected to the maximum stretch We recovered and rationalized the previous results in rubber fatigue: Mars (2002), Saintier (2006), Brunac et al. (2009) Verron E. Configurational Mechanics: a tool to investigate fracture and fatigue of rubber. Rubber Chem. Technol., to appear 8

9 Limits of our approach Influence of a physical defect on mechanical fields? Relevance of configurational stress with defects? The parameter ''flaw size'' is not present 9

10 The present study Continuum Idealized microstructure Model Defect = micro-crack = Small crack in an infinite medium - Collaboration with William V. Mars - Aït-Bachir's thesis (2010) 10

11 Model features small crack in an infinite 2D membrane Crack orientation: N θ Crack length: c Far-field: homogeneous and uniform F λ1 0 0 = 0 λ ( λ1λ 2 ) far-field 2 Near-field: Singularity at crack tip = non-uniform 11

12 Energy release rate T Definition = U c dépl T = lim 0 δc U( c + δc) δc U( c) Practical calculation: J -integral T = q ΣNdΓ Γ 12

13 Method of calculation Crack-based coordinate system Γ: rectangular contour (pathindependence) Small crack R >> c and D>>c - Top, bottom, right: far-field - Left: near-field + far-field 13

14 Details of the derivation Top segment: N = e' 2 D 0 D Σ dl =Σ 0 ( Σ ) dl = 0 D Bottom segment: N = e' 2 = Σ D 14

15 Details of the derivation Right-hand side: N = e' 1 R R Σ 11 dl Left-hand side: N = e' 1 R R R R + left Σ ( ) 11 l dl = ( left Σ ( ) ) 11 Σ11 l dl 15

16 Finally Unbalance of configurational stresses: left vs. right Energy release rate of a small crack = Unbalance of the configurational stress T + ( left ( ) ) = Σ Σ l dl 16

17 Transformation of the result Central symmetry of all mechanical fields w. r. t. center of the crack + ( left ( ) ) T = Σ Σ l dl + 0 = ( left ) T = 2 Σ Σ ( l) dl 17

18 Proportionality with respect to the crack length Scale transformation c kc All the mechanical tensor fields are identical in M 1 and M 2! T(kc) kt(c) Generalization of the results of Rivlin and Thomas (1953) and Yeoh (2002) 18

19 To conclude: link between the two approaches Continuum With a material plane With a small crack Change in energy due to the material displacement of the material plane Energy release rate of the small crack To compare : Σ max θ T ( c) c 19

20 To conclude: link between the two approaches UE EqBE Good agreement for ''small strain'' The near-field is influent for large strain Extension of the small crack approach to multiaxial loading? 20