1. Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method Danas Sutula Prof. Stéphane Bordas Dr. Pierre Kerfriden Alexandre Barthelemy 24/07/2014

2. Content 1.Problem statement 2.Crack growth 3.Discretization by XFEM 4.Implementation 5.Verification 6.Results 7.Summary Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 2

3. Consider a cracked linear-elastic isotropic solid subject to an external load whose quasistatic behavior can be described by the following total Lagrangian form: The solution for u(a) and a(t) are obtained by satisfying the stationarity of L(u,a) during the evolution of t, subject to Δa i ≥ 0 : Problem statement Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 3 Consider a cracked linear-elastic isotropic solid subjected to an external load whose quasistatic behavior can be described by the following total Lagrangian form: The solutions for and are obtained by satisfying the stationarity of during the evolution of, subject to :

4. The solution procedure at time t k consists of 1.solving the variational form for u(a k ) : 2.advancing the fracture fronts, such that Π(u,a k ) → Π(u,a k+1 ) follows the path of steepest descent while satisfying Griffith’s energy balance Problem statement Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 4 The solution procedure at time consists of 1.solving the variational form for : 2.advancing the fracture fronts such that follows the path of steepest descent while satisfying the Griffith’s energy is balance:

5. Crack growth maximum hoop stress Post processing of solution to evaluate SIF Crack growth direction Crack growth criterion Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 5

6. Energy release rate w.r.t. crack increment direction, θ i : The rates of energy release rate: Updated directions: Crack growth energy minimization Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 6

7. The discrete potential energy is given by: Energy release rate w.r.t. crack increment direction θ i : The rates of the energy release rate: Crack growth energy minimization Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 7, where:

8. Discretization XFEM Approximation function Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 8 singular tip enrichment discontinuous enrichment standard part

9. Implementation Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 9

10. Implementation Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 10

11. Verification energy release rates Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 11 Π vs. θ F F -θ Test case: square plate with an edge crack with a small kink loaded in vertical tension

12. Verification energy release rates Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 12 G vs. θ Test case: square plate with an edge crack with a small kink loaded in vertical tension F F -θ

13. Verification energy release rates Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 13 (topological enr.) dG/dθ vs. θ Test case: square plate with an edge crack with a small kink loaded in vertical tension F F -θ

14. Verification energy release rates Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 14 (geometrical enr.) dG/dθ vs. θ Test case: square plate with an edge crack with a small kink loaded in vertical tension F F -θ

15. Verification energy min. VS. max-hoop Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 15 G min(Π) /G hoop vs. θ F F θ Test case: square plate with an inclined center crack in vertical tension

16. Verification energy min. VS. max-hoop Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 16 F F Test case: square plate with an inclined center crack in vertical tension

17. F F p = F Verification energy min. VS. max-hoop Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 17 Test case: square plate with a pressurized inclined center crack in vertical tension

18. F F p = F Verification energy min. VS. max-hoop Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 18 Test case: square plate with a pressurized inclined center crack in vertical tension

19. Verification energy min. VS. max-hoop Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 19 F F Test case: square plate with a pressurized inclined center crack in vertical tension G min(Π) /G hoop vs. θ θ

20. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 20

21. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 21

22. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 22

23. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 23

24. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 24

25. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 25

26. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 26

27. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 27

28. Results 10 crack problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 28

29. Results double cantilever problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 29

30. Results double cantilever problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 30

31. Results 2 edge cracks; plate in vertical tension Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 31

32. Results 2 edge cracks; plate in vertical tension Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 32

33. Results double cantilever problem Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 33

34. Results 2 edge cracks; plate in vertical tension Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 34

35. Results 2 edge cracks; internal pressure loading Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 35

36. Results 3 cracks; center crack pressure loading Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 36

37. Results Edge crack in a PMMA beam with 3 holes Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 37

38. Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 38 Results Edge crack in a PMMA beam with 3 holes

39. Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 39 Results Edge crack in a PMMA beam with 3 holes

40. Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 40 Results Edge crack in a PMMA beam with 3 holes

41. Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 41 Results Edge crack in a PMMA beam with 3 holes

42. Results 2 edge cracks and 2 holes (Khoeil et al. 2008) Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 42

43. Results 2 edge cracks and 2 holes (Khoeil et al. 2008) Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 24

44. Results 2 edge cracks and 2 holes (Khoeil et al. 2008) Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 44

45. A robust approach to determining multiple crack growth directions based on the principle of minimum energy Limitations undermining the max. hoop-stress criterion, e.g. traction free cracks, absence of body loads etc., are overcome The energy minimization approach is characterized by mode-1 dominance at the end of each time-step (i.e. on convergence) The criteria lead to fracture paths that are, in the global sense, in very close agreement - virtue of self-similar crack growth Better accuracy and faster convergence of the fracture path can be obtained by taking a bi-section of the interval that is bounded by the respective criteria. Summary Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 45

46. Acknowledgement Danas Sutula (IMAM) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM 46 Danas Sutula, Stéphane Bordas and Pierre Kerfriden would like to acknowledge the support of SOITEC, with particular thanks to Alexandre Barthelemy (RND)