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Danas Sutula Prof. Stéphane Bordas Dr. Pierre Kerfriden Alexandre Barthelemy

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. Content. Problem statement Crack growth Discretization by XFEM Implementation

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Danas Sutula Prof. Stéphane Bordas Dr. Pierre Kerfriden Alexandre Barthelemy

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  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 Problem statement Crack growth Discretization by XFEM Implementation Verification Results Summary Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  3. Problem statement • 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 : 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 Δai ≥ 0: Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  4. The solution procedure at time consists of • solving the variational form for : • advancing the fracture fronts such that follows the path of steepest descent while satisfying the Griffith’s energy is balance: Problem statement • The solution procedure at time tkconsists of • solving the variational form for u(ak): • advancing the fracture fronts, such that Π(u,ak) → Π(u,ak+1) follows the path of steepest descent while satisfying Griffith’s energy balance Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

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

  6. Crack growthenergy minimization Energy release rate w.r.t. crack increment direction, θi: The rates of energy release rate: Updated directions: Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

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

  8. DiscretizationXFEM discontinuous enrichment singular tip enrichment standard part Approximation function Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  9. Implementation Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  10. Implementation Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

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

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

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

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

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

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

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

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

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

  20. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  21. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  22. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  23. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  24. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  25. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  26. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  27. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  28. Results10 crack problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  29. Resultsdouble cantilever problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  30. Resultsdouble cantilever problem Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  31. Results2 edge cracks; plate in vertical tension Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  32. Results2 edge cracks; plate in vertical tension Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  33. Results2 edge cracks; internal pressure loading Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  34. Results3 cracks; center crack pressure loading Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  35. ResultsEdge crack in a PMMA beam with 3 holes Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  36. ResultsEdge crack in a PMMA beam with 3 holes Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  37. ResultsEdge crack in a PMMA beam with 3 holes Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  38. ResultsEdge crack in a PMMA beam with 3 holes Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  39. ResultsEdge crack in a PMMA beam with 3 holes Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  40. Results2 edge cracks and 2 holes (Khoeil et al. 2008) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  41. Results2 edge cracks and 2 holes (Khoeil et al. 2008) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  42. Results2 edge cracks and 2 holes (Khoeil et al. 2008) Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

  43. Summary 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. Global energy minimization for multi-crack growth in linear elastic fracture using XFEM

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

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