1. Crack propagation on highly heterogeneous composite materials Miguel Patrício

2. Motivation Macroscopic view : - a (foot)ball (round object) Microscopic view : - thick round-ish skin - fissures and cracks - collection of molecules - simple (?) problem - not so accurate -complicated problem - accurate

3. Motivation Macroscopic view : - a (foot)ball (round object) Microscopic view : - thick round-ish skin - fissures and cracks - collection of molecules Best of both worlds???

4. Model crack propagation Macroscopic view Microscopic view Matrix Inclusions

5. Problem formulation “Determine how (and whether) a given crack will propagate.” - Where to start?

6. Problem formulation “Determine how (and whether) a given crack will propagate.” - What makes the problem complicated?

7. Simplify “Determine how (and whether) a given crack will propagate.” - Microstructure - Crack propagation (how) ??? Assume: Static crack

8. Starting point - Static crack is part of the geometry “Determine whether a given crack will propagate in a homogenised medium.” What homogenised medium?

9. Microstructure to macrostructure ? Macrostructure Microstructure

10. Microstructure to macrostructure Homogenisation Macrostructure Microstructure

11. Homogenisation Macrostructure Microstructure Assume: There exists a RVE

12. Mathematical homogenisation Macrostructure Microstructure Assume: There exists a RVE Periodical distribution

13. Mathematical homogenisation Microstructure Linear elastic materials: Hook’s law Elasticity tensors

14. Mathematical homogenisation Microstructure averaging procedure ( and )

15. Example -0.5 0.5 0.5 Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio

16. Homogenised solution Example Exact solution Horizontal component of the displacements

17. Mathematical homogenisation - “Sort of” averaging procedure - Loss of accuracy - Alternatives do exist (heterogeneous multiscale method, multiscale finite elements…) - Periodic structures (but not only) - Simplifies problem greatly

18. Crack propagation (homogeneous case) Assume: pre-existent static crack homogeneous material

19. Crack propagation (homogeneous case) Question: will the crack propagate?

20. Crack propagation (homogeneous case) Question: will the crack propagate? (one possible) Answer: look at the SIFs Crack tip

21. How to compute the SIFs? Crack propagation (homogeneous case) Question: will the crack propagate? Why look at the SIFs? - Solve elasticity problem (FEM) - Determine the stresses - Crack will propagate when - Direction of crack propagation - Compute + how?

22. Step by step - Pull the plate - Compute displacements and stresses - Check propagation criterion compute SIFs - If the criterion is met, compute the direction of propagation Increment crack (update geometry) -What length of crack increment?

23. Example - FEM discretisation (ABAQUS) - Crack modelled as a closed line - Open crack (after loading):

24. Example

25. Crack propagation - Homogeneous media - how and whether the crack will propagate - Pre-existing crack - Incrementation approach - What about heterogeneous media?

26. Crack propagation - What about heterogeneous media? Idea: employ homogenisation and apply same procedure Bad

27. Local effects

28. Crack tip in material ACrack tip in material B Crack tip in homogenised material

29. Crack propagation - What about heterogeneous media? Idea: employ homogenisation and apply same procedure Bad Because the local structure may not be neglected when the SIFs are computed

30. FEM will not work!!! Crack propagation (composite material) Assume: pre-existent static crack composite material

31. Domain decomposition Assume: pre-existent static crack composite material - Partition computational domain - Instead of one heavy problem, solve many light problems

32. - Allows for a complex problem to be divided into several subproblems Domain decomposition - Schwarz procedure dates back to the XIX century - Parallelization may be implemented - Deal with different problems where different phenomena exists - May overlap or not

33. Homogenisable Hybrid Approach Homogenisable Schwarz (overlapping) Homogenisation Crack

34. Hybrid approach for the SIF Layered material Crack

35. Hybrid approach for the SIF Crack - Employ homogenisation far away from the crack - Use Schwarz overlapping scheme

36. - Uses homogenisation where possible; resolves heterogeneous problem where necessary Hybrid approach - Combines homogenisation and domain decomposition - More than one micro region may be considered - Accuracy depends on the accuracy of homogenisation or, on other words, on how much the material is homogenisable in the macro region - Why not domain decomposition? - Why not homogenisation?

37. - Domain decomposition divides problem in subproblems Summary - Homogenisation yields macroscopic equations - Fracture propagation can be implemented by incrementing the crack - Hybrid approach combines these two techniques

38. Main open question Assume: pre-existent static crack composite layered material - What happens when the crack hits the interface between the layers?

39. A few references

40. Model crack propagation Macroscopic view Microscopic view Linear elastic homogeneous plate Plate composed by linear elastic homogeneous constituents Matrix Inclusions Layered material ? Other micro- structure