1. Crack Shape Evolution William T. Riddell Civil and Environmental Engineering

2. Outline Introduction Work by Mahmoud – the PPP. Beyond the PPP –Complex crack shapes –Non-planar cracks. Summary and Conclusions.

3. Key References Mahmoud [EFM 1988a] Mahmoud [EFM 1988b] Mahmoud [EFM 1990] Mahmoud [EFM 1992] Gera and Mahmoud [EFM 1992] Newman and Raju [NASA TM 1979, EFM 1988]

4. Simple Question How can we quantify the evolution of a crack shape?

5. Cracked Configuration Crack size and shape defined by a and c Remaining crack front interpolated as elliptical quadrants Bending and uniform tension loading considered a c

6. Simulating Crack Growth Assume a cyclic load history Assume initial a, c Calculate K( ) Calculate K a and K c Calculate a and c Update a and c Newman and Raju Paris model 4 different approaches

7. Crack Lengths vs Time

9. a and c define size and shape for semi-elliptical cracks

10. Outline Introduction Work by Mahmoud – the PPP. Beyond the PPP –Complex crack shapes –Non-planar cracks. Summary and Conclusions.

11. Simulating Crack Growth Assume a cyclic load history Assume initial a, c Calculate K( ) Calculate K a and K c Calculate a and c Update a and c Newman and Raju Paris model 4 different approaches Mahmoud reformulated so time was not explicitly in formulation a/t (size) and a/c (shape) remaining variables.

12. Resulting Differential Equation da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o

13. Resulting Differential Equation da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o da dc = F(a,c) G(a,c) a(t o ) = a o a/c(t o ) = a o /c o

14. Evolution of Crack Shape

15. Other initial sizes and shapes

16. Preferred Propagation Pattern Cracks evolve toward same path, regardless of initial size and shape

17. Mahmouds Idea Mathematicized System is autonomous, i.e., equations do not explicitly contain t. The a-c plane is the phase plane Curve developed by solution to equations is a trajectory Mahmouds equations are asymptotically stable. da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o

18. Mahmouds Idea Mathematicized System is autonomous, i.e., equations do not explicitly contain t. The a-c plane is the phase plane Curve developed by solution to equations is a trajectory Mahmouds equations are asymptotically stable. da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o

19. Mahmouds Idea Mathematicized System is autonomous, i.e., equations do not explicitly contain t. The a-c plane is the phase plane Curve developed by solution to equations is a trajectory Mahmouds equations are asymptotically stable. da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o

20. Mahmouds Idea Mathematicized System is autonomous, i.e., equations do not explicitly contain t. The a-c plane is the phase plane Curve developed by solution to equations is a trajectory Mahmouds equations are asymptotically stable. da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o

21. Mahmouds Idea Mathematicized System is autonomous, i.e., equations do not explicitly contain t. The a-c plane is the phase plane Curve developed by solution to equations is a trajectory Mahmouds equations are asymptotically stable. da dt = F(a,c) dc dt = G(a,c) a(t o ) = a o c(t o ) = c o

22. So What? You do not have a chance of predicting crack growth rate if you do not know crack shape. There is significantly less variation in the way that crack shape evolves, compared to crack growth rate. Comparison of crack shape evolution is one way to evaluate the accuracy of a crack growth simulation.

23. Quantification of Error Defined residual R i Compare predicted and observed evolution of shape. Evaluate 4 different ways to find K A and K C Evaluated empirical relationships from Portch [1979], Kawahara and Kurihara [1975], and Iida [17]. R i = pred. (a/c) i – obs. (a/c) i

24. Extract Values of K Value at specific a c a c Weighted average over Reduce K c by a factor

25. Refined Question How can crack shape be quantified? –Graphically describe evolution as 2D problem in phase plane –Quantify differences between predicted and observed shapes.

26. Mahmouds Conclusions PPPs depend on Paris exponent, m. PPP change as loading ranges from pure tension to pure bending. Point values with factor best overall prediction. Kawahara and Kuriharas equation best of three empirical equations considered.

27. Outline Introduction Work by Mahmoud – the PPP. Beyond the PPP –Complex crack shapes –Non-planar cracks. Summary and Conclusions.

28. More Complex Planar Cracks 4 point bend specimens Off-center initial notch

29. Crack Shape Evolution Crack size and shape not easily characterized by two values BA

30. Crack Lengths vs Cycles Crack lengths defined as distance traveled by crack tip crack tip 1 crack tip 2 a 1 a 2

31. Crack Shape Evolution Slight divergence in crack shape Small variation compared to rates crack tip 1 crack tip 2 a 1 a 2

32. Extension to Complex Shapes Concept of PPP applies to the shape of planar cracks, not just 2 crack lengths. Concept of phase plane is useful, even when 2 DOF do not completely characterize crack shape.

33. Non-Planar Crack Growth Tests by Pook and Greenan [1984] Side View Bottom View 25 mm 360 mm 75 mm 25 mm 300 mm P = 18 to 79 kN R = 0.1

34. 3D Simulations compared to Pooks data Experiments and Simulations

35. 3D concept of PPP L.P. Pook On Fatigue Crack Paths, IJF 1995 –There can exist a surface that attracts crack growth

36. Non-Planar Crack Growth Initial notch at angle to principal stresses

37. Photo of Resulting Face Factory roof facets near top of initial notch A B

38. Path of Crack Tips Good agreement for crack tip path. Not much progress toward attracting surface until crack tips pass edges and crack becomes a through crack.

39. Application to Non-Planar Cracks Attracting surface and phase plane both useful concepts. Even though its very hard to quantify non- planar crack shape, 2 DOF can help evaluate simulations. Both PPP and Attracting Surface describe tendency for cracks to evolve certain ways, despite initial crack size and shape.

40. Outline Introduction Work by Mahmoud – the PPP. Beyond the PPP –Complex crack shapes –Non-planar cracks. Summary and Conclusions.

41. Summary and Conclusions Mahmoud presented concept of PPP for evolution of aspect ratio for a cracked configuration. This is an example of an autonomous system that is asymptotically stable. – there are rules for stability based on eigenvalues. PPP can be used to evaluate fidelity of simulations when compared to experiments. Concept of a-c phase plane is useful for more complex geometries as well.

42. Summary and Conclusions Pook presented concept of attracting surface for non-planar crack growth. Attracting surface is straightforward to visualize under symmetric load cases. Under some real cases, crack growth is non-planar. The attracting surface is less obvious here.

43. Parting Question There is a mathematical way to show Mahmouds PPP are asymptotically stable. Is there an analogous way to identify, define, or show an attracting surface exists?