Geometrical Aspects of 3D Fracture Growth Simulation (Simulating Fracture, Damage and Strain Localisation: CSIRO, March

1. Geometrical Aspects of 3D Fracture Growth Simulation(Simulating Fracture, Damage and Strain Localisation: CSIRO, March 2010) John Napier CSIR, South Africa University of the Witwatersrand, South Africa

2. Acknowledgements Dr Rob Jeffrey, CSIRO Dr Andrew Bunger, CSIRO

3. OUTLINE • Target applications. • Displacement discontinuity approach to represent fracture growth. • Projection plane scheme: Search rules and linkage elements. • Application to (i) tensile fracture (ii) brief comments on shear fracture. • Explicit crack front growth construction. • Application to tensile fracture. • Conclusions and future work.

4. TARGET APPLICATIONS • Fracture surface morphology (fractography). • Fracture growth near a free surface. • Hydraulic fracture propagation. • Fatigue fracture growth. • Rock fracture and slip processes near deep level mine excavations and rock slopes. • Mine-scale seismic source modelling.

5. KEY QUESTIONS • How should complex crack front evolution surfaces be represented spatially in a computational model? • What general principles apply to 3D tensile crack front propagation? e.g. “no twist” and “tilt only” postulates (Hull, 1999). • To what extent does roughness/ fractal fracture affect fracture surface evolution? • Can complex shear band structures be replaced sensibly by equivalent displacement discontinuity surfaces?

6. 3D fracture surface complexity

7. Tensile fracture structures: • “Fractography”: Crack surface features such as river lines and “mirror/ mist/ hackle” markings are extremely complex. • The spatial discontinuity surface is not restricted to a single plane. • Different surface features may arise with “slow” vs. “fast” dynamic crack growth. • Crack front surfaces may disintegrate under mixed mode loading over all scales.

8. River line pattern from mixed mode I/ III loading. (Hull, Fractography, 1999) Propagation direction ~0.1 mm

9. Coal mine roof spall (From Ortlepp: “Rock fractures and rockbursts – an illustrative study”, 1997)

10. Shear fracture structures: • Complex substructures – overall “localised” damage region in narrow bands. • Multiple damage structures on multiple scales. • Differences between “slow” vs. “fast” deformation mechanisms on laboratory, mine-scale and geological-scale structures is unclear.

11. (From Scholz “The mechanics of earthquakes and faulting”)

12. West Claims burst fracture (From Ortlepp: “Rock fractures and rockbursts – an illustrative study”, 1997)

13. West Claims burst fracture detail (From Ortlepp: “Rock fractures and rockbursts – an illustrative study”, 1997)

14. Displacement Discontinuity Method • Natural representation for material dislocations. • Require host material influence functions (complicated for orthotropic materials and for elastodynamic applications). • Small strain unless geometry re-mapping used. • Only require computational mesh over crack surfaces. • Crack surface intersections require special consideration.

15. Displacement Discontinuity Method (DDM) - displacement vector integral equation:

16. DDM – stress tensor integral equation:

17. Assume element surfaces are planar. Allow constant or high order polynomial variation in each element with internal collocation. Edge singularity unresolved problem in some cases – not necessarily square root behaviour near corners or near deformable/ damaged excavation edges. Element shape functions

19. Shape function weights:

20. Overall element DD variation:

21. Full-space influence functions – radial integration over planar elements:

22. Influence evaluation: • Radial integration scheme most flexible for planar elements of general polygonal or circular shapes. • Can combine both analytical and numerical methods for radial and angular components respectively. • Half-space influences developed.

23. Projection plane strategy • Reduce geometric complexity. • Allow for fracture surface morphology: e.g. front deflections, river line features. • Construct a mapping of the evolving fracture surface offset from an underlying projection plane. • Cover the projection plane with contiguous tessellation cells.

24. Additional assumptions • Assume that the fracture is represented by a single, flat discontinuity element within each growth cell. • Assume a simple constitutive description for tensile fracture or shear slip vs. shear load in each growth element. • Need to postulate ad hoc rules to decide on the orientation of the local discontinuity surface in each growth cell.

25. Projection plane growth cells Z Variable Vertex elevations to determine growth element position and tilt within projection prism Y Fixed cell boundaries in X-Y projection plane X Possible “linkage” element perpendicular to projection plane

26. Edge connected search: Existing edge Z New element test orientations Existing element Y Cell boundaries in X-Y projection plane X

27. Edge search distance factor, Rfac: New element orientation Existing element Search radius = Rfac X element effective dimension

28. Search along growth cell axis: Selected element centroid and orientation Z Existing element vertices Y Growth cell centroid X Search line perpendicular to projection plane

29. Implications: • Must consider whether linking, plane-normal bridging cracks need to be defined. • Cannot efficiently represent inclinations relative to the projection plane cells greater than ~ 60 degrees. • Require assumptions concerning the choice of cell facet boundary positions. • Fracture intersection will require special logic.

30. Initial investigation • Assume that the projection plane is tessellated by a random Delaunay triangulation or by square cells. • Test tension and shear growth initiation rules. • Determine fracture surface orientation using (a) an edge-connected search strategy in tension and (b) growth cell axis search strategy in shear.

31. Incremental element growth rules • Introduce a single element in each growth step. • Determine the optimum tilt angle, using a growth potential “metric” such as maximum tension or maximum distance to a stress failure “surface”, evaluated at a specified distance from each available growth edge. • Re-solve the entire element assembly following each new element addition. • Stop if no growth element is found with a “positive” growth potential metric.

32. Parallel element growth rules • Introduce multiple elements in each growth step. • Determine the optimum tilt angles at all available growth edges using the growth potential “metric” evaluated at a specified distance from all available growth edges. • Select the best choice within each growth cell prism. • Accept all growth cell elements having a “positive” growth potential metric. • Re-solve the entire element assembly following the addition of the selected growth elements. • Stop when no further growth is possible.

33. EXAMPLE 1:Mixed mode loading crack front evolution – simulation of “river line” evolution.

34. Mixed mode loading Z Y X Crack front Inclined far-field tension in Y-Z plane

46. EXAMPLE 2:SHEAR FRACTURE SIMULATION

47. SHEAR BAND PROPERTIES • Shear band structures have complicated sub-structures but have intensive localised damage in a narrow zone. • Multiple deformation processes (tension, “plastic” failure, crack “bridging”, particle rotations) arise in the shear zone. • Can these complex structures be represented by a single, equivalent discontinuity surface with appropriate constitutive properties?

48. Preliminary tests: • Shear fracture growth with projection plane: Search along growth cell axis. Growth cell tessellation; triangular vs. square cells. Incremental growth initiation. Coulomb failure: Initial and residual friction angle = 30 degrees.

49. Shear loading across projection plane: Z X-Y projection plane 30 MPa 200 MPa X Angle = 20 degrees

50. PROJECTION PLANE: TRIANGULAR GROWTH CELLS