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) ~0.1 mm Propagation direction

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. Element shape functions 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.

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

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

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

28. Search along growth cell axis: X Y Z Growth cell centroid Existing element vertices Selected element centroid and orientation 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 Y X Z 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: X Z X-Y projection plane Angle = 20 degrees 200 MPa 30 MPa

50. PROJECTION PLANE: TRIANGULAR GROWTH CELLS

54. PROJECTION PLANE: SQUARE GROWTH CELLS

58. Explicit crack front growth construction.

59. Curvilinear fracture surface construction Represent crack surface using flat triangular elements (constant or cubic polynomial). Search around each crack front boundary segment to determine growth direction according to a specified criterion. Advance the crack front using local measures of advance “velocity”. Construct new edge positions and add new crack surface elements in 3D. Re-solve crack surface discontinuity distributions. Return to step 2.

60. Local crack front coordinate system: F T N Crack edge F = Crack front direction T = Edge tangent N = Crack surface normal

61. Search around each edge segment for maximum tensile stress σ θθ Existing element New element orientation, F Search radius = R0 Element edge

62. TENSILE GROWTH Search for maximum tensile stress ahead of current space surface crack edges. Construct incremental edge extension triangulations: NeutralContraction Expansion

63. EXAMPLE 1: CRACK GROWTH NEAR A FREE SURFACE Simple maximum tension growth rule. Constant elements. Half-space influence functions. No horizontal confinement.

68. Inclined starter crack Inclination angle = 5 degrees relative to Y- axis.

75. EXAMPLE 2: OVERLAPPED CRACK GROWTH INTERACTION Two cracks with internal pressure. Square element initial crack shape. Tensile growth rule. Constant elements.

79. EXAMPLE 3 Starter crack with step jog. Possible mechanism for surface “river line” structure/ fracture “lance” development.

89. EXAMPLE 4: CONE CRACK SIMULATION Central rigid “punch” load in annular region. Effect of fracture growth mode on cone angle: (1) Tensile mode only. (2) Shear mode followed by tensile growth.

96. Mixed mode crack initiation Initial growth direction with maximum ESS = shear stress – shear resistance Subsequent growth steps at maximum tensile stress

100. EXAMPLE 5: FRACTURE-FAULT PLANE INTESECTION Circular starter crack Fault plane orthogonal to fracture plane No pore pressure on fault

110. CONCLUSIONS A simplified 3D projection plane construction can accommodate non-planar tensile fracture surface development and crack front fragmentation. The underlying tessellation shapes may prevent fully detailed simulation of “river line” or “mirror/ mist/ hackle” features. Some form of “fractality”/ “randomness” seems to be necessary to effect a computational simulation of surface features such as river lines.

111. Conclusions (continued) Fracture edge profile tilt angles are reduced when “link” elements are introduced to maintain the fracture surface continuity. Shear fracture simulation can be accommodated using the projection plane approach but requires a number of ad hoc assumptions. Single shear fracture surface orientations appear to be more coherent when represented using non-connected growth cells (axial growth search).

112. Conclusions (continued) An explicit 3D crack edge growth construction method has been devised using the displacement discontinuity method. This appears to be useful for analysing relatively simple tensile growth structures (near-surface fractures, cone cracks, multiple fracture surface growth interaction). The treatment of fracture intersections is a significant problem. The explicit front growth approach can be useful to analyse and highlight 3D interface crossing mechanisms that are not revealed in 2D.

113. Conclusions (continued) Explicit shear fracture growth rules need further investigation. (In particular the effect of slip-weakening on effective shear surface propagation directions).

114. Future developments The projection plane construction allows for the implementation of fast, hierarchical solution schemes for large-scale problems. Coupling of fluid flow into evolving 3D fractures will be explored (Anthony Peirce). Investigation of near-surface crack growth simulation will be continued (Lisa Gordeliy, Emmanuel Detournay). Simulations of 3D shear failure and elastodynamic fracture growth analysis can be investigated in deep level mining problems. It is necessary to include more general power law edge tip shapes in crack front simulations.