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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 Acknowledgements Dr Rob Jeffrey, CSIRO Dr Andrew Bunger, CSIRO 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. 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. 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? 3D fracture surface complexity 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. River line pattern from mixed mode I/ III loading. (Hull, Fractography, 1999) Propagation direction ~0.1 mm Coal mine roof spall (From Ortlepp: “Rock fractures and rockbursts – an illustrative study”, 1997) 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. (From Scholz “The mechanics of earthquakes and faulting”) West Claims burst fracture (From Ortlepp: “Rock fractures and rockbursts – an illustrative study”, 1997) West Claims burst fracture detail (From Ortlepp: “Rock fractures and rockbursts – an illustrative study”, 1997) 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. Displacement Discontinuity Method (DDM) - displacement vector integral equation: u k ( P) Gijk ( P, Q) Di (Q)n j (Q)dS Q Di (Q) ui (Q) ui (Q) DD vector at point Q n j (Q) unit normal vector at point Q Crack surface (piecewise smooth patches) u k ( P) displaceme nt vector at point P Gijk ( P, Q) displaceme nt field influence tensor (Implicit summation on repeated indices) DDM – stress tensor integral equation: kl ( P) ijkl ( P, Q) Di (Q)n j (Q)dS Q kl ( P) stress tensor components at point P ijkl ( P, Q) stress field influence tensor (Implicit summation on repeated indices) 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. Element collocation point layouts (a) 10 point triangular (b) 9 point quadrilateral element element Shape function weights: n Wi ( x, y) (aik x bik y cik ) k 1 i ( x, y) Wi ( x, y) (1 S N ( x, y)) / N N S N ( x, y) Wk ( x, y) k 1 Overall element DD variation: N DE D ( x, y) i i i 1 N ( x, y) 1 i 1 i Full-space influence functions – radial integration over planar elements: I pq (k , l ; z ) sin cos p q R ( ) dk 1 d 0 ( z ) 2 2 1/ 2 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. 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. 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. 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 Edge connected search: Existing edge Z Existing element New element test orientations Y Cell boundaries in X-Y projection plane X Edge search distance factor, Rfac: New element orientation Existing element Search radius = Rfac X element effective dimension Search along growth cell axis: Z Selected element centroid and orientation Existing element vertices Y Growth cell centroid X Search line perpendicular to projection plane 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. 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. 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. 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. EXAMPLE 1: Mixed mode loading crack front evolution – simulation of “river line” evolution. Mixed mode loading Z Y X Crack front Inclined far-field tension in Y-Z plane Starter crack and projection plane growth cell tessellations 15 Y Growth cells Starter crack 10 5 0 -10 -5 0 5 10 15 20 X -5 -10 -15 200 incremental growth steps (no link elements) 10 Z 8 6 Y 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 X -6 -8 200 incremental growth steps (with link elements) 10 Z 8 6 Y 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 X -6 -8 Incremental growth - Section plot at X = 4 4 Z 3 No link elements With link elements 2 1 0 -8 -6 -4 -2 0 2 4 Y -1 -2 -3 20 parallel growth steps (no link elements) 10 Z 8 6 Y 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 -4 X -6 -8 -10 20 parallel growth steps (with link elements) 10 Z 8 6 Y 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 -4 X -6 -8 -10 Parallel growth - Section plot at at X = 6 6 5 No link elements 4 With link elements 3 2 1 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -2 -3 -4 -5 -6 20 parallel growth steps - plan view (with link elements) 12 9 6 Rough crack front 3 0 -12 -9 -6 -3 0 3 6 9 12 -3 Ad hoc crack front "smoothing" using -6 filler elements -9 -12 15 parallel growth steps - plan view (with smoothing and link elements) 12 9 6 3 0 -12 -9 -6 -3 0 3 6 9 12 -3 -6 -9 -12 15 parallel growth steps (with smoothing and link elements) 10 Z 8 6 Y 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 X -6 -8 -10 Effect of crack front smoothing - section plot at X = 6 6 5 With front smoothing 4 No front smoothing 3 2 1 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -2 -3 -4 -5 -6 EXAMPLE 2: SHEAR FRACTURE SIMULATION 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? 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. Shear loading across projection plane: Z X-Y projection plane 30 MPa 200 MPa X Angle = 20 degrees PROJECTION PLANE: TRIANGULAR GROWTH CELLS Triangular cells; Plan view (Axial growth) 8 6 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 -6 -8 Triangular cells; Oblique view (Axial growth) 10 Z 8 6 4 Y 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 X -6 -8 -10 Triangular cells; Y-axis (Axial growth) 8 Z 6 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 X 12 -2 -4 -6 -8 PROJECTION PLANE: SQUARE GROWTH CELLS Square cells; Plan view (Axial growth; 200 steps) 8 Y 6 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 X -2 -4 -6 -8 Square cells; axial search (200 steps) 10 8 Z 6 4 Y 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 X -6 -8 -10 Square cells; Y-axis view (Axial growth; 200 increments) 8 Z 6 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 X 12 -2 -4 -6 -8 Explicit crack front growth construction. 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. Local crack front coordinate system: T N F F = Crack front direction T = Edge tangent N = Crack surface normal Crack edge Search around each edge segment for maximum tensile stress σθθ New element orientation, F Existing Search radius = R0 element Element edge TENSILE GROWTH • Search for maximum tensile stress ahead of current space surface crack edges. • Construct incremental edge extension triangulations: Neutral Contraction Expansion EXAMPLE 1: CRACK GROWTH NEAR A FREE SURFACE • Simple maximum tension growth rule. • Constant elements. • Half-space influence functions. • No horizontal confinement. Near surface crack growth (8 growth steps; H = 4; R0 = 2; constant elements) 10 8 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 Free surface -4 -6 Near surface crack growth (10 growth steps; H = 4; R0 = 2; constant elements) 10 8 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 -4 -6 Oblique view 1 10 8 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 -4 -6 -8 Constant vs High-order elements (X-Z section) 8 Z Constant High order cubic 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 X -2 -4 -6 Inclined starter crack • Inclination angle = 5 degrees relative to Y- axis. Tilted start crack (8 growth steps; plan view) 12 Y 10 8 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 X -2 -4 -6 -8 -10 -12 Tilted start crack (8 growth steps; side view) 10 Z 8 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 X -2 -4 -6 Effect of starter crack tilt on growth path (X-Z section plot) 8 Tilt angle = 0 degrees Z Tilt angle = 5 degrees 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 X -2 -4 -6 Distance from inclined crack circumference to free surface 2.5 Distance to free surface (m) 2 1.5 1 0.5 0 0 60 120 180 240 300 360 Angular position (degrees) Estimated stress intensity factors around crack circumference 35 KI - flat starter KII - flat starter 30 KI - inclined starter KII - inclined starter Stress intensity (MPa.m^1/2) 25 20 15 10 5 0 0 60 120 180 240 300 360 -5 Angular position (degrees) -10 Estimated mode III stress intensity around inclined crack circumference 0.8 0.6 Stress intensity (MPA.m^1/2) 0.4 0.2 0 0 60 120 180 240 300 360 -0.2 Angular position (degrees) -0.4 -0.6 -0.8 EXAMPLE 2: OVERLAPPED CRACK GROWTH INTERACTION • Two cracks with internal pressure. • Square element initial crack shape. • Tensile growth rule. • Constant elements. Y-axis view - 3D crack overlap (Two cracks with internal pressure; 8 tensile growth steps) 10 Z 5 0 -15 -10 -5 0 5 10 15 20 X -5 -10 Plan view: Pressurised crack growth fronts 12 Series1 Y Starters 10 Step 1 Step 2 8 Step 3 Step 4 6 Step 5 Step 6 4 Step 7 Step 8 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 X -2 -4 -6 -8 -10 -12 Crack overlap - Oblique angle (8 tensile growth steps) 15 10 Z 5 Y 0 -25 -20 -15 -10 -5 0 5 10 15 20 X -5 -10 EXAMPLE 3 • Starter crack with step jog. • Possible mechanism for surface “river line” structure/ fracture “lance” development. Tensile growth - start crack with edge step 10 8 Z Far-field tensile stress in Z-axis direction 6 Y 4 2 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 X -6 -8 -10 Tensile growth from edge step (6 growth steps - plan view) 8 Y 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 X -2 -4 -6 -8 Tensile growth from edge step (6 steps) 8 Z Growth start edge 6 Y 4 2 0 -8 -6 -4 -2 0 2 4 6 8 -2 -4 X -6 -8 Tensile growth from edge step 5 Z 4 3 2 1 0 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -1 Y X -2 -3 -4 -5 Tensile growth from edge step 2 Z 1.5 1 0.5 0 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Y -0.5 -1 -1.5 -2 Section plots in Y-Z plane 3 X = 1.0 Z X = 2.0 X = 3.0 2 Growth start edge 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Y -1 -2 -3 Starter crack with two steps: inclined stress field in Y-Z plane 10 Z 8 6 Y 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 -4 X -6 -8 10 degrees -10 Growth from edge with two steps (inclined field stress; plan view) 16 Y 12 8 4 0 X -16 -12 -8 -4 0 4 8 12 16 -4 -8 -12 -16 Growth from edge with two steps (inclined field stress) 4 Z 3 2 1 0 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Y -1 -2 Approximate tensile stress field direction -3 -4 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. Annular region for cone crack growth 12 10 Zero stress 'Rigid' punch 8 6 4 2 0 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -2 -4 -6 -8 -10 -12 Stress around starter crack vertex (R0 = 0.2) 600 Constant elements 400 Cubic elements 200 0 -200 -150 -100 -50 0 50 100 150 200 Stress (MPa) -200 -400 -600 -800 -1000 Angle from crack plane (degrees) Cone crack: X-axis view (Tension growth; cubic elements) 3 2 1 0 -4 -3 -2 -1 0 1 2 3 4 -1 Cone crack: X-Z Section plot (Tension growth; Cubic elements) 3 2 Cone angle ~ 45 degrees 1 0 -4 -3 -2 -1 0 1 2 3 4 Rigid punch on free surface -1 Cone crack: (Tensile growth; Cubic elements in step 1; R0 = 0.2) 5 Axes Punch region Growth elements 4 3 2 1 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 Cone crack: (Tension growth; Cubic elements in step 1; R0 = 0.2) 5 Axes Punch region 4 Growth elements 3 2 1 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 Mixed mode crack initiation • Initial growth direction with maximum ESS = shear stress – shear resistance • Subsequent growth steps at maximum tensile stress Mixed shear and tensile growth modes (CONE03; X-axis view) 1.5 1 0.5 0 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 -0.5 -1 Initial tensile growth angle ~ 22.5 degrees -1.5 Cone crack - Oblique view (Mixed mode growth rules) 2 1 0 -4 -3 -2 -1 0 1 2 3 4 -1 -2 -3 Cone crack - Oblique view (Mixed mode growth rules) 2 1 0 -4 -3 -2 -1 0 1 2 3 4 -1 -2 -3 EXAMPLE 5: FRACTURE-FAULT PLANE INTESECTION • Circular starter crack • Fault plane orthogonal to fracture plane • No pore pressure on fault Fracture growth towards fault plane (plan view) 8 Y 6 4 Fault position (not mobilised) 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 X -2 -4 -6 -8 Fracture growth towards fault plane (early intersection) 8 6 4 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 -6 -8 Fracture growth towards fault plane (later intersection) Y 8 6 4 2 X 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 -6 -8 Oblique view of mobilised fault elements 8 Z Y 6 Fault elements 4 2 X 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -2 -4 -6 -8 Mobilised fault elements (X-axis view) 6 Z 4 2 Y 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 Penetration of fault plane before mobilisation? -6 Principal stress field in Y-Z plane 0.2 m ahead of fault 10 Z 8 6 4 2 Y 0 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 -2 -4 -6 -8 -10 Principal stress fielf in Y-Z plane 0.2 m ahead of fault (with pore pressure) 10 Z 8 6 4 2 Y 0 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 -2 -4 -6 -8 -10 Stress values 0.2 m ahead of fault (dry fault) 6 Fracture intersection line 4 Txx Tyy Tzz 2 Stress (MPa) 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Y-coordinate (m) -2 -4 -6 Stress values 0.2 m ahead of fault (pore pressure on fault) 4 Fracture intersection line 2 0 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Y-coordinate (m) -2 Stress (MPa) -4 -6 -8 Txx Tyy Tzz -10 -12 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. 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). 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. Conclusions (continued) • Explicit shear fracture growth rules need further investigation. (In particular the effect of slip-weakening on effective shear surface propagation directions). 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.

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