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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 ( )
                                dk 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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