Geometrical Aspects of 3D Fracture Growth Simulation

(Simulating Fracture, Damage and Strain Localisation: CSIRO, March 2010)

John NapierCSIR, 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)

~0.1 mm

Propagation direction

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:

indices) repeatedon summation (Implicit

tensorinfluence fieldnt displaceme ),(

point at nt vector displaceme )(

patches)smooth (piecewise surfaceCrack

point at vector normalunit )(

point at vector DD )()()(

)()(),()(

QPG

PPu

QQn

QQuQuQD

dSQnQDQPGPu

ijk

k

j

iii

Qjiijkk

DDM – stress tensor integral equation:

indices) repeatedon summation (Implicit

tensorinfluence field stress ),(

point at components tensor stress )(

)()(),()(

QP

PP

dSQnQDQPP

ijkl

kl

Qjiijklkl

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 element

(b) 9 point quadrilateral element

Shape function weights:

)(),(1

ikikik

n

ki cybxayxW

NyxSyxWyx Nii /)),(1(),(),(

N

kkN yxWyxS

1

),(),(

Overall element DD variation:

N

iiiD yxDE

1),(

N

ii yx

1

1),(

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

qppq zlkI cossin);,(

dz

dkR

2/122

1)(

0 )(

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

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

Edge connected search:

X

Y

Z

Cell boundaries in X-Y projection plane

Existing element

Existing edge

New element test orientations

Edge search distance factor, Rfac:

Existing element

New element orientation

Search radius = Rfac X element effective dimension

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

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

Y

X

Z

Crack front

Inclined far-field tension in Y-Z plane

Starter crack and projection plane growth cell tessellations

-15

-10

-5

0

5

10

15

-10 -5 0 5 10 15 20

Growth cells

Starter crack

X

Y

200 incremental growth steps (no link elements)

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10

X

Y

Z

200 incremental growth steps (with link elements)

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10

X

Y

Z

Incremental growth - Section plot at X = 4

-3

-2

-1

0

1

2

3

4

-8 -6 -4 -2 0 2 4 Y

Z

No link elements

With link elements

20 parallel growth steps (no link elements)

-10

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

X

Y

Z

20 parallel growth steps (with link elements)

-10

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

X

Y

Z

Parallel growth - Section plot at at X = 6

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

No link elements

With link elements

20 parallel growth steps - plan view (with link elements)

-12

-9

-6

-3

0

3

6

9

12

-12 -9 -6 -3 0 3 6 9 12

Rough crack front

Ad hoc crack front "smoothing" using filler elements

15 parallel growth steps - plan view (with smoothing and link elements)

-12

-9

-6

-3

0

3

6

9

12

-12 -9 -6 -3 0 3 6 9 12

15 parallel growth steps (with smoothing and link elements)

-10

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X

Y

Z

Effect of crack front smoothing - section plot at X = 6

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

With front smoothing

No front smoothing

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:

X

ZX-Y projection plane

Angle = 20 degrees

200 MPa30 MPa

PROJECTION PLANE: TRIANGULAR GROWTH CELLS

Triangular cells; Plan view (Axial growth)

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Triangular cells; Oblique view (Axial growth)

-10

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X

Y

Z

Triangular cells; Y-axis (Axial growth)

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12X

Z

PROJECTION PLANE: SQUARE GROWTH CELLS

Square cells; Plan view (Axial growth; 200 steps)

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12X

Y

Square cells; axial search (200 steps)

-10

-8

-6

-4

-2

0

2

4

6

8

10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X

Y

Z

Square cells; Y-axis view (Axial growth; 200 increments)

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12X

Z

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:

F

T

N

Crack edge

F = Crack front directionT = Edge tangentN = Crack surface normal

Search around each edge segment for maximum tensile stress σθθ

Existing element

New element orientation, F

Search radius = R0

Element edge

TENSILE GROWTH

• Search for maximum tensile stress ahead of current space surface crack edges.

• Construct incremental edge extension triangulations:

Neutral ContractionExpansion

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)

-6

-4

-2

0

2

4

6

8

10

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

Free surface

Near surface crack growth(10 growth steps; H = 4; R0 = 2; constant elements)

-6

-4

-2

0

2

4

6

8

10

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

Oblique view 1

-8

-6

-4

-2

0

2

4

6

8

10

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

Constant vs High-order elements (X-Z section)

-6

-4

-2

0

2

4

6

8

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

X

Constant High order cubic

Z

Inclined starter crack

• Inclination angle = 5 degrees relative to Y-axis.

Tilted start crack (8 growth steps; plan view)

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 X

Y

Tilted start crack (8 growth steps; side view)

-6

-4

-2

0

2

4

6

8

10

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14X

Z

Effect of starter crack tilt on growth path (X-Z section plot)

-6

-4

-2

0

2

4

6

8

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

Tilt angle = 0 degrees

Tilt angle = 5 degrees

X

Z

Distance from inclined crack circumference to free surface

0

0.5

1

1.5

2

2.5

0 60 120 180 240 300 360

Angular position (degrees)

Dis

tan

ce t

o f

ree

surf

ace

(m)

Estimated stress intensity factors around crack circumference

-10

-5

0

5

10

15

20

25

30

35

0 60 120 180 240 300 360

Angular position (degrees)

Str

ess

inte

nsi

ty (

MP

a.m

^1/

2)

KI - flat starter

KII - flat starter

KI - inclined starter

KII - inclined starter

Estimated mode III stress intensity around inclined crack circumference

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 60 120 180 240 300 360

Angular position (degrees)

Str

ess

inte

nsi

ty (

MP

A.m

^1/

2)

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

-5

0

5

10

-15 -10 -5 0 5 10 15 20X

Z

Plan view: Pressurised crack growth fronts

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18

Series1

Starters

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

X

Y

Crack overlap - Oblique angle (8 tensile growth steps)

-10

-5

0

5

10

15

-25 -20 -15 -10 -5 0 5 10 15 20

Y

X

Z

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

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

X

Y

ZFar-field tensile stress in Z-axis direction

Tensile growth from edge step (6 growth steps - plan view)

-8

-6

-4

-2

0

2

4

6

8

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X

Y

Tensile growth from edge step (6 steps)

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Growth start edge

X

Y

Z

Tensile growth from edge step

-5

-4

-3

-2

-1

0

1

2

3

4

5

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

X

Z

Y

Tensile growth from edge step

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Y

Z

Section plots in Y-Z plane

-3

-2

-1

0

1

2

3

-5 -4 -3 -2 -1 0 1 2 3 4 5

X = 1.0

X = 2.0

X = 3.0

Growth start edge

Y

Z

Starter crack with two steps: inclined stress field in Y-Z plane

-10

-8

-6

-4

-2

0

2

4

6

8

10

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

X

Y

Z

10 degrees

Growth from edge with two steps (inclined field stress; plan view)

-16

-12

-8

-4

0

4

8

12

16

-16 -12 -8 -4 0 4 8 12 16X

Y

Growth from edge with two steps (inclined field stress)

-4

-3

-2

-1

0

1

2

3

4

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Y

Z

Approximate tensile stress field direction

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

-8

-6

-4

-2

0

2

4

6

8

10

12

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

Zero stress

'Rigid' punch

Stress around starter crack vertex (R0 = 0.2)

-1000

-800

-600

-400

-200

0

200

400

600

-200 -150 -100 -50 0 50 100 150 200

Angle from crack plane (degrees)

Str

ess

(MP

a)

Constant elements

Cubic elements

Cone crack: X-axis view (Tension growth; cubic elements)

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4

Cone crack: X-Z Section plot (Tension growth; Cubic elements)

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4

Cone angle ~ 45 degrees

Rigid punch on free surface

Cone crack: (Tensile growth; Cubic elements in step 1; R0 = 0.2)

-3

-2

-1

0

1

2

3

4

5

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Axes

Punch region

Growth elements

Cone crack: (Tension growth; Cubic elements in step 1; R0 = 0.2)

-3

-2

-1

0

1

2

3

4

5

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Axes

Punch region

Growth elements

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

0.5

1

1.5

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

Initial tensile growth angle ~ 22.5 degrees

Cone crack - Oblique view (Mixed mode growth rules)

-3

-2

-1

0

1

2

-4 -3 -2 -1 0 1 2 3 4

Cone crack - Oblique view (Mixed mode growth rules)

-3

-2

-1

0

1

2

-4 -3 -2 -1 0 1 2 3 4

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

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Fault position (not mobilised)

X

Y

Fracture growth towards fault plane (early intersection)

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Fracture growth towards fault plane (later intersection)

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X

Y

Oblique view of mobilised fault elements

-8

-6

-4

-2

0

2

4

6

8

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X

Y

Z

Fault elements

Mobilised fault elements (X-axis view)

-6

-4

-2

0

2

4

6

-10 -8 -6 -4 -2 0 2 4 6 8 10

Y

Z

Penetration of fault plane before mobilisation?

Principal stress field in Y-Z plane 0.2 m ahead of fault

-10

-8

-6

-4

-2

0

2

4

6

8

10

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16

Y

Z

Principal stress fielf in Y-Z plane 0.2 m ahead of fault(with pore pressure)

-10

-8

-6

-4

-2

0

2

4

6

8

10

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16

Y

Z

Stress values 0.2 m ahead of fault (dry fault)

-6

-4

-2

0

2

4

6

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Y-coordinate (m)Str

ess

(MP

a)

Txx Tyy Tzz

Fracture intersection line

Stress values 0.2 m ahead of fault (pore pressure on fault)

-12

-10

-8

-6

-4

-2

0

2

4

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Y-coordinate (m)

Str

ess

(MP

a)

Txx Tyy Tzz

Fracture intersection line

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.