Numerical Analysis of Blast Induced Fracturing of Hard Rocks - - PowerPoint PPT Presentation

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Numerical Analysis of Blast Induced Fracturing of Hard Rocks - - PowerPoint PPT Presentation

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Hotel Caesar Park, Rio de Janeiro, Brazil August 2-6, 2004 Numerical Analysis of Blast Induced Fracturing of Hard Rocks Araken Lima,Celso Romanel and Deane Roehl Civil Engineering Department Motivation Rock blasting plan

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SLIDE 1

Numerical Analysis of Blast Induced Fracturing of Hard Rocks

Araken Lima,Celso Romanel and Deane Roehl

Civil Engineering Department

Hotel Caesar Park, Rio de Janeiro, Brazil August 2-6, 2004

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SLIDE 2

Motivation

  • Rock blasting plan
  • Prediction of fracture extention
  • Formation of blocks
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SLIDE 3

Rock blasting

Detonation energy = stress waves + gas pressure + others (temperature, flying rock fragments, air displacements) Explosives: TNT – high stress waves and low gas production ANFO – high gas production and low energy in stress waves shock wave propagation - µs (dynamic) gas pressurization - ms (quasi-static)

Energy partitioning during rock blasting (Whittaker, Singh e Sun [1992]).

Blasthole pressure Volume of expanding gases or time of gas expansion

p1 p2 p3 1 1 2 2 3 3 4 4

Shock wave energy for driving the detonation in the explosive Kinetic and strain shock wave energy. Energy release during crack propagation Noise, heat and other wasted energy

AB C D

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SLIDE 4

Rock blasting Blast induced P-waves

  • crushing around the borehole
  • dense radial cracks (4-8 radii)
  • dominant cracks propagate

Number of dominant cracks Ghosh e Daemen (1995) - 8 a 12 Song e Kim (1995) - 10 a 12

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SLIDE 5

Numerical Model 2D Adaptive finite element model

  • Loading: blast induced stress waves
  • Rock material: homogeneous, isotropic

linear elastic up to breakage

  • Crushed and dense fracture zones are neglected
  • Mixed mode fracture (P- and S-waves)
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SLIDE 6

Fracture model

  • Mixed mode I-II fracture criteria (static loading)
  • Dynamic fracture criteria - mode I Grady e Lipkin (1980)
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SLIDE 7

Fracture model

  • Crack propagation direction – maximum tensile tangential

stress

Mode I KI = KIC, KII = 0 →θm=0 º Mode II KI = 0, KII = KIIC →θm= -70,53º

  • Crack growth in a time interval

∆a = c . ∆t c crack growth velocity (smaller than half shear wave speed Grady & Kipp, 1979)

          +         ±         = θ 8 4 1 4 1 arctan 2

2 2 II I II I m

K K K K

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SLIDE 8

Finite element model Adaptive finite element mesh

  • Mesh generation : recursive spatial enumeration techniques

quadtree inner domain boundary triangulation

(Araújo 1999)

  • Maximum element size

Wave reflection on element interface due to change in element size Efficient frequency transmission: Celep e Bazant [1983] and Mullen e Belytschko [1982]: 1/10 smallest wave length

  • State variable mapping { }

[ ]{ } { } [ ]{ } { } [ ]{ }

q N u q N u q N u

t t P P n t t t P P n t t t P P n t

  • ∆

− ∆ − ∆ −

= = = . ) , ( . ) , ( . ) , ( η ξ η ξ η ξ

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SLIDE 9
  • Singular quarter point elements Henshell & Shaw (1975)

Barsoum (1976)

  • Finite element rosette 8 elements
  • Stress intensity factors

Displacement correlation technique (Shih, de Lorenzi and German, 1976) Modified crack closure method (Raju, 1987)

Finite element model

Crack tip

l/4 l/4 3l/4 3l/4

x y j i

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SLIDE 10

Finite element model

x y L L x y

j j-2 j-1 j+1 j+2

element rosette at crack tip

( )

2 1

4 2 1

− − −

⋅ ⋅ ⋅ ⋅       + =

j j I

v v L K π κ µ

( )

2 1

4 2 1

− − −

⋅ ⋅ ⋅ ⋅       + =

j j II

u u L K π κ µ

Stress intensity factors Displacement correlation technique (Shih, de Lorenzi and German, 1976)

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SLIDE 11

Finite element model Stress intensity factors Modified crack closure method (Raju, 1987)

Normal stress distribution at crack tip

δa

( ) ( ) [ ] ( ) ( ) [ ]

{ }

l l m m y l l m m y I

v v t v v t F v v t v v t F a G

j i

′ ′ ′ ′

− + − ⋅ + − + − ⋅ ⋅ − =

22 21 12 11

2 1 δ

( ) ( ) [ ] ( ) ( ) [ ]

{ }

l l m m x l l m m x II

u u t u u t F u u t u u t F a G

j i

′ ′ ′ ′

− + − ⋅ + − + − ⋅ ⋅ − =

22 21 12 11

2 1 δ

2 3 6

11

π − = t

20 6

12

− = π t

2 1

21 =

t

1

22 =

t

Nodal forces at singular elements

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SLIDE 12

Finite element model Contact - penalty formulation

Node to edge contact gap function

( )

u g λ p g(u) − = ⇒ ≤ 0

Linearized surface Master node Element boundary

g(u)

1 2 3

Fracture surface penetration

( )

2 2

B A C y B x A u g + + ⋅ + ⋅ =

[ ]

R d K K

c

= ⋅ +

( )

∑

=

∇ ⋅ ∇ ⋅ =

nc j j T j

g g with

1

λ

c

K

Fracture closure control

Equilibrium Normal to contact forces

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SLIDE 13

Dynamic model

  • Numerical integration of the equation of motion

Wilson θ implicit algorithm

  • Pressure pulse on hole wall- Duvall (1953)

rock constants α and β (Aimone, 1982).

[ ] { } [ ] { } [ ] { } ( ) { }

t P w K w C w M = ⋅ + ⋅ + ⋅

  • (

)

t t

e e p t p

⋅ − ⋅ −

− ⋅ =

β α

) (

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Dynamic model Viscous dampers on the boundary- radiation condition

explosion Stress waves u w

u c a

L

⋅ ⋅ ⋅ = ρ σ w c b

T

  • ⋅

⋅ ⋅ = ρ τ w c a

L

⋅ ⋅ ⋅ = ρ σ u c b

T

⋅ ⋅ ⋅ = ρ τ

u c b w c a

T L

  • ⋅

⋅ ⋅ = ⋅ ⋅ ⋅ = ρ τ ρ σ

R Ku u C u M = + +

  • time (s)

time (s) velocity (m/s)

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SLIDE 15

Dominant Cracks Detonation hole with 4, 8, 10, 12 and 16 dominant cracks

Gosh/Daemem (1995) 8-12 cracks Song/Kim (1995) 10-12 cracks

  • hole size a0=5 cm
  • material properties: ρ = 28 Mg/m3, E = 41 GPa,ν = 0,25

KID = 1,65 MPa⋅m1/2 KIID =1,03MPa⋅m1/2

fracture propagation velocity: 1210 m/s

  • t=100 µs hole pressure 715, 8 MPa
  • finite element meshes

4 dominant fractures 16 dominant fractures

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SLIDE 16

Dominant Cracks

σ1

8 cracks

σ3 σ1 σ3

4 cracks

Principal stresses

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SLIDE 17

Dominant Cracks

σ1

12 cracks

σ3 σ1 σ3

10 cracks

Principal stresses

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SLIDE 18

Dominant Cracks

σ1 σ3

16 cracks

Principal stresses

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SLIDE 19
  • 1 Blasthole with 9 dominant cracks
  • 1 Blasthole with 8 dominant cracks
  • 2 Blastholes with 8 dominant cracks

Examples

Granite: E = 60 GPa ν = 0.25 ρ = 2.80 Mg/m3 KID= 1.65 MPa.m1/2 KIID=1.03 MPa.m1/2 radius = 2.54 cm

initial crack length 1/3 hole radius

Borehole: penalty parameter 107 penetration tol 103

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SLIDE 20

Example 1: Blasthole with 9 dominant cracks Time increments ∆t = 20 µs to 50 µs Blasthole 40 cm free surface 00

0o 45o 90o 135o 157.5o 202.5o 225o 270o 315o

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SLIDE 21

Blasthole with 9 dominant cracks

t = 3 µs

4811 nodes and 2292 elements

σ1 σ1

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Blasthole with 9 dominant cracks

t = 17 µs

4811 nodes and 2292 elements

σ1 σ1

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SLIDE 23

Blasthole with 9 dominant cracks

t = 37 µs

4783 nodes and 2260 elements

σ1 σ1

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SLIDE 24

Blasthole with 9 dominant cracks

t = 182 µs

6942 nodes and 3277 elements

free surface

σ1

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SLIDE 25

Blasthole with 9 dominant cracks

t = 297 µs

8266 nodes and 3891 elements

free surface free surface

σ1

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SLIDE 26

Blasthole with 9 dominant cracks

t = 330 µs

7127 nodes and 3325 elements

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SLIDE 27

Blasthole with 9 dominant cracks

t = 405 µs

7777 nodes and 3637 elements

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SLIDE 28

Blasthole with 9 dominant cracks

t = 545 µs

8995 nodes and 4199 elements

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SLIDE 29

Blasthole with 9 dominant cracks

t = 680 µs

10047 nodes and 4687 elements

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SLIDE 30

Blasthole with 9 dominant cracks

t = 970 µs

14154 nodes and 6594 elements

cracks Plate border

Experimental resuts on glass plate Porter(1970)

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SLIDE 31

Example 2: Blasthole with 8 dominant cracks Blasthole 40 cm free surface 00

0o 45o 90o 135o 180o 225o 270o 315o

Time increments ∆t = 50 µs to 125 µs

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Blasthole with 8 dominant cracks

t = 15 µs

1 2 3 4 5 6 7 8

σ1 σ1

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Blasthole with 8 dominant cracks

t = 523 µs

(without fracture interaction control)

Stress σ1 Horizontal displacement free surface free surface

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Blasthole with 8 dominant cracks

t = 523 µs

(with fracture interaction control – penalty method)

free surface free surface Stress σ1 Horizontal displacement

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Blasthole with 8 dominant cracks

t = 523 µs

No contact control Contact control

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SLIDE 36

Example 3: 2 Blastholes with 8 dominant cracks

Crack propation speed: 1464 m/s

Blasthole 1 Blasthole 2 40 cm 40 cm free surface Simultaneous blasting

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SLIDE 37

free surface 5 13 14 4 free surface 3 15 2 16

2 Blastholes with 8 dominant cracks

t = 217,7 µs t = 289,7 µs

Crack pattern

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free surface 2 Blastholes with 8 dominant cracks

Adaptive mesh t = 538,2 µs

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free surface 2 Blastholes with 8 dominant cracks

Crack pattern t = 538,2 µs

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Example 4: 2 Blastholes with 8 dominant cracks

Crack propation speed: 1464 m/s

Blasthole 1 Blasthole 2 40 cm 40 cm free surface

∆t=123.9 µs

Sequential blasting

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SLIDE 41

free surface 2 Blastholes with 8 dominant cracks

Crack pattern t = 123,9 µs

1 2 1 2

Sequential blasting

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2 Blastholes with 8 dominant cracks

2 1 13 5 12 14

t=439,9 µs t=199,4 µs t=285,4 µs t=311,9 µs

1 13

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2 Blastholes with 8 dominant cracks

t=439,9 µs t=311,9 µs t=199,4 µs t=285,4 µs

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End Remarks

Finite element tool for the analysis of dynamic fracture of hard rock by blast-induced stress waves. Special features

  • Direction of fracture is not predetermined
  • Determination of stress intensity factors (mixed mode)
  • Crack closure and fracture interaction is incorporated
  • Silent boundaries for simulation of stress wave radiation condition
  • Efficient algorithm for adaptive mesh generation
  • Quarter point quadratic element rosetts at crack tip
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SLIDE 45

Further deveplopments

  • new algorithm for adaptive mesh generation
  • include gas pressure on fracture walls
  • automatic time step definition
  • parallel processing
  • fracture initiation/dominant cracks
  • dynamic mixed mode criteria

→ in progress

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SLIDE 46

Gas pressure

gas pressure on fracture walls

SIG1, t= 0.315755145 s mesh 4096 elements, 8695 nodes