Application of Mechanical Crack Model to Numerical Study of Rock Mass Behavior

Abstract

Rock is a very complex and heterogeneous material, containing structural flaws due to geologic generation process. Because of those structural flaws, deformation and failure of rock when subjected to differential compressive stresses is non-linear. To simulate the non-linear behavior of rock, mechanical crack models, that is, sliding and shear crack models have been used in several studies. In those studies, non-linear stress- strain curves and various behaviors of rock including the changes of effective elastic moduli (E₁, E₂, ν₁, ν₂, G₂) due to crack growth were simulated (Kemeny, 1993; Jeon, 1996, 1998). Most of the studies have mainly focused on the verification of the mechanical crack model with relatively less attempt to apply it to practical purposes such as numerical analysis for underground and/or slope design. In this study, the validity of mechanical crack model was checked out by simulating the non-linear behavior of rock and consequently it was applied to a practical numerical analysis, finite element analysis commonly used.

Keywords : Crack model, Non-linear behavior, Critical crack length, FEM, Slidng crack, Shear crack

.

Application of Mechanical Crack Model to Numerical Study of Rock Mass Behavior

*¹( )

*² ,

*¹ Park, Do-hyun

*² Jeon, Seok-won

. , (Sliding crack model) (Shear crack model) .

- (E1, E2, ν1, ν2, G2)

(Kemeny, 1993; Jeon, 1996, 1998).

.

- ,

.

: , , , , ,

1.

, ,

.

. (strain

hardening), (strain softening),

(dilatancy) ,

, (interaction), (coalescence) .

.(Cotterell & Rice, 1980; Costin, 1985; Sammis & Ashby, 1986; Nemat- Nasser & Horii, 1982; Wang & Kemeny, 1993; Jeon, 1998).

(sliding crack model) (shear crack model)

.

(Kemeny & Cook, 1991), (Myer et al., 1992), (Wang & Kemeny, 1993)

.

-

.

(fracture network) ,

.

.

-

.

,

. ,

-

.

,

.

, ,

.

2.

Simmons Richter (1976 )

(microcrack) 3 2 1

(opening) ,

(aspect ratio) 10^-2 10^-3

10^-5 . 100 μm

.

1987 Kemeny Cook .

(sliding crack model) (shear crack model) .

2.1 (Extensile crack model)

. .

(interaction)

.

, K1 .

KI= - σ² πℓ (1)

2l^o＝ , θ＝

μ＝ , τ* ＝

τ*

.

τ* = σ¹(sin θcos θ- μcos²θ)

- σ²(sin θcos θ+μsin²θ) (2)

, K^I , K^IC 2ℓ

, σ^c .

KI= KIC (1)

σ^c=

+ (3)

2ℓ

, ε^crack Castigliano's theorem .

ε^crack= ×

ln - σ² - 1 (4)

, KI .

2ℓoτ*cos θ

πℓ √

(KIC+σ²√πℓ)√πℓ 2ℓocos θ sin θcos θ- μcos²θ σ2(sin θcos θ+ μcos²θ)

sin θcos θ- μcos²θ

8ℓo2

cosθ(sinθcos θ+ μcos²θ) νE′

2π*cosθ π

ℓ ℓo

ℓ ℓo

[ ( ) ]

√

KI= -σ² 2btan (5)

KI= KIC (5) 1 2ℓ

, σ^c Castigliano's theorem ε^crack

ε^ν .

σ^c=

+ (6)

ε^crack= [τ*cos θ×

ℓn - σ² ℓn

(7)

ε^ν= σ²ℓn

- τ* cos θℓn (8)

2.2 (Shear crack model)

.

.

KI = KIII= 0

KII= τ* πℓ (9)

G= G1 (9)

ε^crack

ε^v .

σ^c= (10)

ε^crack= (11)

ε^v = 0 (12)

2.3 1

. N M

. i

l0i θi . ε^crack(θⁱ,

θ^oi,ℓi) ε^v(θⁱ, θ^oi,ℓi) .

2ℓoτ*cosθ bsinπℓ

b

πℓ

( )

KIC+σ² 2btan

2ℓocosθ sinθcos θ- μcos²θ

πℓ

(

[ )

( )

σ2(sinθcos θ+ μcos²θ) sinθcos θ- μcos²θ 16ℓo2

cosθ(sinθcos θ-μcos²θ) νE′π

tan πℓ

( )

( )

tanπ 1+ ℓ 4 b

( )

tanπ 1+ ℓo

4 b

( )

b

ℓo

]

16ℓo2

πE′V b ℓo

( )

πℓ

( )

( )

ℓo

tanπ 1+ ℓ 4 b

( )

tanπ 1+ ℓo

4 b

( ) ]

GcE

(1-ν²)πℓ+ σ²(sinθcos θ+ μcos²θ) sinθcos θ- μcos²θ 2ℓ^o2πτ*(sinθcos θ+ μcos²θ)

νE′

ℓ ℓ^o

2

-1

] [( )

√

:

ε = ε + ε

=

(13)

ε = ε + ε

=

(14)

:

ε =

(15)

ε = 0 (16)

2.4

,

1 (transverse

isotropy) .

. 2

5 E₁, E₂, ν₁, ν₂, G₂ , -

.

=

(17)

(17) 3 -

2 -

.

= [D]ε (18)

= EC

, EC =

- [D]

. (stress

transformation) [D]

total

axial elastic

axial crackl

axial

N

i=1

(1-ν²) E

ν

σ¹- σ1-ν ²

]

[

N

i=1

(1-ν²) E

ν

σ1- σ1-ν 2

]

[

total v

total axial

total v

elastic

volume crackl

volume

N

i=1

(1+ν)(1-2ν)(σ¹+σ²)

E + ε∑ ν(θⁱ, θ^oi,ℓi)

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 E1

-ν² E1

-ν¹ E1

-ν2

E1

1 E2

-ν2

E1

-ν1

E1

-ν2

E1

1 E1

1 G2

1 G2

2(1+ν1) E1

σ11

σ22

σ33

τ12

τ23

τ13

ε11

ε22

ε33

γ12

γ23

γ13

ν² 0

ν² (1-ν¹) 0

0 0

E1-νE2

(1+ν1)E2 22

G² EC σ11

σ22

τ12

ε11

ε22

γ12

E2

1-ν¹- νE2

E1 22

.

[D*] =[R] [D] [R]^T (19)

[R] (rotation matrix) .

cosβ sinβ 0

[R] = -sinβ cosβ 0 (20)

0 0 1

Castigliano's theorem 5

.

E¹= (21)

E²= (22)

G²= (23)

ν²= (24)

ν¹= (25)

3. -

-

2 .

,

. ,

(random number generation) .

2

.

. .

.

.

n

i=1

16ℓoi2(sc-μc²)²c² πV

1+ ℓnℓi

ℓoi

∑

n

i=1

16ℓoi2(sc-μc²)²c² πV

1+ ℓnℓi

ℓoii

∑

n

i=1

8ℓoi2(sc-μc²)c

ν+ V ℓi

ℓoii

∑

E

n

i=1

16ℓoii2(sc-μc²)²c² πV

1+ ℓnℓi

ℓoii

∑

ν

n

i=1

2πℓoi2

1+ V

2

{ ( )

}

∑

E

n

i=1

ℓoi2

V(1+ν) 1+

2

-1)

-1

{ ( )

}

∑

G

( )

1.

,

.

. - .

5 , E₁, E₂, ν₁, ν₂, G₂ .

1 ,

, (uniform distribution)

.

3 -

- .

0 MPa, 10 MPa

. 35 MPa

..

4 35 MPa

(active crack) , . 400 MPa

. 700 MPa

700 MPa

5 .

. 20 MPa

20

MPa ,

20 MPa

. 6

Initial crack length 67 - 150 μm

Cohesion 3.3 - 7.5 MPa

Crack orientation Random

Coefficient of friction 0.3 Shear fracture energy 800 Joules/m² Mode I fracture toughness 0.4 MPa·m^1/2

E, ν 40 GPa, 0.2

Crack interval (lo/b) 0.15 Number of sliding cracks 1,250

Number of shear cracks 1,250 Volume (unit thickness) 0.000005 m² 1. -

2. -

.

20 MPa 20 MPa

0 . 20 MPa

100 MPa 0

. 5

.

4. (Critical crack length)

.

(negative exponential distribution) 300 μm 300 μm

300 μm 300μm .

.

4.

3. -

6.

5.

.

(log-normal distribution) (negative exponential distribution)

. 2 .

Hadley Westerly granite

655 (negative

exponential distribution)

7 .

0 MPa 8 (a)

-

, (b) 40 μm, 45 μm, 50 μm,

55 μm - . (a)

40 μm -

.

(b) 40 μm -

40 μm

Initial crack length 0 - 250 μm

Cohesion 3.3 - 7.5 MPa

Crack orientation Random

Coefficient of friction 0.3 Shear fracture energy 800 Joules/m² Mode I fracture toughness 0.4 MPa·m^1/2

E, ν 40 GPa, 0.2

Crack interval (lo/b) 0.15 Number of sliding cracks 2000 Numer of shear cracks 2000 Volume (unit thickness) 0.000001 m² 2.

7.

(b)

8. (Westerly granite)

(a)

.

40 μm -

. 2

40 μm 40 μm

, 40 μm .

5.

, . .

. 9 .

5.1 ( / )

Fig. 5.4 . 5 cm×10

cm 1/4 200 .

. 0.002 mm

.

11 (a)

E1 'X'

. (b)

E1 .

.

45°

. . 12

. 11

,

5.2 ( )

. 13

. ( x, y

40 MPa, K = 1) (x

30 MPa, y 10 MPa K＝ 3

) .

. 14

.

10. (F=0)

(a)

(b) 11.

, E1

(a) (b)

12.

Initial crack length 0.05 - 0.1 m

Cohesion 3.3 - 7.5 MPa

Crack orientation Random

Coefficient of friction 0.3 Mode I fracture toughness 0.5 MPa·m^1/2

E, ν 20 GPa, 0.25

Crack density 500/m²

3.

15

. K＝

3 30 MPa

x

.

13.

(a)

(b)

14. (a) (b)

(a) (b)

17.

(a) (b)

15. (a) (b)

(a) (b)

16.

. 16

.

.

. 17 .

.

6.

,

. -

.

. .

1) - : -

, .

.

2) :

Hadley Westerly granite

49 μm .

3) :

,

.

.

, .

, .

.

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