Rock rheology – time dependence of dilation and stress around a tunnel

P. VánHAS, Institute of Particle and Nuclear Physics

and Montavid Research Group, Hungary

Z. SzarkaUniversity of Miskolc, Institute of Mathematics

and Montavid Research Group, Hungary

– Poynting-Thomson: not one of the many– Unloading – rod– Unloading – circular tunnel– Conclusions

EuRock’06, Liège

Rock rheology – the Poynting-Thomson modell

fFv div balance of momentum

ξξDξD :)(ˆ),( ff Helmholtz free energy

.01

ξ:ξDD:D

DF fT s

Universal – independent of microstructure

EF

Entropy production

.0)32( 00 ξ:ξD:IEIT KG

ITF 0

Ideal elastic

)0(tr TIED 0 )0(tr E

IEITF 00 32 KGE

,3

,

,2

2212

1211

ooo LK

LL

LLG

ξEξ

ξEET

Small deformation

Isotropic

,3

,

,2

2212

1211

ooo LK

LL

LLG

ξEξ

ξEET

0

,0,0,02

2211

2211

12

LLL

LLL

.3

,22

ooo

d

LK

G

EEETT

,22

,,112

11

111

1

222212

2222

GLLLL

LLL d

,

,0,0

Gd

Poynting-Thomson =Hooke+Kelvin-Voigt

.3

,22

oo K

G

EETT

Opening of 1D tunnel – released rod

F

l

Three stages:primary: equilibrium is linear elastic Opening: new initial conditionsecondary: processes after the opening

A) Released rod – primary stage

Elastic equilibrium

F

l

.3

,2

oAAo

AA

K

G

ET

p=F/A

GK

GK

GKp

z

y

x

oAAA

2

1

3

100

02

1

3

10

001

3

1

300

00

00

IED

B) Released rod – openingF

l

p=F/A

BB ET 2

Fast, “quasielastic”

.

100

010

002

16

12

G

G

pG

GA

B

TD

Finally:

2 4 6 8 10 12 14

t

0.1

0.2

0.3

0.4

0.5

-1 1 2 3t

0.5

1

1.5

2

V=(0.1,0.2,0.5,1,3,100)

V=0.5

jump: 045.1

t

ε

F

l

B) Released rod – secondary

Initial condition

Finally:

100

010

002

16

)(t

Gt

G

B eG

G

pet

DD

Bt DD )0(

.3

,22

oo K

G

EETT

F

l

.0)(

,3

,22

IT

EETT

o

oo

div

K

G

Opening of a circular tunnel

Three stages:primary: equilibrium is

linear elastic opening: new initial conditionsecondary: processes after the opening

p

R

r

pr

Cylindrical symmetry

0

rrrr

).(2

)(2)(

,)3/2()3/4(

)3/2()3/4(

r

rrr

r

rrr

G

KK

GKGK

01

12

2

r

u

r

u

rr

u

to.

)()(),( 2

1 r

tCrtCtru

Initial conditions:

.)(

)(),( 21 r

tCrtCtru

.23

3

2

,23

3

2

22

22

r

Rp

r

R

KG

p

r

Rp

r

R

KG

p

AB

ArB

Asymptotic condition:

.1),(,1),(

,2

),(,2

),(

,2

),(

22

22

2

r

Rptr

r

Rptr

r

R

G

ptr

r

R

G

ptr

r

R

G

prtru

r

AAr

A

Time dependent radial displacement:

.2

)1(2

),(

,2

)1(2

),(

22

22

r

Re

p

r

Re

G

ptr

r

Re

p

r

Re

G

ptr

tt

A

tt

Ar

Time dependent deformation fields:.

)()(),( 2

1 r

tCrtCtru

.2

)1(2

),(22

r

Re

p

r

Re

G

prtru

tt

A

Radial displacement :

2 3 4 5rR0.01

0.02

0.03

0.04

u

G=10000MPa, K=30000MPa, p=1, R=1m, η=400000 MPaht=10h, α=9hβ =11h

.2

)1(2

),(22

r

Re

p

r

Re

G

prtru

tt

A

PT – jumpPT – no jumpIdeal elastic

Time dependent stress field:

),,(1),(

),,(1),(

2

2

trr

Rptr

trr

Rptr

t

tr

.)()(

1)(

)(

)(

)(

)(),(

2

t

tt

r

eG

G

eG

Ge

G

r

Rptr

50 100 150 200

t

0.2

0.4

0.6

0.8

1

50 100 150 200t

0.2

0.4

0.6

0.8

1

mG

mG

sG

GPaGs

G

26.2

,85.2

,5.26,1,0

22

10

TT

EETT

jump

Remark to the presentation of F. Sandrone, J.P. Dudt, V. Laiouse and F. Dexcoeudres

Convergence (m)

t (day)

Conclusions

– Poynting-Thomson is not just one of them

– Separate timescales: initial jump– Analytical solution can be generalized:

elliptical, lining, etc.– Measurements of the material parameters– Numerics

mesh: gradient generalization (thermodynamics!)– Practice:

stops and starts in tunnel excavations swelling?

F

l

Thank you for your attention!