On a Class of Contact Problems in Rock Mechanics

Exadaktylos George, Technical University of Crete

http://minelab.mred.tuc.gr

Acknowledgements

We would like to thank the financial support from the EU 5th Framework Project “Integrated tool for in situ characterization of effectiveness and durability of conservation techniques in historical structures” (DIAS) with Contract Number: DIAS-EVK4-CT-2002-00080 http://minelab.mred.tuc.gr

A class of plane contact problems in Rock Mechanics includes:1. The interaction of a thin liner with a circular

opening in an elastic, isotropic, homogeneous rock (design of tunnel support)

2. The indentation of rocks (design of cutting tools (contact stresses), characterization of elasticity of rocks)

3. The cutting of rocks (design of cutting tools, characterization of strength of rocks)

Mixed Boundary Value Problems

Why considering these 3 problems simultaneously ? Because in underground construction the rock

excavation precedes every other work (design of proper cutting tools, i.e. picks, discs, drag bits etc. & operational parameters of machine).

Rock cutting gives precious information for the strength of the intact rock whereas indentation for its elasticity at mesoscale (~ 1mm -100 mm).

Support must be manufactured by considering rock deformability and strength.

Problem #1: Elastic interaction of a thin shell in perfect contact with a circular opening

Add BC’s

0

d

dT

R

1

0R

T

tt

nn

, Equilibrium

eqn’s for the shell (Kirchhoff-Love)

r1121

1

r

r21

1

11E

1

hO1

hET

)(

)( ),(

Constitutive relations

hRr0i tn ,

Rrr )(

Method of solution

Kolosov-Muskhelishvili complex variable method

0zzz

z

d

i2

1z

2

1

4

1

zzzz

00

2121

00

)(,)()(

,)(

,,

,)()(,)()(

2

tn0

z

d

i2

1

z

d

i2

1

dz

i

i2

1z

)(

)()(

)()()(

dr

du

1

2i

1i

1

1t t

sn

)(

1)

2)

3)

Numerical implementation

1n21j1n2ij2

t1

1tt

1n2

1t

j

n

nj j

nj

j

,...,,)/(exp

,/

/

54601 ./ 31Rh // 01 21 ,

Ivanov (1976)

System of (8n+4) eqns with (8n+4) unknowns

n=20

Comparison with classical analytic solution by Savin (1961) for ‘welded’ elastic ring Discre-pancy ?

Other References: Einstein and Schwartz (1979), Bobet (2001)

Why choosing the Complex Variable technique ?

...

)(

2

1

z

di

i2

1

z

digg

i2

1

z

d

i2

1z

21

21

Stress Intensity factors at crack tips: KI, KII

Interaction of 2 straight cracks with supported holeSystem of (16n+8) eqns

with (16n+8) unknowns

31Rh //

11 /

Problem #2: Rock indentation by DIAS portable indentor

wkFn

LFk

23 / 1

where k is the ‘penetration stiffness’ with dimensions

Elasticity of mtl from back-analysis of indentation test data (analytical solution by Lur’e, 1964)

Surface waves

Indentation

Ø=2.5 mm

Recurrent loading-unloading cycles Ø=2.5 mm

More complex σ-ε paths ?

Problem #3: Rock cutting by drilling

2nd generation of DFMS with tripod

3rd generation of DFMS [light instrument with jackleg (like jackhammer)]

Ø=5 mm

WOB-Torque measurements Normal & tangential forces during drilling are

linearly constrained

Each point is a test with different cutting depth δ

Numerical modeling of rock cutting by drilling (Stavropoulou, 2005)

Gioia marble

0

20

40

60

80

100

1 10 100 1000

Grain size [μm]

Pe

rce

nt p

ass

ing

[%]

intact rock

drilling dust

DIAS EU R&D Project 2003-2005 (http://minelab.mred.tuc.gr/dias)

comminution

Elasto-visco-plastic cutting model (FLAC2D)

vx

,...),,,,,,,,,( cEFfF ns

Comparison of numerical simulations with experimental drilling data (Stavropoulou, 2005)

Remark #1: Initiation of strain localization

Remark #2: c,φ for numerical modeling estimated from triaxial compression tests in lab (ψ=0o)

An approach to design structures in brittle rock masses:

2. Elasticity & strength of intact rock (L = .001 – 0.1 m)

- Fast drilling/indentation/ acoustic measurements

3. Rock transected by cracks (L=.1 – 100 m)

- LEFM (fast algorithms) for stress analysis, KI,KII,KIII estimations & check of micromech – damage models !

- Stiffness and strength of joint walls (another contact problem)

4. Support

- DIAS measurements

- Modeling

Hoek, Kaiser & Bawden (1995)

1. Excavation

END

System of complex integro-differential eqns Note from the

1st eqn that for the limit of zero relative rigidity or thickness of the shell the radial and tangential stresses vanish

t

td

dt2i

t

dt

i

1

t

d

i

1d

t

i

i

1

td

dt

t

d

i

12

tn2

tn

,Ret

,)(

)()(

)()()()(Re

i

)(Re

,)()()(

,)()(

tt

0tf11

1tftf

1d

d1

R

h

E

E

0tf1R

h

E

Etf1

R

h

E

E1

i

3

21

211

221

112

1

Boundary element method

Limit for relative rigidity of the thin shell tending to infinity

0d

d

d

d

0tf11

1tftf

1d

d1

R

h

E

E

1

0tf1R

h

E

Etf1

R

h

E

E1

r

EE

3

21

211

rEE

221

112

1

1

1

/

/

)()()(

)()(

Frictional contact of a gently dipping rigid slider with an elastic half-plane

Boundary conditions

Ltcttfu

ttP

TPT

y

yxyy

xy

)()(

,tan)()(,

,,tan

)tan()(

)()(

x

utf

ctttf

y

Lt0tt yxy )()(

1)

2)

0xP0xP

x

u

x

u

x

x

y

x

y

x

)(lim)(lim

,limlim

3)

tan,)( 00

b

a

0 PTPdttP

Analytical solution (Muskhelishvili, 1963)

Normal force varies proportionally with indentation depth

21a0

1

1a

1

a214P0 /,tan)tan(,

)(

a21

a21

0

tt

t4a81P

1

atP

)(

)(

)cos()(

Graphical illustration of the solution

Remark #1

Remark #2

o5 o5tanφ=0,

tanφ=0.5

tanφ=1

tanφ=0,

tanφ=0.5

tanφ=1

)( 1

a214P0

Remark #3