On a Class of Contact Problems in Rock Mechanics Exadaktylos George, Technical University of Crete http :// minelab . mred . tuc . gr
Jan 15, 2016
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