Relation of rock mass characterization and damage
Ván P.13 and Vásárhelyi B.23
1RMKI, Dep. Theor. Phys., Budapest, Hungary, 2Vásárhelyi and Partner Geotechnical Engng. Ltd. Budapest, Hungary,
3Montavid Thermodynamic Research Group, Budapest, Hungary
– Deformation moduli and strength of the rock mass– Thermo-damage mechanics– Summary, conclusions and outlook
3 – Nicholson and Bieniawski (1990)
8 – Sonmez et. al. (2004)
– Carvalcho (2004)
– Zhang and Einstein (2004)
0 20 40 60 80 1000
20
40
60
80
100
120
Deformation modulus of the rock mass (Erm) – formulas containing the elastic modulus of intact rock (Ei)
Hoek & Diederichs, 2006
22.82RMR
2
i
rm e 0.9 RMR0.0028 EE
01 EE 1.91-0.0186RQD
i
rm
0.4a
i
rm s E
E
9100-RMR
e s
320
15RMR
e-e610.5 a
0.25
i
rm s E
E
• Yudhbir et al. (1983) )
• Ramamurthy et al. (1985)
• Kalamaras & Bieniawski (1993)
• Hoek et al. (1995)
• Sheorey (1997)
Unconfined compressive strength of the rock mass (cm) – formulas containing the strength of intact rock (c)
Zhang, 2005
cm/c = exp(7.65((RMR-100)/100)
cm/c = exp((RMR-100)/18.5)
cm/c = exp((RMR-100)/25)
cm/c = exp((RMR-100)/18)
cm/c= exp((RMR-100)/20)
Using a damage variable - D
Intact rock: D=0Fractured rock at the edge of failure: D= Dcr
rock mass quality measure = damage measure
crDDRMR 1100
1001 RMRDD cr
Intact rock Fractured rock
RMR scales 100 0
Damage scales 0 Dcr
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Damage factor (D)
E rm
/Ei
Zhang & Einsten (2004)
Nicholson & Bieniawski (1990) Sonmez et al. (2004)
Carvalho (2004)
Equation: A
Nicholson & Bieniawski (1990) 4.358
Zhang & Einstein (2004) 4.440
Sonmez et al. (2004) 2.624
Carvalho (2004) 2.778
Deformation modulus: exponential form
AD-
i
rm E
E e
6100
RMR
es
3.9364.167
Hoek-Brown constantfor disturbed rock mass:
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
damage (D)
cm/
c
Kalamaras & Bieniawski, (1993)
Yudhbi et al. (1983)
Ramamurthy et al. (1985)
Hoek et al. (1995)Sheorey (1997)
Uniaxial strength: exponential form
D-
i
rm Be Equation B
Yudhbi et al. (1983) 7.650
Ramamurthy et al. (1985) 5.333
Kalamaras & Bieniawski (1993) 4.167
Hoek et al. (1995) 5.556
Sheorey (1997) 5.000
Thermo-damage mechanics I.
Thermostatic potential: Helmholtz free energy
linear elasticity:
damaged rock:
Energetic damage:
Consequence:
EFEF
2
)(2
)(?),( DEFDFD
),( DFDF
The energy content of more deformed rock mass is more reduced by damage.
0
2
2),( FEeDF i
D
iEE )0(
Thermo-damage mechanics II.
Deformation modulus
…exponential, like the empirical data.
Strength – thermodynamic stability (Ván and Vásárhelyi, 2001)(convex free energy, positive definite second derivative)
Di
DRM eEFE
1
0
2
2),( FEeDF i
D
0)det(2
2
0
22
2
FFEE
EEeF
ii
iiD
2)det(0
2
0222
iD EFeF
00 22 FeEF Dicmcm
cic E
D
c
cm e
…exponential, like the empirical data.
Equation: a
Nicholson & Bieniawski (1990) 22.95
Zhang & Einstein (2004) 22.52
Sonmez et al. (2004) 38.11 (25.41)
Carvalho (2004) 36.00 (24.00)
Equation: b
Yudhbi et al. (1983) 13.07
Ramamurthy et al. (1985) 18.75
Kalamaras & Bieniawski (1993) 24.00
Hoek et al. (1995) 18.00
Sheorey (1997) 20.00
Summary
a100-RMR
i
rm E
E e
b100-RMR
c
cm e
? a ba = 22.52…(25.41)…38.11
average: 30 (or 23.7, with disturbed)
b = 13.07…24.00
average: 18.76 (or 20.19, without Yudhbi)
Damage model:
Empirical relations:
average:
MRb c
i
cm
rm
i
rm
c
cm E E EE a
MR=Modification Ratio
100)100(2
100)100)((
c
i
cm
rm E E
RMRRMRAB
eMReAD-
i
rm E
E e
BD-
i
rm e
0,...,1 AB
Conclusions
Outlook
• Linear elasticity:
damage: scalar, vector, tensor, …
• Nonideal damage and thermodynamics:damage evolutiondamage gradients
εεεε :2
)( 2 trF
Thank you for your attention!
Real world –Second Law is valid
Impossible world –Second Law is violated
Ideal world –there is no dissipation
Thermodinamics - Mechanics
Damage model:
Empirical relations:
average:
MRb c
i
cm
rm
i
rm
c
cm E E EE a
MR=Modification Ratio
100)100(2
100)100)((
c
i
cm
rm E E
RMRRMRAB
eMReAD-
i
rm E
E e
BD-
i
rm e
0,...,1 AB
Conclusions
q
i
rm
c
cm
EE
A
BADBD ee
ABq
Zhang, 2009