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