Hoek-Brown model for description of short and long term be. rocks

Transcription

1 Hoek-Brown model for description of short and long term behavior of rocks Andrzej Truty CUT & ZACE Services

2 Hoek s web page Here we can find lot of useful information on basic H-B model and its calibration. Hoek's web page: ww.rockscience.com

3 General remarks Strain decomposition: dε = dε e + }{{} dε p short term + }{{} dε vp long term Short term behavior of rocks is described by an elasto-plastic H-B model that may include <hardening><softening><variable dilatancy> dε p Long term behavior is described by an additional Lemaitre type of the creep law dε vp Actual version (till ZSoil 2012) of H-B model approximates the true H-B (ZSoil 2012) The new version (ZSoil 2013) is based on the 2002 edition of H-B model and can be calibrated using given GSI index Notation: all stresses are effective and positive in compression

4 Experimental evidence: triaxial test r res R LINEAR PRE PEA AK POST PEA AK P RESIDUA AL 1 Strain softening and strong dilatancy are well visible for low confining pressures σ 3 For high confining pressures both phenomena dissappear under triaxial test conditions

5 Handling strain softening Strain softening requires certain kind of regularization to avoid mesh dependency Nonlocal approach will be used (same as for M-C model) It will be possible to use relatively large elements and still cancel parasitic mesh dependency

6 GSI index GSI - Geological strength index it considers shape of intact rock layers and joints Rough classification 1 Massive Brittle Rocks (70 < GSI < 90) 1 2 Jointed Strong Rocks (50 < GSI < 65) 3 Jointed Intermediate Rocks (40 < GSI < 50) 4 Very Weak Rocks GSI < 30 1 J.J.Crowder and W.F. Bowden. Review of post-peak parameters...

7 Hoek-Brown (2002) edition Yield surface: f (σ 1, σ 3 ) = σ 1 σ 3 σ ci ( m b σ 3 σ ci + s σ ci - intact rock compressive strength other parameters ( are usually ) related to the GSI index GSI 100 m b = m i exp D ) a a = (exp ( GSI/15) exp ( 20/3)) ( 6 ) GSI 100 s = exp 9 3 D m b is reduced value of m i one (for intact rock) D = factor that depends on degree of disturbance (1.0 means highly disturbed and 0.0 undisturbed)

8 How parameters a and s depend on GSI and D D=0.0 D=0.2 D=0.4 D=0.6 D=0.8 D=1.0 a GSI s GSI f t = s σ ci m b f c = σ ci s a

9 Dilatancy: ψ = ψ (σ 3, γ p ) In the basic setup we may assume that ψ=const. (bad choice) Triaxial tests indicate that for σ 3 = σ ψ dilatancy is neglible 90 o This value can be strain dependent f t 0 3 NB. σ ψ is a material parameter

10 Dilatancy: ψ = ψ (σ 3, γ p ) ψ (σ 3 = 0) = ψ o f γ ψ (γp ) 1 f r res 0 p

11 Hardening/Softening: how to include it? Following the disscussion in the recent publicity on H-B, parameters selected for the softening are (same thing for the pre-peak hardening) 1 m b (γ p ) 2 s(γ p ) 3 a(γ p ) To include hardening and softening two values of the accumulated plastic deviatoric strain are defined 1 γ r - deviatoric strain at peak on q ε triax curve 2 γ res - deviatoric strain at residual state on q ε triax curve NB. Both hardening and softening are optional (can easily be excluded); standard H-B parameters m b, s, a correspond to the peak quantities

12 Hardening/Softening: how to include it? Parameter Linear Pre-peak Post-peak m b mb o mb r mb res s s o s r s res a a o a r a res In the linear region the initial H-B surface is defined by m o b, so, a o In the pre-peak region linear interpolation is used for pairs of m o b mr b, so s r, a o a r In the post-peak region 3-rd polynomial is used to interpolate pairs of mb r mres b, sr s res, a r a res

13 User interface

14 Customizing H-B model NB. Yellow cells should be defined by the user

15 Lemaitre creep law dε vp = ( A o exp B ) exp (b γ ) R T ( g(σ) = exp σ σ ) o σ ref ( q qo g(σ) σ ref ) n ( ) ε vp m q eq σ m = m(γ ) = 2 (m o m 1 ) γ 3 3 (m o m 1 ) γ 2 + m o γ p γ if γ = p γ r γ r 1 if γ p > γ r This law does not produce any dilatancy

16 Example: creep test 1 kn/m2 X LTF (t) ɛ xx ɛ yy kn/m2 ɛ[ ] t[h] Initial stresses σ o = { 1000, 1000, 0, 1000}

17 Example: relaxation test Uy=1.0 * LTF (t) 1000 σ xx kn/m2 σ[kpa] t[h] Initial stresses σ o = { 1000, 1000, 0, 1000}

18 Example: triaxial compression (with pre-peak hardening) Triaxial compression with 3 confining stresses σ 3 = 0, 3, 6 MPa σ 3 =0 MPa σ 3 =3 MPa σ 3 =6 MPa 40 σ 1 σ 3 [kpa] ɛ[ ]

19 Example: triaxial compression (without pre-peak hardening) Triaxial compression with 3 confining stresses σ 3 = 0, 3, 6 MPa σ 3 =0 MPa σ 3 =3 MPa σ 3 =6 MPa 40 σ 1 σ 3 [kpa] ɛ[ ]

20 Stability analysis using H-B model Here we will use notion of Stress Level SF = SL f (σ 1, σ 3 ) = σ 1 σ 3 1 ( SF σ σ 3 ci m b + s σ ci f (σ 1, σ 3 ) = σ 1 σ 3 [ (SF 1 ) 1 a ) a ] a ( ) a σ 3 σci m b + s σ ci Modified form of H-B with m b and s parameters σ 3 f (σ 1, σ 3 ) = σ 1 σ 3 σ ci (mb + s σ ci 1 mb = m b SF a 1 s = s SF a ) a

21 Stability: moderate slope M C SF = 1.38 Const. =10 o H B SF = 1.58 Const. =10 o

22 Conclusions H-B model should improve predictions for rocks Model can easily be customized to basic/advanced version Model can be used for predictions of short and long term behavior of rocks Finally...it is the true H-B version...