Constitutive modelling of the thermo-mechanical behaviour of soils

Transcription

1 First Training Course: THM behaviour of clays in deep excavation with application to underground radioactive waste disposal Constitutive modelling of the thermo-mechanical behaviour of soils Lyesse LALOUI & Bertrand FRANCOIS Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland

2 Constitutive modelling of the behaviour of soils The constitutive analysis in environmental geo-mechanics is based on the effects of the environmental load (temperature, suction, ) on: The (induced) STRAINS The DEFORMABILITY (change) The STRENGTH (change) MECHANICAL PROPERTIES LL / BF 2

3 Contents 1. Introduction 2. Principal experimental observations 3. Thermo-mechanical framework 4. Constitutive modelling ACMEG-T 5. Numerical simulations ATLAS experiment 6. Conclusions LL / BF 3

4 55 mm 110 mm 100 Å Idealised arrangement of clay particles on the basis of a high-resolution transmission electron microscope computer image (after Veblen et al. 1990). Micro-scale Sample of soil tested in a triaxial cell. Macro-scale (continuum) The thermo-mechanical phenomena observed at the macro-scale are a consequence of the micro-structural changes The understanding of micro-structural behaviour is required LL / BF 4

5 Unified Soil Classification System Soils that have more than 50% by weight passing the N 200 sieve (grain size of 0.06 mm) Sand Silt Sand/clay mix Clay Low plasticity High plasticity Granular materials Fine grained materials Cohesive soils Adsorbed water (solid/fluid interface) plays a major role LL / BF 5

6 Micro-structural forces Surface electrical charges: -Proportional to the surface areas of the grains (diminish with the square of the particule diameter) Dominant for cohesive soils Self weight of the grains -Proportional to the volumes of the grains (diminish with the cube of the particule diameter) Dominant in granular soils Temperature influence is mainly on surface charges LL / BF 6

7 Water in clay Adsorbed water Inter-particle water Inter-agregate water Inter-particle water Inter-lamellar water Adsorbed water Inter-agregate water Inter-lamellar water Schematic illustration of the two main types of water in saturated soils: i) free water, mainly in the inter-aggregate space and ii) adsorbed water located in the inter-particle and inter-lamellar spaces. From soil constitutive point of view: Adsorbed water is considered as a part of the solid constituent Free water is taken into account only through the THM coupling LL / BF 7

8 Diffuse double layer theory Clay particle : Surface negative charge Water : Anions + Cations Effect of the electrical potential on water Closer to the particle is the water, higher is his attraction to the particle and lower is his mobility (adsorbed water) Effect of the electrical potential on the particles interaction Clay-water system (Mitchell, 1976) Closer are the particles, greater is the repulsive forces acting between them (overlapping of the double layer) Temperature effects - Dilation of clay minerals Modification of the equilibrium between attractive and repulsive forces Failure of some interparticle ties Facilitation of particles rearrangement LL / BF 8

9 2. Principal experimental observations 1. Introduction 2. Principal experimental observations 3. Thermo-mechanical framework 4. Constitutive modelling 5. Numerical simulations 6. Conclusions LL / BF 9

10 Hypothesis Saturated fine grained soil Infinitesimal strains Temperature range between 5 C and 90 C (no phase change) Results in which changes of state cannot be achieved by homogeneous deformations are excluded Cambridge parameters in triaxial conditions σ 1 σ 3 p w T σ 3 p = (2 σ + σ ) / 3 q = σ σ LL / BF 10

11 Thermo-mechanical loading paths Plane 1 : Thermal paths at constant mechanical stresses Plane 2 : Mechanical paths at different constant temperatures LL / BF 11

12 Overconsolidated state (OCR=12) 1 st cycle: 0-A-B 2 nd cycle: B-C-B Normally consolidated state 1 st cycle: nd cycle: Heating-Cooling cycles 3 rd cycle: B-C-B 3 rd cycle: A C Temperature, T [ C] Irreversible thermal strains: Thermo-Plasticity 20 dilatation B Volumetric strain, ε [%] v 4 compaction LL / BF 12 Thermal cycle on Kaolin clay under constant isotropic compression (after Cekerevac & Laloui, 2005)

13 Thermally induced effects Influence of overconsolidation ratio on the thermal strain OCR = 12 OCR = 6 OCR = 2 OCR = 1.5 OCR = dilation compaction Thermal volum etric strain, ε T [%] v Temperature increase on Kaolin clay under constant isotropic compression (Cekerevac & Laloui, 2004) LL / BF 13

14 Mechanical behaviour Isotropic path!compressibility independent from temperature!heating applied prior to loading produces a densification of the sample at constant isotropic pressure. Void ratio [-] After Campanella & Mitchell (1968) T=25 C T=39 C T=51 C LL / BF Consolidation stress (MPa) NC saturated Illite

15 Dependency of p c from temperature p c Initial state p T Isotropic plane LL / BF 15

16 Dependency law for thermal evolution of p c p'c (T)/ p'c (T 0 ) [-] T/T 0 [-] γ The preconsolidation pressure decreases non-linearly with increased temperature: { 1 γ log [ / ]} p = p T T c c0 T 0 p c0 preconsolidation pressure at reference temperature (T 0 ) T 0 - minimal testing temperature (Laloui & Cekerevac, 2003) LL / BF 16

17 20 C 90 C LL / BF 17 Thermally induced effects on stress-strain behaviour Deviator stress, q [kpa] Volumetric strain, ε v [%] 750 NC at 90 C at 22 C Kaolin clay (Cekerevac & Laloui, 2004) Axial strain, ε 1 [%] NC Deviator stress, q [kpa] Void ratio, e [-] Critical state OCR= Initial state NC Mean effective stress, Thermo-mechanical p' behaviour [kpa]

18 Deviatoric path : critical state Friction angle at critical state Temperature has a very slight influence on the friction angle at critical state LL / BF 18

19 Deviatoric path : Yield surface determination CSL at 22 C q/p' e [-] T = 22 C The pseudo-elastic limit shrinks with increase in temperature. 0.2 T = 90 C Isotropic thermal effect p'/p' e [-] Kaolin clay (Laloui et al., 2003) LL / BF 19

20 Thermo-mechanical yield limits q Deviatoric plane at T = T 0 p c p Deviatoric plane at T = T 1 > T 0 T p ct Isotropic plane LL / BF 20

21 Thermo-hydro-mechanical coupling Undrained paths Total pore pressure [kpa] initial effective stress = kpa initial void ratio = from Plum & Esrig, σ' 1 -σ' 3 [MPa] M=1.045 (at T= 92 C) C from Hueckel & Pellegrini M=0.87 (at T=21 C) 6 Temperature [ C] (σ' 1 +2σ' 3 )/3 [MPa] Newfields clay Boom clay!heating results in a significant pore pressure increase!the pore pressure growth can induce failure in the sample LL / BF 21

22 4. Constitutive modelling 1. Introduction 2. Principal experimental observations 3. Thermo-mechanical framework 4. Constitutive modelling 5. Numerical simulations 6. Conclusions LL / BF 22

23 Summary of thermo-mechanical features of soils to be incorporated in the constitutive modelling Thermal behaviour Reversible thermoelasticity Irreversible thermoplasticity with a thermal hardening Mechanical behaviour, isotropic path Compressibility independent of T Preconsolidation pressure varies with T Mechanical behaviour, deviatoric path Volumetric strain [%] Void ratio [-] After Campanella & Mitchell Temperature [ C] After Despax T=25 C T=39 C T=51 C Consolidation stress (MPa) Peak strength varies with T (NC soil) Small change of the volumetric strains with T Elastic rigidity varies with T Friction angle variation with T LL / BF 23

24 Total strain rate tensor : Thermo-ElastoPlastic Framework e p ij = ij + ij d ε d ε d ε ACMEG T Model e dε ij Thermo-elastic component LL / BF 24 e ij = ijkl ij + ij dε D dσ β dt ( e ) 1 ij = ijkl kl ij dσ D dε β dt 2 dσij = K G dεv δij 3 βs + 2 Gdεij KdT 3 is the strain rate component that doesn t modify the hardening state of the material δ ij p dε ij Thermo-plastic component p e ij = ij ij dε dε dε q Elastic domain T?? is the thermo-plastic strain rate which can be expressed as the part of the total strain rate which is not recoverable p

25 Thermo-Plastic Component Thermo-plastic strain rate ACMEG T Model Isotropic mechanism in the p - T plane q Deviatoric mechanism in the p - q plane φ(t 0 ) Yield limit at T 1 Yield limit at T 0 φ(t 1 ) p' & Tp ij ε = 2 k = 1 λ k g k σ ' ij p c (T 1 ) p c (T 0 ) LL / BF 25

26 Isotropic thermo-plastic mechanism f = p p r iso c iso ACMEG T Model Mechanical effect Thermal effect Volumetric plastic strain p c0 p c β ln p c log [(T 0 + T)/T 0 )] { βε P }{ γ } f = p p ( T )exp 1 log T T r Ti c0 0 v 0 iso γ p c0 (T) p cm (T 0 ) p c LL / BF 26 { β ε } ( ) exp p p c = p c0 T β ε v { γ } p c0( T) = p c0( T0) 1 log T / T0

27 Isotropic thermo-plastic mechanism Evolution of the preconsolidation pressure due to : ACMEG T Model Mechanical effect Thermal effect T Elastic range Shape of the yield surface for different values of γ σ c0 σ c σ c T Temperature, T [ C] Elastic range γ=0.4 γ=0.3 γ=0.15 γ=0.075 σ c (T 0 ) σ c Preconsolidation pressure, σ' c [kpa] = T { p } p ( T) = p ( T ){ 1 γ log T / T } σ ( ) exp c σ c0 β ε v c0 c0 0 0 LL / BF 27

28 Deviatoric mechanism ACMEG T Model dp' fdev = q Mp'1 b Log = 0 p ' c ( ) M ( T) = M g T T 0 0 Deviatoric yield limit Friction angle = f (T) P { } [ ] { } σ ( T) = σ ( T )exp β ε 1 γ log T T c c0 0 v 0 Preconsolidation pressure = f (T) Modification of the deviatoric yield limit with temperature : dp' = + ( 0 0 ) 1 = 0 p T p' c0 - Friction angle ( T0) exp( βεv ) 1 γ log T 0 Dependency f q M of g( T T-) Preconsolidation p b Log pressure Td LL / BF 28

29 Advanced Constitutive Model for Environmental Geomechanics Temperature (ACMEG T Model) LL / BF 29

30 Mechanical hardening induced by heating T T 1 T 0 σ c (T 1 ) σ c (T 0 ) σ c σ σ c (T 1 ) σ c (T 0 ) = Stress state LL / BF 30

31 T T 1 T 0 Mechanical hardening induced by heating p c (T 1 ) p c (T 0 ) p c But the stress state is outside of the yield limit (f > 0) Inadmissible Mechanical hardening of the isotropic yield limit: dε p v = λ iso g p ' iso ACMEG T Model σ The stress state is now on the yield limit (f = 0) p c (T 1 ) p c (T 0 ) = Stress state LL / BF 31

32 Plastic flow rule ACMEG T Model Isotropic mechanism dε g λ piso, iso iso ii = λiso = σ ii 3 g dε λ λ p ' piso, giso dεd = λiso = 0 q piso, iso v = iso = iso pdev, g 1 dev q q 1 dεij = λdev = λdev + M σij Mp ' σij p ' 3 with Deviatoric mechanism ( σ ij ) 3 p ' if i = j q 2q = σ 3σ ij ij if i j q pdev, g dev 1 q dεv = λdev = λdev M p ' Mp ' p' pdev, gdev 1 & εd = λdev = λdev q Mp' LL / BF 32

33 Determination of the plastic multipliers ACMEG T Model Consistency equations df Solving the two consistency equations k p ( σ, T, ε ) = 0 v Two equations with two unknowns 2 p v & ε = k = 1 λ k g k p ' 1 q dfdev = σ ' fdev dσ ' + T fdev dt + p fdev dev M iso 0 ε λ + λ = v Mp ' p ' 1 q dfiso p ' fiso dp' T fiso dt r fiso dr p = fiso λdev M λiso 0 ε + = v Mp ' p ' LL / BF 33

34 2 Thermo-hydro-mechanical coupling Undrained paths ACMEG T Model T 3 1 T Thermo-mechanical paths in which the water is not able to drain out from the soil Pore-pressure generation THM coupling in anelastic porous media: 2 phases (water, solid); 3 fields (THM) Mass conservation of each constituent + homogenisation (averaging procedure): t p w 1 = nβ f ( ε t e v + ε t Tp v + [ nβ' + (1 n)( β' +ςt) ξ] T ) f s0 Water Solid grains t water compressibility Thermal coefficients LL / BF 34

35 5. Numerical simulations 1. Introduction 2. Principal experimental observations 3. Thermo-mechanical framework 4. LTVP constitutive model 5. Numerical simulations 6. Conclusions LL / BF 35

36 Thermo-mechanical paths Mechanical loading at constant temperature σ σ T ε Thermal loading σ T T ε LL / BF 36

37 Thermal loading Isotropic mechanism Normally consolidated sample ACMEG T Model Hardening of the monotonic surface Heating induces plastic compaction LL / BF 37

38 Thermal loading Isotropic mechanism Slightly overconsolidated sample ACMEG T Model Heating induces an elastic dilation followed by a plastic compaction LL / BF 38

39 Thermal loading Isotropic mechanism Highly overconsolidated sample ACMEG T Model Heating induces an elastic dilation LL / BF 39

40 Thermal loading Isotropic mechanism 100 TBoom 9 80 Elastic [Kref, Gref, n] [MPa], [MPa], [-] Plastic [β, d, p co ] [-], [-], [MPa] Hardening [c] [-] Domain [r éla ] [-] 150, 130, , 1.3, Volumetric strain [%] Experimental results Simulation OCR = 1 Simulation OCR = 2 Simulation OCR = 6 Thermal [β s0, γ] [ C -1 ], [-] , 0.18 Simulation of an heating-cooling cycle for different degrees of overconsolidation (OCR=1, 2 and 6) (experimental test of Baldi et al.,1991) LL / BF 40

41 Thermo-mechanical paths Mechanical loading at constant temperature σ σ T ε Thermal loading σ T T ε LL / BF 41

42 Mechanical loading at constant temperature Isotropic mechanism ACMEG T Model First loading = Plastic loading Ti f = p' σ r c m r i = & ε 1 p > σ T v c 0 0 ' ( ) increases O = Initial state log σ A ε v LL / BF 42

43 Mechanical loading at constant temperature Isotropic mechanism ACMEG T Model Unloading = Elastic unloading & ε p σ T v c 0 = 0 ' ( ) = constant O = Initial state log σ B A ε v LL / BF 43

44 Mechanical loading at constant temperature Isotropic mechanism ACMEG T Model Elastic reloading Inside the elastic nuclei c r i = & ε constant p σ T v c 0 = 0 ' ( ) = constant O = Initial state log σ B C A ε v LL / BF 44

45 Mechanical loading at constant temperature Isotropic mechanism ACMEG T Model Plastic reloading Inside the monotonic surface & ε = & ε > 0 σ ' ( T ) increases p p, cyc v v c r c i 0 increases O = Initial state log σ B C D A E ε v LL / BF 45

46 Mechanical loading at constant temperature Isotropic mechanism ACMEG T Model Plastic loading Hardening of the monotonic surface m r i = & ε 1 p > σ T v c 0 0 ' ( ) increases O = Initial state log σ B C D A E ε v F LL / BF 46

47 Mechanical loading at constant temperature Isotropic mechanism Experimental result (Jamin 2003) Simulation Elastic [K ref, n] [MPa], [-] Plastic [β, d, σ co ] [-], [-], [kpa] Hardening [c] [-] Domain [r éla ] [-] 460, 1 22, 2, Mean effective stress [Pa] Result of simulation of mechanical compression tests on silty sand at constant temperature LL / BF 47

48 Mechanical loading at constant temperature Pontida clay (1/3) Drained triaxial tests for 2 OCR values (experiment from Baldi et al., 1991) Deviatoric stress [MPa] OCR = 5 T=20 C OCR = 12.5 Experiment Model Axial strain [%] Deviatoric stress [MPa] T=95 C OCR = 12.5 OCR = 5 Experiment Model Axial strain [%] LL / BF 48

49 Mechanical loading at constant temperature Pontida clay (2/3) Drained triaxial tests for 2 OCR values (experiment from Baldi et al., 1991) Volumetric strain [%] Experiment Model T=20 C OCR = 12.5 OCR = Axial strain [%] Volumetric strain [%] T=95 C Experiment Model OCR = 12.5 OCR = Axial strain [%] LL / BF 49

50 Mechanical loading at constant temperature Pontida clay (3/3) Undrained creep and heating test on an initially anisotropic sample (experiment from Hueckel & Pellegrini, 1991) LL / BF 50 Pore pressure [MPa] Numerical simulation T=68 C T=56 C T=39.5 C T=22 C Experiment T=79 C Axial deformation [%] 1->2: undrainded loading 2->3: creep test during 48h 3->4: undrained thermal loading 2 4 Deviatoric stress [MPa] Experiment Numerical simulation Mean pressure [MPa] 3 2 1

51 Thermo-mechanical paths Mechanical loading at constant temperature σ σ T ε Thermal loading σ T T ε Thermo-mechanical loading σ σ,t T ε LL / BF 51

52 Thermo-mechanical loading Isotropic mechanism LL / BF 52 T ( C) Tboom 10 T ( C) p' (MPa) p' (MPa) 2 8 Tboom 11 (Experiment from Baldi et al., 1991) 3 Mean pressure, p' (MPa) Mean pressure, p' (MPa) 10 Numerical simulation 8 Experimental results H (mm) 10 Numerical simulation 3 8 Experimental results Temperature, T ( C) Temperature, T ( C) H (mm)

53 Thermo-mechanical undrained loading The ACMEG T Model parameters for the Kaolin-MC clay are determined on the following experimental paths : - Triaxial shear test in NC state at T=20 C - Triaxial shear test in OCR=2.2 state at T=20 C - Thermal loading path: T= 20 C to 90 C in NC state - Triaxial shear test in NC state at T=90 C Deviatoric stress, q [kpa] OCR = 2.2 Sig3=89 KPa Numerical simulation Experimental results OCR = 1 Sig3=196 KPa Axial strain, ε [%] 1 Pore pressure [kpa] Axial strain, ε [%] 1 Experimental results (Kuntiwattakul, 1991) OCR=1 OCR=2.2 Variation of void ratio [-] Temperature [ C] LL / BF 53

54 Thermo-mechanical undrained loading Numerical back-prediction 90 T ( C) 100 numerical predictions experimental results T=90 C Initial states p (kpa) T=20 C Deviatoric stress, q [kpa] Numerical simulation (20 C) Numerical simulation (90 C) Experimental result (90 C) Experimental result (20 C) Axial strain, ε [%] 1 T=20 C Pore pressure [kpa] T=90 C Axial strain, ε [%] 1 LL / BF 54

55 ATLAS experiment - Presentation Global view of the HADES-URF in Mol Bernier et al. (2007) The ATLAS experiment (horizontal plane) De Bruyn and Labat (2002) LL / BF 55

56 ATLAS experiment - Presentation VIDEO LL / BF 56

57 ATLAS experiment Thermal loading LL / BF 57

58 ATLAS experiment Material parameter (Boom Clay) Experiment from Baldi et al. (1991) LL / BF 58

59 ATLAS experiment Comparison with experiment Comparisons between numerical predictions and measurement in the two instrumented boreholes Temperature Pore water pressure LL / BF 59

60 ATLAS experiment Results LAGAMINE FE code LL / BF 60

61 ATLAS experiment Results LAGAMINE FE code LL / BF 61

62 ATLAS experiment Results (stress paths) LAGAMINE FE code LL / BF 62

63 ATLAS experiment Results (Temperature) LAGAMINE FE code VIDEO LL / BF 63

64 ATLAS experiment Results (Pore water pressure) LAGAMINE FE code VIDEO LL / BF 64

65 ATLAS experiment Results (Mean effective stress) LAGAMINE FE code VIDEO LL / BF 65

66 ATLAS experiment Results (Deviatoric stress) LAGAMINE FE code VIDEO LL / BF 66

67 ATLAS experiment Results (Volumetric plastic strain) LAGAMINE FE code VIDEO LL / BF 67

68 ATLAS experiment Results (Displacements) LAGAMINE FE code VIDEO LL / BF 68

69 Conclusions! Thermally induced effects on soils are complex including non-linearity and plasticity! The presented ACMEG-T constitutive model is able to reproduce the main features of the behaviour of soils under thermo-mechanical loading! The ACMEG-T model associated with the LAGAMINE FE code allows to reproduce the observed THM behaviour at the field scale. LL / BF 69