Geo-E2010 Advanced Soil Mechanics L Wojciech Sołowski. 07 March 2017

Transcription

1 Geo-E2010 Advanced Soil Mechanics L Wojciech Sołowski 07 March 2017

2 Soil modeling: critical state soil mechanics and Modified Cam Clay model

3 Outline 1. Refresh of the theory of lasticity 2. Critical state soil mechanics 3. Modified Cam Clay Model a. Formulation b. Predictions c. Good & bad of the model Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 3

4 Plasticity refresh

5 Theory of Plasticity - Summary 1. Elastic Strain e dε e dε q = 1 K d 3G dq 2. Yield surface f (, q, 0 ) = 0 3. Plastic Potential g(, q, ζ ) = 0 4. Flow rule g g dε dλ = ; dεq = dλ q

6 5. Hardening law 6. Plastic deformations 7. Total deformations q q q d d d ε ε ε ε ε ε + = = ), ( + = dq d q g q f q g f g q f g f q g g f d d q q ε ε ε ε e e q q q d d d d d d ε ε ε ε ε ε = + Theory of Plasticity - Summary

7 Some comments on the last exercise

8 Drained test τ φ =30 deg c =3 kpa 1 3 ν = 0.3 E =10 MPa. σ 1 = σ 2 = σ 3 =100 kpa σ F = ( σ σ ) ( σ + σ )sinφ 2c cosφ = Deartment of Civil and Environmental Engineering Advanced Course in Soil Mechanics. W. Sołowski 8

9 Drained test c*=6.23 kpa q 1 η = ν = 0.3 E =10 MPa. σ 1 = σ 2 = σ 3 =100 kpa F = q η c = * 0 Deartment of Civil and Environmental Engineering Advanced Course in Soil Mechanics. W. Sołowski 9

10 Drained test q We cannot go away from the yield surface, hence dq/d=1.2 c*=6.23 kpa 1 η = ν = 0.3 E =10 MPa. σ 1 = σ 2 = σ 3 =100 kpa F = q η c = * 0 Deartment of Civil and Environmental Engineering Advanced Course in Soil Mechanics. W. Sołowski 10

11 Critical State Soil Mechanics

12 Critical state soil mechanics To learn / refresh : - what is critical state - how soil behaves when over-consolidated / dense and when normally consolidated / loose - volumetric behaviour of soil - oedometer, drained triaxial tests - semi logarithmic scale - how we can aroximate the elastic behaviour of soil in the semi-logarithmic lot ( secific volume) - how we can aroximate the elasto-lastic behaviour of soil in the semi-logarithmic lot ( secific volume) Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 12

13 Plastic Behaviour of Soils Mohr-Coulomb in Princial Stress Sace σ 3 σ 1 =σ 2 = σ 3 σ 2 σ 1./ Mohr Coulomb failure surface is a irregular hexagon in the rincial stress sace

14 Salient Features and Drawbacks of MC Model./ It is assumed that the material behaves elastically until the failure surface is reached../ In reality, lastic deformation begins well before the failure conditions are met. ' + H E G D B 0 A C F ε v

15 Drucker et al. (1957) model./ Drucker et al. (1957) roosed the introduction of a yield surface bounded by a ca which is allowed to exand and increase in size with increasing stress level../ The behaviour of the soil is urely elastic only inside the failure surface and the ca. σ 3./ If the stress increase is directed outwards the ca, lastic deformations are generated and the ca exands. σ 2 σ1

16 Cam clay models In Cambridge University, several researchers tried to imrove the Drucker model to describe the behaviour of reconstituted Kaolin clay, on which a large library of axisymmetric tests was available for normally consolidated and overconsolidated secimens. In articular, Schofield and Wroth roosed the model which goes by the name (Original) Cam Clay model A real revolution in the soil modelling field. Soon afterwards Burland (1967) & Roscoe & Burland (1968) resented the: Modified Cam Clay model which we will discuss in detail

17 Formulation of Modified Cam Clay model I n t he stress sace, a yield function f behaviour is elastic when f < 0. is defined so that the soil The surface described by f is the so-called yield locus and is formulated in function of the stress tensor and scalar values called hardening arameters. These scalar values are related to the size of the yield surface in the stress sace, as it will be clarified. Moreover, inside the yield locus the soil behaviour is described by the exonential law: ν = ν 1 κ ln where ν is the secific volume, exressed in terms of void ratio by: κ is the sloe of the recomression ν 1+ e line in ν-ln' lane

18 Formulation of Modified Cam Clay model: elasticity ν = ν 1 κ ln (c) Muir Wood

19 Formulation of Modified Cam Clay model: elasticity ν 1+ e ν = ν 1 κ ln Void ratio and volumetric strains are correlated by: dεv = de = dν 1+ e ν (c) Muir Wood κ is the sloe of the unloading reloading (url) line in ν-ln' lane Other elastic arameter : Poisson ratio or shear modulus G

20 Clay behaviour Mitchell & Soga 2005 Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 20

21 Clay behaviour Glacial till Mitchell & Soga 2005 Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 21

22 Modified Cam Clay volumetric behaviour Starting oint of Cam-Clay theory: Logarithmic isotroic comression and unloading Primary loading: e e 0 = λ ln Unloading / reloading: e e 0 = κ ln c v=1+e c κ 1 1 λ ln iso-ncl c

23 Behaviour of clays under triaxial states Isotroic Loading

24 Modified Cam Clay For other than isotroic stress aths from the origin there is another (arallel) NCL q failure ln K 0 1 λ iso iso-ncl K 0 -NCL failure-ncl

25 Clay behaviour Glacial till Mitchell & Soga 2005 Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 25

26 Behaviour of clays under triaxial states Shearing (drained) Volumetric deformations during shearing (dilatancy): Initial elastic contraction v Followed by an elastolastic behaviour: NC soils or slightly OC (OCR 1 to 4), a volumetric contraction is observed OC soils (OCR > 4), a volumetric exansion is observed ε 1

27 Clay behaviour Glacial till Mitchell & Soga 2005 Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 27

28 Behaviour of clays under triaxial states Shearing (undrained) Pore ressure during shearing : Initial elastic (ositive) change of ore water ressure Followed by elastolastic changes: NC soils or slightly SC (OCR 1 to 4), ositive changes of the ore water ressure are observed w NC or slightly SC OC soils (OCR > 4), negative changes of the ore water ressure are observed hightly OC ε 1

29 Clay behaviour Glacial till Mitchell & Soga 2005 Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 29

30 Behaviour of clays under triaxial states Stage II: Shearing The final values of the void ratio, at constant volume, show a unique line arallel to the virgin comression line: v v = Γ λ ln ln

31 Behaviour of clays under triaxial states Shearing (drained) Linear relationshi between q and on the critical states: q q = M M and M is constant

32 Mechanical behaviour of clays under triaxial states Shearing (drained)

33 Behaviour of clays under triaxial states Stress aths with σ 3 σ 1 = K C =const. q In articular, the oedometric test corresonds to ε H = 0 K C = K 0 Notation: K 0 = σ 3 σ 1 when lateral deformation is zero

34 Stress aths with σ 3 σ 1 = K C =const. q v K 0, Isotroic virgin comression line The relationshi of (virgin) volumetric comression: dv = λ d And, hence, NC lines are arallel to isotroic virgin comression line and critical state line Yield surface exands as a function of lastic deformations K 0 ln

35 Modified Cam Clay - lasticity The shae of the yield function is an ellise, symmetrical with resect to the axis. The equation of the yield function is: ( ) 2 2 f q M = 0 q M sloe of the critical state line (CSL) 0/2 0 '

36 Modified Cam Clay Model

37 Modified Cam Clay To learn: - formulation of Modified Cam Clay (MCC) model - model arameters - yield surface - elastic law - elasto-lastic formulation, hardening, softening - lastic otential & flow rule, elasto-lastic matrix* - how to set the value of the hardening arameter 0 - how to choose arameters for the model (also exercises) - MCC model rediction of soil behaviour of soil behaviour in isotroic loading, K 0 loading, shearing in critical state - MCC model redictions on triaxial aths (drained / undrained) for normally consolidated soil and for overconsolidated soil - ros and cons for Modified Cam Clay model Deartment of Civil Engineering Advanced Soil Mechanics. W. Sołowski 37

38 Introduction

39 MCC model MCC is a simle model does not reroduce ALL the features of the soil Same shae of the yield surface (changing its size) Changes of the yield surface are only deendant on the volumetric deformation: VOLUMETRIC HARDENING MODEL The actual size of the yield surface deends on the revious loading-unloading history

40 Invariants Variables used within the model (Cambridge formulation): = 1 (σ + 2σ ), q = σ 1 σ 3 ε ε = 2 (ε ε ) = ε 1 + 2ε 3, q Good redictions on laboratory tests on reconstituted soils, not as good when redicting real (in situ) behaviour of the soil

41 Invariants Cambridge formulation: Work conjugate air for change in size: = 1 (σ + 2σ ) ε = ε 1 + 2ε 3 Volumetric work: Work conjugate air for change in shae: q = σ 1 σ 3 ε = 2 (ε ε ) q Distortional work: δw = 'δε δw q = qδε q

42 Invariants The total work done by an stress increment should be equivalent to one calculated with Cambridge variables : δw = σ ' 1 δε 1 + 2σ ' 3 δε 3 = 'δε + qδε q = δw +δw q? δw = δw +δw q = 'δε + qδε q = 1 2 = (σ ' 1 + 2σ ' 3 )(δε 1 + 2δε 3 )+(σ ' 1 σ ' 3 ) (δε 1 δε 3 )= 3 3 = 1 σ ' δε + 2 σ ' δε + 2 σ ' δε + 4 σ ' δε + 2 σ ' δε σ ' δε 2 σ ' δε + 2 σ ' δε = = σ ' 1 δε 1 + 2σ ' 3 δε 3 OK!

43 Volumetric deformations: a) Elastic Assuming isotroy and elasticity inside the yield surface. Volumetric and shear deformations are uncouled. K' = volumetric modulus and G' = shear modulus Exressed in terms of effective stresses (and not constants)

44 Volumetric deformations: a) Elastic K' is not constant: K' = K'(') Then: What is the exression of K'(')?

45 Shear strains: a) Elastic

46 Volumetric deformations: b) Plastic q yl 2 L yl 1 K A B v M ' ncl iso-ncl url 1 K v url 2 M L A B ' Triaxial tests on Winnieg clays [Graham et al, 1983]

47 Shear strains: b) Plastic If δε δε is known, then δε is also known, as δε q q is known. The vector of lastic deformations (δε,δε q ) can be measured in triaxial tests when crossing the yield surface. f (,q, 0 ) =0 g(,q,ζ ) = 0

48 Shear strains: b) Plastic Ottawa sand (Poorooshasb et al., 1966, 1967) CLAYS: associated lasticity? Winnieg clay (Graham et al., 1983) SANDS: non associated lasticity

49 Formulation of the model Elastic deformations Yield surface Plastic otential and flow rule Elasto-lastic matrix

50 Formulation of the model Elastic deformations G' = constant - Alternative formulation with constant Poisson ratio common; κ = constant

51 Formulation of the model Yield surface q 1 M 0 1 M f (,q, ) q 2 M 2 ( ) = 0 0 0

52 Initial re-consolidation ressure To get initial 0 POP = Pre Overburden Pressure OCR = Over Consolidation Ratio

53 Formulation of the model Yield surface The yield surface can be exressed in terms of η = q/' as follows: η (',q) = 0 M 2 M 2 +η 2 The required derivatives of the model are:

54 Formulation of the model Plastic otencial Modified Cam-clay assumes associated lasticity, so: g = f q 2 M 2 ( ) = 0 0 As a consequence the flow rule (that gives the lastic strain rates) is:

55 Formulation of the model Hardening law (gives the change in the size of the yield surface) As we know, δε = λ κ δ 0 And then: 0 ε v = v 0 0 ε = 0 q λ κ 0 δ = δε v λ κ 0 0 These equations define how the size of the yield surface changes (through the variation of the hardening arameter o ) as a function of the lastic volumetric strains Notice that only deendent on lastic volumetric deformation

56 Formulation of the model Elasto-lastic comliance matrix It is only used within elasto-lastic states Is symmetric f = g Its determinant is 0, as the volumetric deformations and the shear strains are related:

57 Deformations under an alied stress ath q B A 0 A 0 B 0 B > 0 A ELASTO-PLASTIC

58 Deformations under an alied stress ath q B A 0 B 0 A 0 B < 0 A ELASTIC

59 Hardening law q δε q dε < 0 dε r dε = 0 dε = η > M dε r 0 2 λ κ d 0 v 0 η = M dε r dε > 0 0 η < M δε η < M dε > 0 d 0 > 0 Yield surface exands η > M dε < 0 d < 0 0 Yield surface contracts η = M dε = 0 d 0 = 0 Yield surface constant

60 Modified Cam Clay Model Predictions of soil behaviour

61 Normally consolidated clays: Test CD q

62 Normally consolidated clays : test CD

63 Lightly overconsolidated clays: test CD q

64 Lightly overconsolidated clays: test CD

65 Highly overconsolidated clays: test CD q

66 Highly overconsolidated clays: test CD

67 Normally consolidated clays: test CU q TTE TTT

68 Normally consolidated clays: test CU

69 Lightly overconsolidated clays: test CU q TTE TTT

70 Lightly overconsolidated clays: test CU

71 Highly overconsolidated clays: test CU q TTE TTT

72 Hightly overconsolidated clays: test CU

73 Modified Cam Clay Model Drawbacks

74 Drawbacks of MCC Comutation roblems In undrained triaxial test on a heavily overconsolidated soil, after the stress oint reaches the yield surface (above M line), due to the negative direction of volumetric lastic strain vector, the yield surface contracts. This henomenon is referred to as strain softening. Even though the constitutive model is erfectly able to model this asect of mechanical behaviour, strain softening may give serious roblems in a finite element analysis: mesh deendency and roblems with convergence. That can be overcome with good coding & algorithms, but many leading codes still struggle and diverge or give erroneous results!

75 Drawbacks of MCC Strength rediction in undrained conditions MCC model assumes Drucker-Prager failure condition, which overestimates undrained q 1 M 2c u strength in triaxial extension Better redictions if Mohr Coulomb failure 1 0 or Lode angle deendency is M introduced Real soils are anisotroic and both the σ y α shae and size of the yield surface would need to change (see e.g. Wheeler et al. 2003, Can. Geotech. J. for S- CLAY1 model) σ z σ x

76 Drawbacks of MCC K 0 rediction Given MCC assumes an associated flow rule, the model redicts unrealistically high K 0 values in normally consolidated range This has been fixed e.g. in the Soft Soil q Tension cut-off 1 M (not critical state) Mohr Coulomb failure model by de-couling the volumetric yield surface (ca) from the failure line Consequently, in the Soft Soil model, M as become a shae coefficient and no α longer corresonds to the critical state σ y line Alternatively, the anisotroic S-CLAY1 model also gives good K0 rediction Non-associated flow rule is also an otion σ z σ x which will hel with that issue

77 Positives of MCC Few arameters - 0 is the only initial arameter - M, κ, λ, G are the soil constants - generally we need to know void ratio or secific volume of soil at given reference stress usually denoted by N and c Qualitative rediction of soil behaviour - relicates many behaviour not ossible to relicate in e.g. Mohr-Coulomb model BUT Model is too simle to redict all soil behaviour accurately (real soil is anisotroic, structured, etc. etc.)

78 Thank you

79 Basic Concets of Plasticity (reeat from the last lecture)

80 Plastic Behaviour of Soils Idealization of elasto-lastic behaviour, different models σ σ σ σ Yield limit deends on (effective) stresses ε ε ε ε Rigid Perfectly Plastic Elasto-lastic erfect lasticity Elasto-lastic hardening Elasto-lastic softening dε = dε e + dε total elastic lastic

81 Y 0 = yield stress Y F = failure stress On softening models

82 Some Basic Concets Strains (ε) Total strains Elastic strains (recoverable on unloading) Plastic strains (not recoverable on unloading) Total strains = Elastic strains + Plastic strains Stresses (σ) Total stresses = Effective stresses + Pore Pressures

83 Some Basic Concets Stresses are related to elastic strains even in nonlinear theories Stresses are stresses - there is nothing like elastic stress and lastic stress. We talk mainly in terms of effective stress.

84 Elasto-Plastic Models Stress Stress Ideal lastic Strain Strain hardening Strain An elastic law A criterion for yielding (Yield function/surface) The direction of lastic flow (Flow rule) Does the yield function change due to lastic flow? If yes, how? (Hardening/Softening rule)

85 Yield Surface Used to delimit the elastic domain σ 1 PLASTIC On the surface IMPOSSIBLE STATE Outside It is a generalization of the 1D case Yield limit (1-D) Yield surface (2D- 3D) σ 3 ELASTIC Inside yield surface σ 2 F( σ, ξ ) = 0 F(σ,hij i )=0 i

86 Yield Surface F σ ({ }, h) = 0 F ({ σ } + { d σ },h + dh) = 0

87 Yield Surface Fixed yield surface F (σ ij ) = 0 Perfect lasticity Exanding yield surface F(σ ij,h i ) = 0 Hardening lasticity Contractive yield surface F(σ ij,h i ) = 0 Softening lasticity The exansion or contraction of the YS is controlled by the hardening (or softening) arameters h i The stress state must be either inside the surface or on the surface (stress states outside the surface are not allowed). Stress inside the surface F(σ ij,h i ) < 0 elastic strain only Stress on the surface F(σ ij,h i ) = 0 elastic and lastic strain

88 Yield Surface The YS is often exressed in term of the stresses or stress invariants. ',q are tyical stress variables used to describe soil behaviour and, also, to define the YS Therefore tyical exression of the YS are as follows: f ( ij h) σ, = 0 f, q, = 0 ( ) 0 where is a tyical hardening arameter (h) used in geotechnical models. The hardening arameter(s) control the exansion or contraction of the YS. 0

89 Flow Rule In one-dimensional roblem, it is clear that lastic strains take lace along the direction of alied stress σ 1, ε 1 ε In 2D or 3D we need to make a hyothesis regarding the direction of lastic flow (relative magnitude of lastic strain increments) σ 3, ε 3

90 Plastic Potential and Flow Rule Plastic Deformations To evaluate lastic deformations the existence of a lastic otential (g or G ) is assumed. The lastic otential rovides the direction of the lastic strain: ' ( ij, ) 0 g σ ζ = g(, q, ζ ) = 0 ζ where is the arameter that control the size of the lastic otential It is also necessary to define the flow rule g g dε dλ = ; dεq = dλ q

91 Plastic Potential and Flow Rule Plastic Deformations In general, dε ij control the magnitude of lastic deformation g = dλ σ ij control the direction of the lastic deformations: the vector of the lastic deformations is normal to the g = constant surfaces g(, q, ζ ) = 0

92 Plastic Potential and Flow Rule Yield Surface (f) and Plastic Potential (g) are generally different functions If f g => associated lasticity The comonents of the lastic deformations are related, i.e. there is a couling, which is defined by the flow rule The lastic deformations deend on the stress state rather than the increment of the stresses alied

93 Plastic Potential and Flow Rule FLOW RULE ASSOCIATED FLOW RULE NON ASSOCIATED The flow rule defines direction of lastic strain increment So, we know the lastic-strain direction, but how we can determine the magnitude?

94 Hardening rule It is necessary to rovide a descrition of the variation of the size and/or osition of the yield surface during lastic deformations (i.e. how the YS evolve during yielding) = ( ε, ε ) 0 0 d 0 = dε + dε ε q 0 0 εq q q YS o '

95 Consistency condition The lastic state is reached when the stress state is on the surface: f, q, = 0 ( ) 0 It is assumed that once yield occurs (i.e. f = 0), the stresses must remain on the yield surface during lastic deformation. This constraint is enforced by the consistency condition as follows: df = 0

96 Consistency condition f f f df = d + dq + q 0 d 0 d 0 g = dε + dε = dλ + ε ε ε ε q q q g dλ q d ε = dλ f ( q ) Now we can determine the magnitude of the lastic strain,, = 0 0 g { σ '} f f f 0 g 0 g df = d + dq + dλ + dλ = q 0 ε εq q dλ = f 0 d f f d + dq q 0 g 0 g + ε εq q 0 0

97 Plastics Deformations dε = f f f f d + dq d + dq q g q dε q = f 0 g 0 g f 0 g 0 g ε εq q 0 ε εq q g q dε 1 = d ε q f 0 g 0 g + 0 ε ε q q f g f g q d f g f g dq q q q Plastic modulus, H H = 0 erfect lasticity and Eq. above not valid! H > 0 lasticity with hardening H < 0 lasticity with softening